UniformSampleCone, y

Percentage Accurate: 57.7% → 98.1%

Time: 14.4s

Alternatives: 17

Speedup: 4.5×

Specification

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

Math

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

FPCore

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

Initial Program: 57.7% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.1% accurate, 0.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.5%

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

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

   3. lift--.f32 N/A

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

   4. sub-neg N/A

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

   5. associate-+l+ N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

   8. lower-+.f32 N/A

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

   9. lift-+.f32 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lift-+.f32 N/A

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

   16. +-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f32 N/A

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

   20. neg-mul-1 N/A

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

   21. lift-*.f32 N/A

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

   22. *-commutative N/A

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

   23. distribute-rgt-out N/A

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

   24. lower-*.f32 N/A

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

4. Applied rewrites 18.6%

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

5. Taylor expanded in ux around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

7. Applied rewrites 98.2%

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

9. Applied rewrites 98.2%

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

10. Final simplification 98.2%

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

Alternative 2: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.5%

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

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

   3. lift--.f32 N/A

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

   4. sub-neg N/A

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

   5. associate-+l+ N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

   8. lower-+.f32 N/A

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

   9. lift-+.f32 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lift-+.f32 N/A

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

   16. +-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f32 N/A

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

   20. neg-mul-1 N/A

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

   21. lift-*.f32 N/A

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

   22. *-commutative N/A

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

   23. distribute-rgt-out N/A

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

   24. lower-*.f32 N/A

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

4. Applied rewrites 17.2%

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

5. Taylor expanded in ux around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

7. Applied rewrites 98.2%

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

9. Applied rewrites 98.2%

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

10. Final simplification 98.2%

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

Alternative 3: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.5%

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

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

   3. lift--.f32 N/A

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

   4. sub-neg N/A

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

   5. associate-+l+ N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

   8. lower-+.f32 N/A

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

   9. lift-+.f32 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lift-+.f32 N/A

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

   16. +-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f32 N/A

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

   20. neg-mul-1 N/A

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

   21. lift-*.f32 N/A

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

   22. *-commutative N/A

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

   23. distribute-rgt-out N/A

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

   24. lower-*.f32 N/A

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

4. Applied rewrites 18.6%

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

5. Taylor expanded in ux around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

7. Applied rewrites 98.2%

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

9. Applied rewrites 98.2%

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

10. Final simplification 98.2%

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

Alternative 4: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.000699999975040555:\\ \;\;\;\;\sqrt{\left(ux - ux \cdot maxCos\right) - \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot ux\right) \cdot \left(maxCos - 1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{2}{ux} - 1\right) \cdot ux\right) \cdot ux} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.000699999975040555)
   (*
    (sqrt
     (-
      (- ux (* ux maxCos))
      (* (* (fma maxCos ux (- 1.0 ux)) ux) (- maxCos 1.0))))
    (* (* (PI) 2.0) uy))
   (* (sqrt (* (* (- (/ 2.0 ux) 1.0) ux) ux)) (sin (* (PI) (* 2.0 uy))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 6.99999975e-4

   1. Initial program 51.4%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. lift--.f32 N/A

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

      4. sub-neg N/A

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

      5. associate-+l+ N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-+.f32 N/A

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

      9. lift-+.f32 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lift-+.f32 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f32 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f32 N/A

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

      20. neg-mul-1 N/A

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

      21. lift-*.f32 N/A

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

      22. *-commutative N/A

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

      23. distribute-rgt-out N/A

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

      24. lower-*.f32 N/A

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

   4. Applied rewrites 21.9%

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

      1. lift--.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. associate--r+ N/A

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

      4. lower--.f32 N/A

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

      5. lift-fma.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. +-commutative N/A

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

      9. lift--.f32 N/A

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

      10. associate-+l- N/A

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

      11. associate--r- N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f32 N/A

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

      14. lift-*.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. lower--.f32 71.2

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

      18. lift-*.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. associate-*r* N/A

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

      21. lift-+.f32 N/A

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

   6. Applied rewrites 70.0%

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

   7. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 96.1

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

   9. Applied rewrites 95.7%

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

   if 6.99999975e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 62.2%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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. lift-+.f32 N/A

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

      3. lift--.f32 N/A

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

      4. sub-neg N/A

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

      5. associate-+l+ N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-+.f32 N/A

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

      9. lift-+.f32 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lift-+.f32 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f32 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f32 N/A

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

      20. neg-mul-1 N/A

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

      21. lift-*.f32 N/A

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

      22. *-commutative N/A

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

      23. distribute-rgt-out N/A

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

      24. lower-*.f32 N/A

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

   4. Applied rewrites 13.5%

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

   5. Taylor expanded in ux around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 97.8%

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

   9. Applied rewrites 97.8%

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

   10. Taylor expanded in maxCos around 0

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

   12. Applied rewrites 91.3%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.000699999975040555:\\ \;\;\;\;\sqrt{\left(ux - ux \cdot maxCos\right) - \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot ux\right) \cdot \left(maxCos - 1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{2}{ux} - 1\right) \cdot ux\right) \cdot ux} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.000699999975040555:\\ \;\;\;\;\sqrt{\left(ux - ux \cdot maxCos\right) - \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot ux\right) \cdot \left(maxCos - 1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{2}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.000699999975040555)
   (*
    (sqrt
     (-
      (- ux (* ux maxCos))
      (* (* (fma maxCos ux (- 1.0 ux)) ux) (- maxCos 1.0))))
    (* (* (PI) 2.0) uy))
   (* (sqrt (* (- (/ 2.0 ux) 1.0) (* ux ux))) (sin (* (PI) (* 2.0 uy))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 6.99999975e-4

   1. Initial program 51.4%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. lift--.f32 N/A

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

      4. sub-neg N/A

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

      5. associate-+l+ N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-+.f32 N/A

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

      9. lift-+.f32 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lift-+.f32 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f32 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f32 N/A

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

      20. neg-mul-1 N/A

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

      21. lift-*.f32 N/A

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

      22. *-commutative N/A

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

      23. distribute-rgt-out N/A

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

      24. lower-*.f32 N/A

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

   4. Applied rewrites 23.3%

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

      1. lift--.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. associate--r+ N/A

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

      4. lower--.f32 N/A

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

      5. lift-fma.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. +-commutative N/A

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

      9. lift--.f32 N/A

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

      10. associate-+l- N/A

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

      11. associate--r- N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f32 N/A

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

      14. lift-*.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. lower--.f32 72.9

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

      18. lift-*.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. associate-*r* N/A

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

      21. lift-+.f32 N/A

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

   6. Applied rewrites 70.6%

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

   7. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 93.2

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

   9. Applied rewrites 95.7%

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

   if 6.99999975e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 62.2%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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. lift-+.f32 N/A

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

      3. lift--.f32 N/A

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

      4. sub-neg N/A

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

      5. associate-+l+ N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-+.f32 N/A

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

      9. lift-+.f32 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lift-+.f32 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f32 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f32 N/A

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

      20. neg-mul-1 N/A

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

      21. lift-*.f32 N/A

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

      22. *-commutative N/A

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

      23. distribute-rgt-out N/A

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

      24. lower-*.f32 N/A

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

   4. Applied rewrites 13.5%

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

   5. Taylor expanded in ux around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 97.8%

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

   8. Taylor expanded in maxCos around 0

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

   10. Applied rewrites 91.2%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.000699999975040555:\\ \;\;\;\;\sqrt{\left(ux - ux \cdot maxCos\right) - \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot ux\right) \cdot \left(maxCos - 1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{2}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.5%

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

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

   3. lift--.f32 N/A

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

   4. sub-neg N/A

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

   5. associate-+l+ N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

   8. lower-+.f32 N/A

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

   9. lift-+.f32 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lift-+.f32 N/A

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

   16. +-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f32 N/A

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

   20. neg-mul-1 N/A

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

   21. lift-*.f32 N/A

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

   22. *-commutative N/A

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

   23. distribute-rgt-out N/A

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

   24. lower-*.f32 N/A

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

4. Applied rewrites 17.3%

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

   1. lift--.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. associate--r+ N/A

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

   4. lower--.f32 N/A

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

   5. lift-fma.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. +-commutative N/A

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

   9. lift--.f32 N/A

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

   10. associate-+l- N/A

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

   11. associate--r- N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. lower--.f32 48.9

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

   18. lift-*.f32 N/A

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

   19. lift-*.f32 N/A

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

   20. associate-*r* N/A

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

   21. lift-+.f32 N/A

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

6. Applied rewrites 50.4%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lift--.f32 N/A

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

   5. associate-+l- N/A

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

   6. lift--.f32 N/A

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

   7. +-lft-identity N/A

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

   8. lift-+.f32 N/A

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

   9. lower--.f32 98.2

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

   10. lift-+.f32 N/A

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

   11. +-lft-identity 98.2

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 98.2

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

8. Applied rewrites 98.2%

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

9. Final simplification 98.2%

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

Alternative 7: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.5%

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

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

   3. lift--.f32 N/A

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

   4. sub-neg N/A

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

   5. associate-+l+ N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

   8. lower-+.f32 N/A

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

   9. lift-+.f32 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lift-+.f32 N/A

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

   16. +-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f32 N/A

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

   20. neg-mul-1 N/A

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

   21. lift-*.f32 N/A

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

   22. *-commutative N/A

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

   23. distribute-rgt-out N/A

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

   24. lower-*.f32 N/A

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

4. Applied rewrites 17.3%

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

   1. lift--.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. associate--r+ N/A

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

   4. lower--.f32 N/A

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

   5. lift-fma.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. +-commutative N/A

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

   9. lift--.f32 N/A

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

   10. associate-+l- N/A

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

   11. associate--r- N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. lower--.f32 49.8

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

   18. lift-*.f32 N/A

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

   19. lift-*.f32 N/A

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

   20. associate-*r* N/A

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

   21. lift-+.f32 N/A

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

6. Applied rewrites 48.6%

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

   1. lift-+.f32 N/A

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

   2. +-lft-identity 48.6

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

   3. lift--.f32 N/A

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

   4. lift-*.f32 N/A

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

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

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

   6. distribute-rgt1-in N/A

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

   7. metadata-eval N/A

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

   8. distribute-neg-in N/A

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

   9. metadata-eval N/A

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

   10. sub-neg N/A

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

   11. lift--.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift--.f32 N/A

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

   14. sub-neg N/A

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

   15. metadata-eval N/A

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

   16. +-commutative N/A

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

   17. distribute-neg-in N/A

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

   18. metadata-eval N/A

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

   19. sub-neg N/A

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

   20. lift--.f32 49.0

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

8. Applied rewrites 49.1%

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

9. Final simplification 49.4%

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

Alternative 8: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.5%

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

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

   3. lift--.f32 N/A

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

   4. sub-neg N/A

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

   5. associate-+l+ N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

   8. lower-+.f32 N/A

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

   9. lift-+.f32 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lift-+.f32 N/A

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

   16. +-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f32 N/A

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

   20. neg-mul-1 N/A

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

   21. lift-*.f32 N/A

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

   22. *-commutative N/A

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

   23. distribute-rgt-out N/A

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

   24. lower-*.f32 N/A

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

4. Applied rewrites 17.4%

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

   1. lift--.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. associate--r+ N/A

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

   4. lower--.f32 N/A

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

   5. lift-fma.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. +-commutative N/A

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

   9. lift--.f32 N/A

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

   10. associate-+l- N/A

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

   11. associate--r- N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. lower--.f32 50.7

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

   18. lift-*.f32 N/A

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

   19. lift-*.f32 N/A

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

   20. associate-*r* N/A

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

   21. lift-+.f32 N/A

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

6. Applied rewrites 48.4%

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

7. Taylor expanded in maxCos around 0

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

   1. lower--.f32 96.1

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

9. Applied rewrites 96.1%

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

10. Final simplification 96.1%

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

Alternative 9: 95.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.000699999975040555:\\ \;\;\;\;\sqrt{\left(ux - ux \cdot maxCos\right) - \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot ux\right) \cdot \left(maxCos - 1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 - ux\right) \cdot ux + ux} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.000699999975040555)
   (*
    (sqrt
     (-
      (- ux (* ux maxCos))
      (* (* (fma maxCos ux (- 1.0 ux)) ux) (- maxCos 1.0))))
    (* (* (PI) 2.0) uy))
   (* (sqrt (+ (* (- 1.0 ux) ux) ux)) (sin (* (PI) (* 2.0 uy))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 6.99999975e-4

   1. Initial program 51.4%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. lift--.f32 N/A

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

      4. sub-neg N/A

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

      5. associate-+l+ N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-+.f32 N/A

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

      9. lift-+.f32 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lift-+.f32 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f32 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f32 N/A

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

      20. neg-mul-1 N/A

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

      21. lift-*.f32 N/A

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

      22. *-commutative N/A

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

      23. distribute-rgt-out N/A

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

      24. lower-*.f32 N/A

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

   4. Applied rewrites 20.9%

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

      1. lift--.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. associate--r+ N/A

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

      4. lower--.f32 N/A

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

      5. lift-fma.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. +-commutative N/A

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

      9. lift--.f32 N/A

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

      10. associate-+l- N/A

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

      11. associate--r- N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f32 N/A

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

      14. lift-*.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. lower--.f32 71.4

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

      18. lift-*.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. associate-*r* N/A

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

      21. lift-+.f32 N/A

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

   6. Applied rewrites 71.5%

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

   7. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 96.1

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

   9. Applied rewrites 95.7%

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

   if 6.99999975e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 62.2%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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. lift-+.f32 N/A

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

      3. lift--.f32 N/A

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

      4. sub-neg N/A

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

      5. associate-+l+ N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-+.f32 N/A

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

      9. lift-+.f32 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lift-+.f32 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f32 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f32 N/A

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

      20. neg-mul-1 N/A

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

      21. lift-*.f32 N/A

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

      22. *-commutative N/A

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

      23. distribute-rgt-out N/A

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

      24. lower-*.f32 N/A

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

   4. Applied rewrites 13.5%

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

   5. Taylor expanded in maxCos around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 91.0

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

   7. Applied rewrites 91.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.000699999975040555:\\ \;\;\;\;\sqrt{\left(ux - ux \cdot maxCos\right) - \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot ux\right) \cdot \left(maxCos - 1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 - ux\right) \cdot ux + ux} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.007499999832361937:\\ \;\;\;\;\sqrt{\left(ux - ux \cdot maxCos\right) - \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot ux\right) \cdot \left(maxCos - 1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(maxCos, -2, 2\right) \cdot ux} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.007499999832361937)
   (*
    (sqrt
     (-
      (- ux (* ux maxCos))
      (* (* (fma maxCos ux (- 1.0 ux)) ux) (- maxCos 1.0))))
    (* (* (PI) 2.0) uy))
   (* (sqrt (* (fma maxCos -2.0 2.0) ux)) (sin (* (PI) (* 2.0 uy))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00749999983

   1. Initial program 53.5%

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

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

      3. lift--.f32 N/A

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

      4. sub-neg N/A

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

      5. associate-+l+ N/A

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

      6. distribute-lft-in N/A

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

      7. *-rgt-identity N/A

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

      8. lower-+.f32 N/A

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

      9. lift-+.f32 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lift-+.f32 N/A

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

      16. +-commutative N/A

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

      17. lift-*.f32 N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f32 N/A

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

      20. neg-mul-1 N/A

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

      21. lift-*.f32 N/A

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

      22. *-commutative N/A

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

      23. distribute-rgt-out N/A

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

      24. lower-*.f32 N/A

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

   4. Applied rewrites 22.0%

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

      1. lift--.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. associate--r+ N/A

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

      4. lower--.f32 N/A

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

      5. lift-fma.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. +-commutative N/A

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

      9. lift--.f32 N/A

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

      10. associate-+l- N/A

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

      11. associate--r- N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f32 N/A

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

      14. lift-*.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. lower--.f32 63.8

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

      18. lift-*.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. associate-*r* N/A

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

      21. lift-+.f32 N/A

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

   6. Applied rewrites 62.4%

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

   7. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 91.2

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

   9. Applied rewrites 92.4%

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

   if 0.00749999983 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 61.6%

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

   3. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 7.3

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

   5. Applied rewrites 7.3%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.007499999832361937:\\ \;\;\;\;\sqrt{\left(ux - ux \cdot maxCos\right) - \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot ux\right) \cdot \left(maxCos - 1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(maxCos, -2, 2\right) \cdot ux} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;1 - t\_0 \cdot t\_0 \leq 0.0003000000142492354:\\ \;\;\;\;\left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos, -2, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot t\_0} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (- 1.0 (* t_0 t_0)) 0.0003000000142492354)
     (* (* (+ uy uy) (PI)) (sqrt (* (fma maxCos -2.0 2.0) ux)))
     (* (sqrt (- 1.0 (* (- 1.0 ux) t_0))) (* (* (PI) 2.0) uy)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))) < 3.00000014e-4

   1. Initial program 34.6%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 31.0

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

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 7.3%

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

   10. Applied rewrites 7.3%

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

   11. Taylor expanded in ux around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

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

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

       4. metadata-eval N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f32 73.5

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

   13. Applied rewrites 72.8%

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

   if 3.00000014e-4 < (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))

   1. Initial program 89.8%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 74.0

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

   5. Applied rewrites 74.0%

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

   6. 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 69.2

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

   8. Applied rewrites 69.2%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) \leq 0.0003000000142492354:\\ \;\;\;\;\left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos, -2, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 5.999999848427251 \cdot 10^{-5}:\\ \;\;\;\;\left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos, -2, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux \cdot maxCos - ux\right) + 1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 5.999999848427251e-5)
   (* (* (+ uy uy) (PI)) (sqrt (* (fma maxCos -2.0 2.0) ux)))
   (*
    (sqrt
     (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- (* ux maxCos) ux) 1.0))))
    (* (* (PI) 2.0) uy))))

Derivation

1. Split input into 2 regimes

2. if ux < 5.99999985e-5

   1. Initial program 33.0%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 30.6

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

   5. Applied rewrites 30.6%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 7.3%

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

   10. Applied rewrites 7.3%

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

   11. Taylor expanded in ux around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

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

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

       4. metadata-eval N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f32 75.2

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

   13. Applied rewrites 74.5%

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

   if 5.99999985e-5 < ux

   1. Initial program 88.4%

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

   3. 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 71.6

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

   5. Applied rewrites 71.6%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. lower--.f32 71.7

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

   7. Applied rewrites 71.7%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 5.999999848427251 \cdot 10^{-5}:\\ \;\;\;\;\left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos, -2, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux \cdot maxCos - ux\right) + 1\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;ux \leq 5.999999848427251 \cdot 10^{-5}:\\ \;\;\;\;\left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos, -2, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - t\_0 \cdot t\_0} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= ux 5.999999848427251e-5)
     (* (* (+ uy uy) (PI)) (sqrt (* (fma maxCos -2.0 2.0) ux)))
     (* (sqrt (- 1.0 (* t_0 t_0))) (* (* (PI) 2.0) uy)))))

Derivation

1. Split input into 2 regimes

2. if ux < 5.99999985e-5

   1. Initial program 33.0%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 30.6

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

   5. Applied rewrites 30.6%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 7.3%

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

   10. Applied rewrites 7.3%

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

   11. Taylor expanded in ux around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

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

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

       4. metadata-eval N/A

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

       5. +-commutative N/A

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

       6. *-commutative N/A

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

       7. lower-fma.f32 71.5

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

   13. Applied rewrites 74.5%

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

   if 5.99999985e-5 < ux

   1. Initial program 88.4%

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

   3. 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 71.6

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

   5. Applied rewrites 71.6%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 5.999999848427251 \cdot 10^{-5}:\\ \;\;\;\;\left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(maxCos, -2, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.2% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.5%

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

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

   3. lift--.f32 N/A

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

   4. sub-neg N/A

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

   5. associate-+l+ N/A

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

   6. distribute-lft-in N/A

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

   7. *-rgt-identity N/A

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

   8. lower-+.f32 N/A

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

   9. lift-+.f32 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lift-+.f32 N/A

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

   16. +-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f32 N/A

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

   20. neg-mul-1 N/A

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

   21. lift-*.f32 N/A

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

   22. *-commutative N/A

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

   23. distribute-rgt-out N/A

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

   24. lower-*.f32 N/A

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

4. Applied rewrites 17.3%

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

   1. lift--.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. associate--r+ N/A

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

   4. lower--.f32 N/A

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

   5. lift-fma.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. +-commutative N/A

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

   9. lift--.f32 N/A

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

   10. associate-+l- N/A

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

   11. associate--r- N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. lower--.f32 49.3

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

   18. lift-*.f32 N/A

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

   19. lift-*.f32 N/A

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

   20. associate-*r* N/A

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

   21. lift-+.f32 N/A

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

6. Applied rewrites 48.5%

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

7. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 79.5

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

9. Applied rewrites 79.3%

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

10. Final simplification 79.5%

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

Alternative 15: 63.3% accurate, 4.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.5%

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

3. 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 47.3

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

5. Applied rewrites 47.3%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 7.1%

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

10. Applied rewrites 7.1%

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

11. Taylor expanded in ux around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

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

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

    4. metadata-eval N/A

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

    5. +-commutative N/A

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

    6. *-commutative N/A

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

    7. lower-fma.f32 63.4

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

13. Applied rewrites 63.3%

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

14. Final simplification 63.4%

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

Alternative 16: 20.4% accurate, 5.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.5%

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

3. 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 47.3

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

5. Applied rewrites 47.3%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 7.1%

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

10. Applied rewrites 7.1%

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

    1. lift--.f32 N/A

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

    2. sub-neg N/A

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

    3. +-commutative N/A

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

    4. neg-mul-1 N/A

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

    5. lower-fma.f32 20.3

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

12. Applied rewrites 20.3%

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

13. Final simplification 20.3%

    \[\leadsto \sqrt{\mathsf{fma}\left(-1, 1, 1\right)} \cdot \left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right) \]
14. Add Preprocessing

Alternative 17: 7.1% accurate, 7.1× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((1.0f - 1.0f)) * (uy + uy);
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((1.0e0 - 1.0e0)) * (uy + uy)
end function

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32(1.0) - Float32(1.0))) * Float32(uy + uy))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((single(1.0) - single(1.0))) * (uy + uy);
end

Derivation

1. Initial program 55.5%

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

3. 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 47.3

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

5. Applied rewrites 47.3%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 7.1%

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

10. Applied rewrites 7.1%

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

12. Applied rewrites 7.1%

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

13. Final simplification 7.1%

    \[\leadsto \sqrt{1 - 1} \cdot \left(uy + uy\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024271 
(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)))))))