UniformSampleCone, y

Percentage Accurate: 57.2% → 98.3%

Time: 10.1s

Alternatives: 12

Speedup: 3.2×

Specification

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

Math

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

FPCore

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

Initial Program: 57.2% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. distribute-rgt-in N/A

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

   5. associate--r+ N/A

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

   6. lower--.f32 N/A

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

   7. lower--.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-+.f32 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

4. Applied rewrites 55.9%

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

5. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

7. Applied rewrites 98.2%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 98.2%

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

11. Taylor expanded in ux around 0

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

13. Applied rewrites 98.2%

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

14. Final simplification 98.2%

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

Alternative 2: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. distribute-rgt-in N/A

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

   5. associate--r+ N/A

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

   6. lower--.f32 N/A

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

   7. lower--.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-+.f32 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

4. Applied rewrites 55.9%

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

5. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

7. Applied rewrites 98.2%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 97.5%

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

11. Final simplification 97.5%

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

Alternative 3: 97.0% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. distribute-rgt-in N/A

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

   5. associate--r+ N/A

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

   6. lower--.f32 N/A

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

   7. lower--.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-+.f32 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

4. Applied rewrites 55.9%

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

5. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

7. Applied rewrites 98.2%

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

9. Applied rewrites 98.2%

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

10. Taylor expanded in maxCos around 0

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

12. Applied rewrites 96.6%

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

13. Final simplification 96.6%

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

Alternative 4: 95.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.6000000186977559 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(1 - maxCos, ux, ux\right) - 2, maxCos, 2 - ux\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.6000000186977559e-6)
   (* (sin (* (* uy 2.0) (PI))) (sqrt (* (- 2.0 ux) ux)))
   (*
    (* (* (PI) uy) 2.0)
    (sqrt (* (fma (- (fma (- 1.0 maxCos) ux ux) 2.0) maxCos (- 2.0 ux)) ux)))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.60000002e-6

   1. Initial program 56.8%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. associate-*r* N/A

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

      8. mul-1-neg N/A

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

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

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lower-pow.f32 N/A

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

      14. lower--.f32 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 97.9%

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

   if 1.60000002e-6 < maxCos

   1. Initial program 60.2%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate--r+ N/A

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

      6. lower--.f32 N/A

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

      7. lower--.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lift-+.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

   4. Applied rewrites 59.9%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in maxCos around 0

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

   10. Applied rewrites 98.4%

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

   11. Taylor expanded in uy around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f32 N/A

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

       5. lower-PI.f32 86.9

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

   13. Applied rewrites 86.9%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 1.6000000186977559 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(1 - maxCos, ux, ux\right) - 2, maxCos, 2 - ux\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \leq 0.013000000268220901:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\mathsf{fma}\left(-1, maxCos - 1, 1\right) - maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\mathsf{fma}\left(ux - \mathsf{fma}\left(ux, maxCos, 1\right), \mathsf{fma}\left(ux, maxCos, 1\right) - ux, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (sqrt (- 1.0 (* t_0 t_0))) 0.013000000268220901)
     (*
      (* (* (PI) 2.0) uy)
      (sqrt (* (- (fma -1.0 (- maxCos 1.0) 1.0) maxCos) ux)))
     (*
      (* (* (PI) uy) 2.0)
      (sqrt
       (fma (- ux (fma ux maxCos 1.0)) (- (fma ux maxCos 1.0) ux) 1.0))))))

Derivation

1. Split input into 2 regimes

2. if (sqrt.f32 (-.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))))) < 0.0130000003

   1. Initial program 34.7%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 30.7

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

   5. Applied rewrites 30.7%

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

      1. unpow1 N/A

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

      2. sqr-pow N/A

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

      3. lower-*.f32 N/A

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

      4. metadata-eval N/A

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

      5. unpow1/2 N/A

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

      6. lower-sqrt.f32 N/A

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

      7. lift-+.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f32 N/A

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

      12. metadata-eval N/A

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

      13. unpow1/2 N/A

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

      14. lower-sqrt.f32 29.9

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

      15. lift-+.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f32 29.9

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

   7. Applied rewrites 29.9%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-sqrt.f32 N/A

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

      8. lift-sqrt.f32 N/A

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

      9. rem-square-sqrt N/A

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

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

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

      11. lift-fma.f32 N/A

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

      12. +-commutative N/A

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

      13. lift--.f32 N/A

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

      14. *-commutative N/A

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

   9. Applied rewrites 30.0%

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

   10. Taylor expanded in ux around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. lower--.f32 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f32 N/A

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

       6. lower--.f32 75.5

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

   12. Applied rewrites 75.5%

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

   if 0.0130000003 < (sqrt.f32 (-.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 87.4%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 76.9

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

   5. Applied rewrites 76.9%

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

      1. unpow1 N/A

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

      2. sqr-pow N/A

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

      3. lower-*.f32 N/A

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

      4. metadata-eval N/A

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

      5. unpow1/2 N/A

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

      6. lower-sqrt.f32 N/A

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

      7. lift-+.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f32 N/A

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

      12. metadata-eval N/A

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

      13. unpow1/2 N/A

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

      14. lower-sqrt.f32 76.4

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

      15. lift-+.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f32 76.4

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

   7. Applied rewrites 76.4%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-sqrt.f32 N/A

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

      8. lift-sqrt.f32 N/A

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

      9. rem-square-sqrt N/A

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

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

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

      11. lift-fma.f32 N/A

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

      12. +-commutative N/A

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

      13. lift--.f32 N/A

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

      14. *-commutative N/A

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

   9. Applied rewrites 76.7%

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

   10. Taylor expanded in uy around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f32 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f32 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f32 N/A

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

       7. lower-PI.f32 N/A

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

       8. lower-sqrt.f32 N/A

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

       9. +-commutative N/A

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

       10. lower-fma.f32 N/A

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

       11. lower--.f32 N/A

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

       12. +-commutative N/A

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

       13. *-commutative N/A

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

       14. lower-fma.f32 N/A

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

       15. lower--.f32 N/A

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

       16. +-commutative N/A

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

       17. *-commutative N/A

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

       18. lower-fma.f32 77.2

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

   12. Applied rewrites 77.2%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \leq 0.013000000268220901:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\mathsf{fma}\left(-1, maxCos - 1, 1\right) - maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\mathsf{fma}\left(ux - \mathsf{fma}\left(ux, maxCos, 1\right), \mathsf{fma}\left(ux, maxCos, 1\right) - ux, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.0% accurate, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.4%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 50.6

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

5. Applied rewrites 50.6%

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

   1. unpow1 N/A

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

   2. sqr-pow N/A

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

   3. lower-*.f32 N/A

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

   4. metadata-eval N/A

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

   5. unpow1/2 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

   11. lift-fma.f32 N/A

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

   12. metadata-eval N/A

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

   13. unpow1/2 N/A

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

   14. lower-sqrt.f32 49.9

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

   15. lift-+.f32 N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. +-commutative N/A

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

   19. lift-fma.f32 49.9

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

7. Applied rewrites 49.9%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-sqrt.f32 N/A

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

   8. lift-sqrt.f32 N/A

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

   9. rem-square-sqrt N/A

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

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

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

   11. lift-fma.f32 N/A

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

   12. +-commutative N/A

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

   13. lift--.f32 N/A

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

   14. *-commutative N/A

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

9. Applied rewrites 50.1%

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

10. Taylor expanded in ux around inf

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. lower--.f32 N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f32 N/A

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

    6. lower-/.f32 N/A

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

    7. lower--.f32 N/A

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

    8. lower-fma.f32 N/A

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

    9. lower--.f32 N/A

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

    10. lower--.f32 N/A

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

    11. lower-/.f32 N/A

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

    12. lower-/.f32 N/A

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

    13. unpow2 N/A

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

    14. lower-*.f32 80.9

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

12. Applied rewrites 80.9%

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

13. Final simplification 80.9%

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

Alternative 7: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9995200037956238:\\ \;\;\;\;t\_1 \cdot \sqrt{1 - t\_0 \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\left(\mathsf{fma}\left(-1, maxCos - 1, 1\right) - maxCos\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))) (t_1 (* (* (PI) 2.0) uy)))
   (if (<= (* t_0 t_0) 0.9995200037956238)
     (* t_1 (sqrt (- 1.0 (* t_0 (- 1.0 ux)))))
     (* t_1 (sqrt (* (- (fma -1.0 (- maxCos 1.0) 1.0) maxCos) ux))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.999520004

   1. Initial program 89.2%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 79.2

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

   5. Applied rewrites 79.2%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower--.f32 76.7

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

   8. Applied rewrites 76.7%

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

   if 0.999520004 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 38.2%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 33.4

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

   5. Applied rewrites 33.4%

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

      1. unpow1 N/A

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

      2. sqr-pow N/A

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

      3. lower-*.f32 N/A

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

      4. metadata-eval N/A

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

      5. unpow1/2 N/A

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

      6. lower-sqrt.f32 N/A

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

      7. lift-+.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f32 N/A

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

      12. metadata-eval N/A

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

      13. unpow1/2 N/A

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

      14. lower-sqrt.f32 32.5

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

      15. lift-+.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f32 32.5

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

   7. Applied rewrites 32.5%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-sqrt.f32 N/A

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

      8. lift-sqrt.f32 N/A

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

      9. rem-square-sqrt N/A

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

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

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

      11. lift-fma.f32 N/A

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

      12. +-commutative N/A

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

      13. lift--.f32 N/A

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

      14. *-commutative N/A

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

   9. Applied rewrites 32.7%

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

   10. Taylor expanded in ux around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. lower--.f32 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f32 N/A

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

       6. lower--.f32 74.2

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

   12. Applied rewrites 74.2%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) \leq 0.9995200037956238:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\mathsf{fma}\left(-1, maxCos - 1, 1\right) - maxCos\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.0% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.4%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 50.6

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

5. Applied rewrites 50.6%

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

   1. unpow1 N/A

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

   2. sqr-pow N/A

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

   3. lower-*.f32 N/A

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

   4. metadata-eval N/A

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

   5. unpow1/2 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

   11. lift-fma.f32 N/A

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

   12. metadata-eval N/A

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

   13. unpow1/2 N/A

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

   14. lower-sqrt.f32 49.9

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

   15. lift-+.f32 N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. +-commutative N/A

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

   19. lift-fma.f32 49.9

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

7. Applied rewrites 49.9%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-sqrt.f32 N/A

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

   8. lift-sqrt.f32 N/A

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

   9. rem-square-sqrt N/A

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

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

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

   11. lift-fma.f32 N/A

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

   12. +-commutative N/A

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

   13. lift--.f32 N/A

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

   14. *-commutative N/A

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

9. Applied rewrites 50.1%

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

10. Taylor expanded in ux around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

12. Applied rewrites 80.9%

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

13. Final simplification 80.9%

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

Alternative 9: 81.0% accurate, 3.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. distribute-rgt-in N/A

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

   5. associate--r+ N/A

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

   6. lower--.f32 N/A

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

   7. lower--.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-+.f32 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

4. Applied rewrites 55.9%

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

5. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

7. Applied rewrites 98.2%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 98.2%

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

11. Taylor expanded in uy around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f32 N/A

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

    5. lower-PI.f32 80.8

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

13. Applied rewrites 80.8%

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

14. Final simplification 80.8%

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

Alternative 10: 75.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\\ \mathbf{if}\;ux \leq 0.00023999999393709004:\\ \;\;\;\;t\_0 \cdot \sqrt{\left(\mathsf{fma}\left(-1, maxCos - 1, 1\right) - maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(ux - 1, 1 - ux, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (PI) 2.0) uy)))
   (if (<= ux 0.00023999999393709004)
     (* t_0 (sqrt (* (- (fma -1.0 (- maxCos 1.0) 1.0) maxCos) ux)))
     (* t_0 (sqrt (fma (- ux 1.0) (- 1.0 ux) 1.0))))))

Derivation

1. Split input into 2 regimes

2. if ux < 2.39999994e-4

   1. Initial program 38.2%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 33.4

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

   5. Applied rewrites 33.4%

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

      1. unpow1 N/A

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

      2. sqr-pow N/A

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

      3. lower-*.f32 N/A

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

      4. metadata-eval N/A

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

      5. unpow1/2 N/A

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

      6. lower-sqrt.f32 N/A

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

      7. lift-+.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f32 N/A

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

      12. metadata-eval N/A

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

      13. unpow1/2 N/A

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

      14. lower-sqrt.f32 32.5

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

      15. lift-+.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f32 32.5

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

   7. Applied rewrites 32.5%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-sqrt.f32 N/A

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

      8. lift-sqrt.f32 N/A

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

      9. rem-square-sqrt N/A

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

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

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

      11. lift-fma.f32 N/A

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

      12. +-commutative N/A

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

      13. lift--.f32 N/A

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

      14. *-commutative N/A

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

   9. Applied rewrites 32.7%

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

   10. Taylor expanded in ux around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. lower--.f32 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f32 N/A

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

       6. lower--.f32 74.2

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

   12. Applied rewrites 74.2%

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

   if 2.39999994e-4 < ux

   1. Initial program 89.2%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 79.2

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

   5. Applied rewrites 79.2%

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

      1. unpow1 N/A

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

      2. sqr-pow N/A

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

      3. lower-*.f32 N/A

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

      4. metadata-eval N/A

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

      5. unpow1/2 N/A

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

      6. lower-sqrt.f32 N/A

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

      7. lift-+.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f32 N/A

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

      12. metadata-eval N/A

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

      13. unpow1/2 N/A

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

      14. lower-sqrt.f32 78.9

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

      15. lift-+.f32 N/A

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f32 78.9

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

   7. Applied rewrites 78.9%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-sqrt.f32 N/A

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

      8. lift-sqrt.f32 N/A

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

      9. rem-square-sqrt N/A

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

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

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

      11. lift-fma.f32 N/A

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

      12. +-commutative N/A

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

      13. lift--.f32 N/A

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

      14. *-commutative N/A

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

   9. Applied rewrites 79.0%

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

   10. Taylor expanded in maxCos around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f32 N/A

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

       4. lower--.f32 N/A

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

       5. lower--.f32 76.3

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

   12. Applied rewrites 76.3%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00023999999393709004:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\mathsf{fma}\left(-1, maxCos - 1, 1\right) - maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\mathsf{fma}\left(ux - 1, 1 - ux, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 48.8% accurate, 4.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.4%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 50.6

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

5. Applied rewrites 50.6%

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

   1. unpow1 N/A

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

   2. sqr-pow N/A

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

   3. lower-*.f32 N/A

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

   4. metadata-eval N/A

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

   5. unpow1/2 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. +-commutative N/A

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

   11. lift-fma.f32 N/A

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

   12. metadata-eval N/A

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

   13. unpow1/2 N/A

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

   14. lower-sqrt.f32 49.9

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

   15. lift-+.f32 N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. +-commutative N/A

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

   19. lift-fma.f32 49.9

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

7. Applied rewrites 49.9%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-sqrt.f32 N/A

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

   8. lift-sqrt.f32 N/A

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

   9. rem-square-sqrt N/A

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

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

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

   11. lift-fma.f32 N/A

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

   12. +-commutative N/A

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

   13. lift--.f32 N/A

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

   14. *-commutative N/A

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

9. Applied rewrites 50.1%

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

10. Taylor expanded in maxCos around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f32 N/A

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

    4. lower--.f32 N/A

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

    5. lower--.f32 49.0

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

12. Applied rewrites 49.0%

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

13. Final simplification 49.0%

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

Alternative 12: 7.1% accurate, 5.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.4%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 50.6

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

5. Applied rewrites 50.6%

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

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 7.1%

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

10. Applied rewrites 7.1%

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

Reproduce

herbie shell --seed 2024353 
(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)))))))