UniformSampleCone, y

Percentage Accurate: 57.4% → 98.2%

Time: 13.5s

Alternatives: 15

Speedup: 5.4×

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: 57.4% 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: 98.2% accurate, 0.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.8%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. associate-+l- N/A

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

   4. lower--.f32 N/A

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

   5. lower--.f32 55.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 55.9

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

4. Applied rewrites 55.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift--.f32 N/A

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

   4. sub-neg N/A

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

   5. distribute-lft-in N/A

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

   6. lower-+.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-+.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. +-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f32 N/A

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

   19. lower-neg.f32 16.9

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

6. Applied rewrites 16.9%

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

7. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   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. lower--.f32 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

9. Applied rewrites 98.3%

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

    1. lift-sin.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. *-commutative N/A

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

    5. associate-*r* N/A

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

    6. *-commutative N/A

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

    7. lift-*.f32 N/A

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

    8. 2-sin N/A

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

    9. lift-sin.f32 N/A

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

    10. lift-cos.f32 N/A

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

    11. associate-*r* N/A

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

    12. lower-*.f32 N/A

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

    13. *-commutative N/A

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

    14. lower-*.f32 98.4

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

11. Applied rewrites 98.4%

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

12. Final simplification 98.4%

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

Alternative 2: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.8%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. associate-+l- N/A

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

   4. lower--.f32 N/A

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

   5. lower--.f32 55.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 55.9

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

4. Applied rewrites 55.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift--.f32 N/A

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

   4. sub-neg N/A

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

   5. distribute-lft-in N/A

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

   6. lower-+.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-+.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. +-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f32 N/A

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

   19. lower-neg.f32 16.9

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

6. Applied rewrites 16.9%

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

7. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   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. lower--.f32 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

9. Applied rewrites 98.3%

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

11. Applied rewrites 98.3%

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

12. Final simplification 98.3%

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

Alternative 3: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.8%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. associate-+l- N/A

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

   4. lower--.f32 N/A

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

   5. lower--.f32 55.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 55.9

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

4. Applied rewrites 55.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift--.f32 N/A

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

   4. sub-neg N/A

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

   5. distribute-lft-in N/A

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

   6. lower-+.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-+.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. +-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f32 N/A

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

   19. lower-neg.f32 16.9

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

6. Applied rewrites 16.9%

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

7. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   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. lower--.f32 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

9. Applied rewrites 98.3%

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

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. *-commutative N/A

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

    6. lift-*.f32 N/A

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

    7. count-2 N/A

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

    8. lift-*.f32 N/A

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

    9. lift-*.f32 N/A

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

    10. distribute-lft-out N/A

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

    11. lower-*.f32 N/A

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

    12. lower-+.f32 98.3

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

11. Applied rewrites 98.3%

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

12. Final simplification 98.3%

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

Alternative 4: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.8%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. associate-+l- N/A

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

   4. lower--.f32 N/A

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

   5. lower--.f32 55.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 55.9

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

4. Applied rewrites 55.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift--.f32 N/A

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

   4. sub-neg N/A

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

   5. distribute-lft-in N/A

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

   6. lower-+.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-+.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. +-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f32 N/A

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

   19. lower-neg.f32 16.9

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

6. Applied rewrites 16.9%

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

7. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. cancel-sign-sub N/A

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

   4. lower-+.f32 N/A

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

9. Applied rewrites 97.6%

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

10. Final simplification 97.6%

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

Alternative 5: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.8%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. associate-+l- N/A

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

   4. lower--.f32 N/A

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

   5. lower--.f32 55.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 55.9

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

4. Applied rewrites 55.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift--.f32 N/A

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

   4. sub-neg N/A

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

   5. distribute-lft-in N/A

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

   6. lower-+.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-+.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. +-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f32 N/A

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

   19. lower-neg.f32 16.9

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

6. Applied rewrites 16.9%

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

   1. lift--.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. associate--r+ N/A

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

   4. *-rgt-identity N/A

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

   5. lift-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-neg.f32 N/A

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

   9. cancel-sign-sub N/A

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

   10. *-commutative N/A

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

   11. lower-+.f32 N/A

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

8. Applied rewrites 43.7%

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

9. Final simplification 44.5%

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

Alternative 6: 96.9% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.8%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. associate-+l- N/A

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

   4. lower--.f32 N/A

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

   5. lower--.f32 55.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 55.9

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

4. Applied rewrites 55.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift--.f32 N/A

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

   4. sub-neg N/A

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

   5. distribute-lft-in N/A

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

   6. lower-+.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-+.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. +-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f32 N/A

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

   19. lower-neg.f32 16.9

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

6. Applied rewrites 16.9%

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

7. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   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. lower--.f32 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

9. Applied rewrites 98.3%

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

10. Taylor expanded in maxCos around 0

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

12. Applied rewrites 83.1%

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

13. Final simplification 83.3%

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

Alternative 7: 82.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - \left(ux \cdot maxCos - ux\right)\\ \mathbf{if}\;ux \leq 0.00026000000070780516:\\ \;\;\;\;\sqrt{2 \cdot ux} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - t\_0 \cdot t\_0} \cdot \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- -1.0 (- (* ux maxCos) ux))))
   (if (<= ux 0.00026000000070780516)
     (* (sqrt (* 2.0 ux)) (sin (* (* 2.0 uy) (PI))))
     (* (sqrt (- 1.0 (* t_0 t_0))) (* 2.0 (* uy (PI)))))))

Derivation

1. Split input into 2 regimes

2. if ux < 2.60000001e-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. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. div-sub N/A

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

      6. associate-+l- N/A

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

      7. lower--.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f32 37.9

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f32 37.9

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

   4. Applied rewrites 37.9%

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

   5. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 36.8

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

   7. Applied rewrites 36.8%

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

   8. Taylor expanded in maxCos around 0

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

   10. Applied rewrites 87.9%

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

   if 2.60000001e-4 < ux

   1. Initial program 92.0%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f32 N/A

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

      5. lower--.f32 92.2

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 92.2

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

   4. Applied rewrites 92.2%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f32 N/A

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

      6. associate--r- N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 92.2

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

   6. Applied rewrites 92.2%

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

   7. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-PI.f32 75.0

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

   9. Applied rewrites 75.0%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00026000000070780516:\\ \;\;\;\;\sqrt{2 \cdot ux} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(-1 - \left(ux \cdot maxCos - ux\right)\right) \cdot \left(-1 - \left(ux \cdot maxCos - ux\right)\right)} \cdot \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.5% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.8%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. associate-+l- N/A

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

   4. lower--.f32 N/A

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

   5. lower--.f32 55.9

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 55.9

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

4. Applied rewrites 55.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift--.f32 N/A

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

   4. sub-neg N/A

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

   5. distribute-lft-in N/A

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

   6. lower-+.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lift-+.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. +-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f32 N/A

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

   19. lower-neg.f32 16.9

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

6. Applied rewrites 16.9%

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

7. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   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. lower--.f32 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

9. Applied rewrites 98.3%

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

10. Taylor expanded in maxCos around 0

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

12. Applied rewrites 92.7%

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

13. Final simplification 92.7%

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

Alternative 9: 76.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - \left(ux \cdot maxCos - ux\right)\\ \mathbf{if}\;ux \cdot maxCos - \left(ux - 1\right) \leq 0.9997950196266174:\\ \;\;\;\;\sqrt{1 - t\_0 \cdot t\_0} \cdot \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{2}{maxCos} - 2\right) \cdot maxCos\right) \cdot ux} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- -1.0 (- (* ux maxCos) ux))))
   (if (<= (- (* ux maxCos) (- ux 1.0)) 0.9997950196266174)
     (* (sqrt (- 1.0 (* t_0 t_0))) (* 2.0 (* uy (PI))))
     (* (sqrt (* (* (- (/ 2.0 maxCos) 2.0) maxCos) ux)) (* (* 2.0 (PI)) uy)))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.99979502

   1. Initial program 91.1%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f32 N/A

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

      5. lower--.f32 91.3

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 91.3

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

   4. Applied rewrites 91.3%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f32 N/A

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

      6. associate--r- N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 91.3

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

   6. Applied rewrites 91.3%

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

   7. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-PI.f32 73.4

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

   9. Applied rewrites 73.4%

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

   if 0.99979502 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 37.4%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 34.2

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

   5. Applied rewrites 34.2%

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

   6. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 76.2

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

   8. Applied rewrites 76.2%

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

   9. Taylor expanded in maxCos around inf

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

   11. Applied rewrites 78.7%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \cdot maxCos - \left(ux - 1\right) \leq 0.9997950196266174:\\ \;\;\;\;\sqrt{1 - \left(-1 - \left(ux \cdot maxCos - ux\right)\right) \cdot \left(-1 - \left(ux \cdot maxCos - ux\right)\right)} \cdot \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{2}{maxCos} - 2\right) \cdot maxCos\right) \cdot ux} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\\ t_1 := ux \cdot maxCos - \left(ux - 1\right)\\ \mathbf{if}\;t\_1 \leq 0.9997950196266174:\\ \;\;\;\;\sqrt{\left(-1 - \left(ux \cdot maxCos - ux\right)\right) \cdot t\_1 + 1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{2}{maxCos} - 2\right) \cdot maxCos\right) \cdot ux} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* 2.0 (PI)) uy)) (t_1 (- (* ux maxCos) (- ux 1.0))))
   (if (<= t_1 0.9997950196266174)
     (* (sqrt (+ (* (- -1.0 (- (* ux maxCos) ux)) t_1) 1.0)) t_0)
     (* (sqrt (* (* (- (/ 2.0 maxCos) 2.0) maxCos) ux)) t_0))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.99979502

   1. Initial program 91.1%

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

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

   5. Applied rewrites 73.3%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f32 N/A

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

      6. associate--r- N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 73.4

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

   7. Applied rewrites 73.4%

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

   if 0.99979502 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 37.4%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 34.2

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

   5. Applied rewrites 34.2%

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

   6. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 76.2

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

   8. Applied rewrites 76.2%

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

   9. Taylor expanded in maxCos around inf

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

   11. Applied rewrites 78.7%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \cdot maxCos - \left(ux - 1\right) \leq 0.9997950196266174:\\ \;\;\;\;\sqrt{\left(-1 - \left(ux \cdot maxCos - ux\right)\right) \cdot \left(ux \cdot maxCos - \left(ux - 1\right)\right) + 1} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{2}{maxCos} - 2\right) \cdot maxCos\right) \cdot ux} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(ux - 1\right) - ux \cdot maxCos\\ t_1 := \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\\ \mathbf{if}\;ux \cdot maxCos - \left(ux - 1\right) \leq 0.9997950196266174:\\ \;\;\;\;\sqrt{1 - t\_0 \cdot t\_0} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{2}{maxCos} - 2\right) \cdot maxCos\right) \cdot ux} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- (- ux 1.0) (* ux maxCos))) (t_1 (* (* 2.0 (PI)) uy)))
   (if (<= (- (* ux maxCos) (- ux 1.0)) 0.9997950196266174)
     (* (sqrt (- 1.0 (* t_0 t_0))) t_1)
     (* (sqrt (* (* (- (/ 2.0 maxCos) 2.0) maxCos) ux)) t_1))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.99979502

   1. Initial program 91.1%

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

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

   5. Applied rewrites 73.3%

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

   if 0.99979502 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 37.4%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 34.2

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

   5. Applied rewrites 34.2%

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

   6. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 76.2

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

   8. Applied rewrites 76.2%

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

   9. Taylor expanded in maxCos around inf

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

   11. Applied rewrites 78.7%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \cdot maxCos - \left(ux - 1\right) \leq 0.9997950196266174:\\ \;\;\;\;\sqrt{1 - \left(\left(ux - 1\right) - ux \cdot maxCos\right) \cdot \left(\left(ux - 1\right) - ux \cdot maxCos\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{2}{maxCos} - 2\right) \cdot maxCos\right) \cdot ux} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\\ t_1 := ux \cdot maxCos - \left(ux - 1\right)\\ \mathbf{if}\;t\_1 \leq 0.999750018119812:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot t\_1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{2}{maxCos} - 2\right) \cdot maxCos\right) \cdot ux} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* 2.0 (PI)) uy)) (t_1 (- (* ux maxCos) (- ux 1.0))))
   (if (<= t_1 0.999750018119812)
     (* (sqrt (- 1.0 (* (- 1.0 ux) t_1))) t_0)
     (* (sqrt (* (* (- (/ 2.0 maxCos) 2.0) maxCos) ux)) t_0))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.999750018

   1. Initial program 92.0%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 74.2

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

   5. Applied rewrites 74.2%

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

   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 69.7

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

   8. Applied rewrites 69.7%

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

   if 0.999750018 < (+.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 34.6

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

   5. Applied rewrites 34.6%

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

   6. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 75.4

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

   8. Applied rewrites 75.4%

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

   9. Taylor expanded in maxCos around inf

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

   11. Applied rewrites 78.0%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \cdot maxCos - \left(ux - 1\right) \leq 0.999750018119812:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(ux \cdot maxCos - \left(ux - 1\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{2}{maxCos} - 2\right) \cdot maxCos\right) \cdot ux} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\\ t_1 := ux \cdot maxCos - \left(ux - 1\right)\\ \mathbf{if}\;t\_1 \leq 0.999750018119812:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot t\_1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot maxCos + 2\right) \cdot ux} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* 2.0 (PI)) uy)) (t_1 (- (* ux maxCos) (- ux 1.0))))
   (if (<= t_1 0.999750018119812)
     (* (sqrt (- 1.0 (* (- 1.0 ux) t_1))) t_0)
     (* (sqrt (* (+ (* -2.0 maxCos) 2.0) ux)) t_0))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.999750018

   1. Initial program 92.0%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 74.2

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

   5. Applied rewrites 74.2%

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

   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 69.7

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

   8. Applied rewrites 69.7%

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

   if 0.999750018 < (+.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 34.6

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

   5. Applied rewrites 34.6%

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

   6. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 75.4

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

   8. Applied rewrites 75.4%

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

   10. Applied rewrites 78.0%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \cdot maxCos - \left(ux - 1\right) \leq 0.999750018119812:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(ux \cdot maxCos - \left(ux - 1\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot maxCos + 2\right) \cdot ux} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.9% accurate, 4.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.8%

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

3. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 47.6

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

5. Applied rewrites 47.6%

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

6. Taylor expanded in ux around 0

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

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

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 65.3

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

8. Applied rewrites 65.3%

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

10. Applied rewrites 67.3%

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

11. Final simplification 67.3%

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

Alternative 15: 63.5% accurate, 5.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.8%

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

3. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 47.6

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

5. Applied rewrites 47.6%

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

6. Taylor expanded in ux around 0

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

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

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 65.3

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

8. Applied rewrites 65.3%

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

9. Taylor expanded in maxCos around 0

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

11. Applied rewrites 65.3%

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

13. Applied rewrites 65.3%

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

14. Final simplification 65.3%

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

Reproduce

herbie shell --seed 2024244 
(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)))))))