UniformSampleCone, y

Percentage Accurate: 57.3% → 98.0%

Time: 16.8s

Alternatives: 10

Speedup: 5.0×

Specification

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

Math

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

FPCore

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

Initial Program: 57.3% 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: 98.0% accurate, 0.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.7%

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

3. Taylor expanded in maxCos around inf

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

   1. lower-*.f32 N/A

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

   2. associate--l+ N/A

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

   3. div-sub N/A

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

   4. lower-+.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower--.f32 57.9

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

5. Applied rewrites 57.9%

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

6. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-+.f32 N/A

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

   16. mul-1-neg N/A

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

   17. lower-neg.f32 N/A

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

   18. sub-neg N/A

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

8. Applied rewrites 98.0%

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

   1. lift-/.f32 N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. div-inv N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*l* N/A

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

   9. lift-/.f32 N/A

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

   10. inv-pow N/A

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

   11. pow-plus N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-+.f32 N/A

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

   16. lower-*.f32 98.2

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

10. Applied rewrites 98.2%

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

    1. lift-neg.f32 N/A

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

    2. lift-+.f32 N/A

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

    3. lift-/.f32 N/A

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

    4. lift-+.f32 N/A

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

    5. lift-*.f32 N/A

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

    6. *-commutative N/A

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

    7. lift-*.f32 N/A

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

    8. associate-*l* N/A

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

    9. lower-*.f32 N/A

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

    10. lift-+.f32 N/A

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

    11. lift-neg.f32 N/A

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

    12. unsub-neg N/A

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

    13. lower--.f32 N/A

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

    14. lower-*.f32 98.3

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

12. Applied rewrites 98.3%

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

13. Final simplification 98.3%

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

Alternative 2: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.7%

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

3. Taylor expanded in maxCos around inf

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

   1. lower-*.f32 N/A

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

   2. associate--l+ N/A

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

   3. div-sub N/A

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

   4. lower-+.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower--.f32 57.9

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

5. Applied rewrites 57.9%

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

6. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-+.f32 N/A

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

   16. mul-1-neg N/A

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

   17. lower-neg.f32 N/A

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

   18. sub-neg N/A

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

8. Applied rewrites 98.0%

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

   1. lift-/.f32 N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. div-inv N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*l* N/A

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

   9. lift-/.f32 N/A

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

   10. inv-pow N/A

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

   11. pow-plus N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-+.f32 N/A

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

   16. lower-*.f32 98.2

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

10. Applied rewrites 98.2%

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

11. Taylor expanded in maxCos around 0

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

13. Applied rewrites 96.6%

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

14. Final simplification 96.6%

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

Alternative 3: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{if}\;uy \cdot 2 \leq 0.0008299999753944576:\\ \;\;\;\;t\_0 \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2 + maxCos \cdot -2}{ux} + maxCos \cdot \left(\left(1 - maxCos\right) \cdot \left(\frac{-1}{maxCos} - -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(ux \cdot \sin t\_0\right) \cdot \sqrt{-1 + \frac{2}{ux}}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy (PI)))))
   (if (<= (* uy 2.0) 0.0008299999753944576)
     (*
      t_0
      (sqrt
       (*
        (* ux ux)
        (+
         (/ (+ 2.0 (* maxCos -2.0)) ux)
         (* maxCos (* (- 1.0 maxCos) (- (/ -1.0 maxCos) -1.0)))))))
     (* (* ux (sin t_0)) (sqrt (+ -1.0 (/ 2.0 ux)))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 8.29999975e-4

   1. Initial program 58.6%

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

   3. Taylor expanded in maxCos around inf

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

      1. lower-*.f32 N/A

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

      2. associate--l+ N/A

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

      3. div-sub N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower--.f32 58.7

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in ux around -inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-+.f32 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f32 N/A

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

      18. sub-neg N/A

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

   8. Applied rewrites 97.9%

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

      1. lift-/.f32 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lift-/.f32 N/A

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

      6. div-inv N/A

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

      7. lift-/.f32 N/A

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

      8. associate-*l* N/A

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

      9. lift-/.f32 N/A

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

      10. inv-pow N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f32 N/A

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

      16. lower-*.f32 98.4

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

   10. Applied rewrites 98.4%

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

   11. Taylor expanded in uy around 0

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

       1. lower-*.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. lower-PI.f32 97.8

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

   13. Applied rewrites 97.8%

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

   if 8.29999975e-4 < (*.f32 uy #s(literal 2 binary32))

   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 maxCos around inf

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

      1. lower-*.f32 N/A

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

      2. associate--l+ N/A

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

      3. div-sub N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower--.f32 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in ux around -inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-+.f32 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f32 N/A

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

      18. sub-neg N/A

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

   8. Applied rewrites 98.2%

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

   9. Taylor expanded in maxCos around 0

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

   11. Applied rewrites 97.7%

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

   12. Taylor expanded in maxCos around 0

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

       1. lower-*.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. lower-sin.f32 N/A

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

       4. lower-*.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. lower-PI.f32 N/A

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

       7. lower-sqrt.f32 N/A

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

       8. sub-neg N/A

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

       9. metadata-eval N/A

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

       10. lower-+.f32 N/A

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

       11. associate-*r/ N/A

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

       12. metadata-eval N/A

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

       13. lower-/.f32 93.9

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

   14. Applied rewrites 93.9%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0008299999753944576:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2 + maxCos \cdot -2}{ux} + maxCos \cdot \left(\left(1 - maxCos\right) \cdot \left(\frac{-1}{maxCos} - -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(ux \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0008299999753944576:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2 + maxCos \cdot -2}{ux} + maxCos \cdot \left(\left(1 - maxCos\right) \cdot \left(\frac{-1}{maxCos} - -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(ux \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0008299999753944576)
   (*
    (* 2.0 (* uy (PI)))
    (sqrt
     (*
      (* ux ux)
      (+
       (/ (+ 2.0 (* maxCos -2.0)) ux)
       (* maxCos (* (- 1.0 maxCos) (- (/ -1.0 maxCos) -1.0)))))))
   (* (sin (* (* uy 2.0) (PI))) (* ux (sqrt (+ -1.0 (/ 2.0 ux)))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 8.29999975e-4

   1. Initial program 58.6%

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

   3. Taylor expanded in maxCos around inf

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

      1. lower-*.f32 N/A

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

      2. associate--l+ N/A

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

      3. div-sub N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower--.f32 58.7

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in ux around -inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-+.f32 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f32 N/A

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

      18. sub-neg N/A

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

   8. Applied rewrites 97.9%

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

      1. lift-/.f32 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lift-/.f32 N/A

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

      6. div-inv N/A

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

      7. lift-/.f32 N/A

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

      8. associate-*l* N/A

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

      9. lift-/.f32 N/A

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

      10. inv-pow N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f32 N/A

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

      16. lower-*.f32 98.4

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

   10. Applied rewrites 98.4%

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

   11. Taylor expanded in uy around 0

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

       1. lower-*.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. lower-PI.f32 97.8

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

   13. Applied rewrites 97.8%

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

   if 8.29999975e-4 < (*.f32 uy #s(literal 2 binary32))

   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 maxCos around inf

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

      1. lower-*.f32 N/A

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

      2. associate--l+ N/A

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

      3. div-sub N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower--.f32 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in ux around -inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-+.f32 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f32 N/A

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

      18. sub-neg N/A

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

   8. Applied rewrites 98.2%

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

   9. Taylor expanded in maxCos around 0

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

       1. lower-*.f32 N/A

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

       2. lower-sqrt.f32 N/A

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

       3. sub-neg N/A

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

       4. metadata-eval N/A

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

       5. lower-+.f32 N/A

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

       6. associate-*r/ N/A

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

       7. metadata-eval N/A

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

       8. lower-/.f32 93.8

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

   11. Applied rewrites 93.8%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0008299999753944576:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2 + maxCos \cdot -2}{ux} + maxCos \cdot \left(\left(1 - maxCos\right) \cdot \left(\frac{-1}{maxCos} - -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(ux \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{if}\;uy \cdot 2 \leq 0.006000000052154064:\\ \;\;\;\;t\_0 \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2 + maxCos \cdot -2}{ux} + maxCos \cdot \left(\left(1 - maxCos\right) \cdot \left(\frac{-1}{maxCos} - -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{ux \cdot \left(2 - maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy (PI)))))
   (if (<= (* uy 2.0) 0.006000000052154064)
     (*
      t_0
      (sqrt
       (*
        (* ux ux)
        (+
         (/ (+ 2.0 (* maxCos -2.0)) ux)
         (* maxCos (* (- 1.0 maxCos) (- (/ -1.0 maxCos) -1.0)))))))
     (* (sin t_0) (sqrt (* ux (- 2.0 maxCos)))))))

Derivation

1. Split input into 2 regimes

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

   1. Initial program 59.5%

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

   3. Taylor expanded in maxCos around inf

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

      1. lower-*.f32 N/A

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

      2. associate--l+ N/A

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

      3. div-sub N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower--.f32 59.8

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

   5. Applied rewrites 59.8%

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

   6. Taylor expanded in ux around -inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-+.f32 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f32 N/A

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

      18. sub-neg N/A

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

   8. Applied rewrites 98.0%

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

      1. lift-/.f32 N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lift-/.f32 N/A

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

      6. div-inv N/A

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

      7. lift-/.f32 N/A

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

      8. associate-*l* N/A

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

      9. lift-/.f32 N/A

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

      10. inv-pow N/A

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

      11. pow-plus N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f32 N/A

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

      16. lower-*.f32 98.3

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

   10. Applied rewrites 98.3%

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

   11. Taylor expanded in uy around 0

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

       1. lower-*.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. lower-PI.f32 95.9

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

   13. Applied rewrites 95.9%

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

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

   1. Initial program 52.4%

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

   3. Taylor expanded in maxCos around 0

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

      1. lower--.f32 51.4

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f32 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f32 N/A

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

      9. lower-sin.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 78.9

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

   8. Applied rewrites 78.9%

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

4. Final simplification 91.5%

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

Alternative 6: 81.3% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.7%

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

3. Taylor expanded in maxCos around inf

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

   1. lower-*.f32 N/A

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

   2. associate--l+ N/A

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

   3. div-sub N/A

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

   4. lower-+.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower--.f32 57.9

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

5. Applied rewrites 57.9%

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

6. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-+.f32 N/A

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

   16. mul-1-neg N/A

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

   17. lower-neg.f32 N/A

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

   18. sub-neg N/A

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

8. Applied rewrites 98.0%

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

   1. lift-/.f32 N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. div-inv N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*l* N/A

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

   9. lift-/.f32 N/A

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

   10. inv-pow N/A

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

   11. pow-plus N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-+.f32 N/A

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

   16. lower-*.f32 98.2

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

10. Applied rewrites 98.2%

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

11. Taylor expanded in uy around 0

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

    1. lower-*.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower-PI.f32 83.2

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

13. Applied rewrites 83.2%

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

14. Final simplification 83.2%

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

Alternative 7: 80.2% accurate, 2.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.7%

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

3. Taylor expanded in maxCos around inf

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

   1. lower-*.f32 N/A

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

   2. associate--l+ N/A

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

   3. div-sub N/A

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

   4. lower-+.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower--.f32 57.9

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

5. Applied rewrites 57.9%

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

6. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-+.f32 N/A

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

   16. mul-1-neg N/A

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

   17. lower-neg.f32 N/A

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

   18. sub-neg N/A

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

8. Applied rewrites 98.0%

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

9. Taylor expanded in maxCos around 0

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

11. Applied rewrites 96.3%

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

12. Taylor expanded in uy around 0

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

    1. lower-*.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower-PI.f32 81.5

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

14. Applied rewrites 81.5%

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

15. Final simplification 81.5%

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

Alternative 8: 77.2% accurate, 3.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.7%

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

3. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. lower-fma.f32 N/A

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

5. Applied rewrites 31.2%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-+.f32 20.5

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 20.5

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

7. Applied rewrites 20.5%

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

8. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. lower--.f32 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-+.f32 20.4

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

10. Applied rewrites 20.4%

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

11. Taylor expanded in ux around inf

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

    1. lower-*.f32 N/A

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

    2. unpow2 N/A

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

    3. lower-*.f32 N/A

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

    4. sub-neg N/A

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

    5. metadata-eval N/A

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

    6. lower-+.f32 N/A

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

    7. associate-*r/ N/A

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

    8. metadata-eval N/A

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

    9. lower-/.f32 78.6

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

13. Applied rewrites 78.6%

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

14. Final simplification 78.6%

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

Alternative 9: 77.2% accurate, 4.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.7%

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

3. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. lower-fma.f32 N/A

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

5. Applied rewrites 31.2%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-+.f32 20.5

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 20.5

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

7. Applied rewrites 20.5%

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

8. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. lower--.f32 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-+.f32 20.4

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

10. Applied rewrites 20.4%

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

11. Taylor expanded in ux around 0

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

    1. lower-*.f32 N/A

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

    2. mul-1-neg N/A

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

    3. unsub-neg N/A

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

    4. lower--.f32 78.6

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

13. Applied rewrites 78.6%

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

Alternative 10: 63.6% accurate, 5.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.7%

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

3. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. lower-fma.f32 N/A

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

5. Applied rewrites 31.2%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lift-+.f32 20.5

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 20.5

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

7. Applied rewrites 20.5%

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

8. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. lower--.f32 N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-+.f32 20.4

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

10. Applied rewrites 20.4%

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

11. Taylor expanded in ux around 0

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

    1. lower-*.f32 64.1

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

13. Applied rewrites 64.1%

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

Reproduce

herbie shell --seed 2024219 
(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)))))))