UniformSampleCone, y

Percentage Accurate: 57.5% → 98.2%

Time: 17.6s

Alternatives: 16

Speedup: 4.0×

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: 57.5% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.1%

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

   1. lift-*.f32 N/A

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

   2. associate-+l- N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-+.f32 N/A

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

   8. lower--.f32 N/A

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

   9. metadata-eval N/A

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

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

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

   11. lower-+.f32 N/A

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

   12. neg-sub0 N/A

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

   13. associate-+l- N/A

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

   14. neg-sub0 N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 57.1%

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

5. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. distribute-lft-out-- N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. lower-+.f32 N/A

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

7. Applied rewrites 98.3%

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

   1. lift--.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. div-sub N/A

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

   4. lift--.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-+l- N/A

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

   8. lower--.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-/.f32 98.4

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

   12. lift-+.f32 N/A

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

   13. lift--.f32 N/A

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

   14. associate-+r- N/A

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

   15. metadata-eval N/A

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

   16. lower--.f32 98.4

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

9. Applied rewrites 98.4%

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

10. Final simplification 98.4%

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

Alternative 2: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.1%

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

   1. lift-*.f32 N/A

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

   2. associate-+l- N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-+.f32 N/A

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

   8. lower--.f32 N/A

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

   9. metadata-eval N/A

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

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

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

   11. lower-+.f32 N/A

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

   12. neg-sub0 N/A

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

   13. associate-+l- N/A

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

   14. neg-sub0 N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 57.1%

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

5. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. distribute-lft-out-- N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. lower-+.f32 N/A

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

7. Applied rewrites 98.3%

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

   1. lift--.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. distribute-neg-in N/A

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

   9. unsub-neg N/A

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

   10. lower--.f32 N/A

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

9. Applied rewrites 98.3%

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

10. Final simplification 98.3%

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

Alternative 3: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.1%

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

   1. lift-*.f32 N/A

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

   2. associate-+l- N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-+.f32 N/A

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

   8. lower--.f32 N/A

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

   9. metadata-eval N/A

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

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

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

   11. lower-+.f32 N/A

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

   12. neg-sub0 N/A

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

   13. associate-+l- N/A

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

   14. neg-sub0 N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 57.1%

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

5. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. distribute-lft-out-- N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. lower-+.f32 N/A

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in uy around inf

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

   1. associate-*l* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-sin.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 N/A

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

   8. lower-sqrt.f32 N/A

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

   9. associate--r+ N/A

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

   10. lower--.f32 N/A

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

10. Applied rewrites 97.9%

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

11. Final simplification 97.9%

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

Alternative 4: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0025599999353289604:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right) + \frac{\left(1 + \left(1 - maxCos\right)\right) - maxCos}{ux}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0025599999353289604)
   (*
    (* 2.0 (* uy (PI)))
    (sqrt
     (*
      (* ux ux)
      (+
       (* (- 1.0 maxCos) (+ maxCos -1.0))
       (/ (- (+ 1.0 (- 1.0 maxCos)) maxCos) ux)))))
   (* (sin (* (* uy 2.0) (PI))) (sqrt (* ux (+ 2.0 (* maxCos -2.0)))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00255999994

   1. Initial program 58.2%

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

      1. lift-*.f32 N/A

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

      2. associate-+l- N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. lower-/.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower-+.f32 N/A

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

      8. lower--.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower-+.f32 N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. lower-*.f32 N/A

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

   4. Applied rewrites 58.1%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 57.9

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

   7. Applied rewrites 57.9%

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

   8. Taylor expanded in ux around -inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. distribute-lft-out-- N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. lower--.f32 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f32 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f32 N/A

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

   10. Applied rewrites 97.1%

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

   if 0.00255999994 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 54.9%

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

      1. lift-*.f32 N/A

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

      2. associate-+l- N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. lower-/.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower-+.f32 N/A

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

      8. lower--.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower-+.f32 N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. lower-*.f32 N/A

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

   4. Applied rewrites 55.0%

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

   5. Taylor expanded in ux around -inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. distribute-lft-out-- N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f32 N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. lower-+.f32 N/A

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

   7. Applied rewrites 97.9%

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

   8. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

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

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

      3. lower-+.f32 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f32 76.6

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

   10. Applied rewrites 76.6%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0025599999353289604:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right) + \frac{\left(1 + \left(1 - maxCos\right)\right) - maxCos}{ux}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.1%

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

   1. lift-*.f32 N/A

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

   2. associate-+l- N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-+.f32 N/A

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

   8. lower--.f32 N/A

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

   9. metadata-eval N/A

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

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

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

   11. lower-+.f32 N/A

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

   12. neg-sub0 N/A

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

   13. associate-+l- N/A

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

   14. neg-sub0 N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 57.1%

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

5. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. distribute-lft-out-- N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. lower-+.f32 N/A

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in maxCos around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-+.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. lower-/.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 93.7

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

10. Applied rewrites 93.7%

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

11. Final simplification 93.7%

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

Alternative 6: 92.2% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.1%

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

   1. lift-*.f32 N/A

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

   2. associate-+l- N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-+.f32 N/A

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

   8. lower--.f32 N/A

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

   9. metadata-eval N/A

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

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

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

   11. lower-+.f32 N/A

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

   12. neg-sub0 N/A

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

   13. associate-+l- N/A

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

   14. neg-sub0 N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 57.1%

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

5. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. distribute-lft-out-- N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. lower-+.f32 N/A

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{\left(ux \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{2 \cdot \frac{1}{ux} - 1}} \]
9. 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 93.5

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

10. Applied rewrites 93.5%

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

Alternative 7: 89.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0025599999353289604:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right) + \frac{\left(1 + \left(1 - maxCos\right)\right) - maxCos}{ux}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0025599999353289604)
   (*
    (* 2.0 (* uy (PI)))
    (sqrt
     (*
      (* ux ux)
      (+
       (* (- 1.0 maxCos) (+ maxCos -1.0))
       (/ (- (+ 1.0 (- 1.0 maxCos)) maxCos) ux)))))
   (* (sin (* (* uy 2.0) (PI))) (sqrt (* 2.0 ux)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00255999994

   1. Initial program 58.2%

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

      1. lift-*.f32 N/A

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

      2. associate-+l- N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. lower-/.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower-+.f32 N/A

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

      8. lower--.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower-+.f32 N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. lower-*.f32 N/A

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

   4. Applied rewrites 58.1%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 57.9

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

   7. Applied rewrites 57.9%

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

   8. Taylor expanded in ux around -inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. distribute-lft-out-- N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. lower--.f32 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f32 N/A

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

      13. sub-neg N/A

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

      14. metadata-eval N/A

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

      15. lower-+.f32 N/A

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

   10. Applied rewrites 97.1%

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

   if 0.00255999994 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 54.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 56.5

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

   5. Applied rewrites 56.5%

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

   6. Taylor expanded in ux around 0

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower-neg.f32 N/A

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

      8. sub-neg N/A

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

      9. lower-+.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. distribute-neg-frac N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f32 73.1

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

   8. Applied rewrites 73.1%

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

   9. Taylor expanded in maxCos around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 73.5

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

   11. Applied rewrites 73.5%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0025599999353289604:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right) + \frac{\left(1 + \left(1 - maxCos\right)\right) - maxCos}{ux}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := 2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{if}\;t\_0 \leq 0.999809980392456:\\ \;\;\;\;t\_1 \cdot \sqrt{1 + t\_0 \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))) (t_1 (* 2.0 (* uy (PI)))))
   (if (<= t_0 0.999809980392456)
     (* t_1 (sqrt (+ 1.0 (* t_0 (- (+ ux -1.0) (* ux maxCos))))))
     (* t_1 (sqrt (* ux (+ 2.0 (* maxCos -2.0))))))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.99980998

   1. Initial program 88.8%

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

   3. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 72.6

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

   5. Applied rewrites 72.6%

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

   if 0.99980998 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 34.3%

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

      1. lift-*.f32 N/A

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

      2. associate-+l- N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. lower-/.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower-+.f32 N/A

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

      8. lower--.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower-+.f32 N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. lower-*.f32 N/A

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

   4. Applied rewrites 34.3%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 31.5

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

   7. Applied rewrites 31.5%

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

   8. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

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

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

      3. lower-+.f32 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f32 77.1

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

   10. Applied rewrites 77.1%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - ux\right) + ux \cdot maxCos \leq 0.999809980392456:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.2% accurate, 2.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.1%

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

   1. lift-*.f32 N/A

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

   2. associate-+l- N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-+.f32 N/A

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

   8. lower--.f32 N/A

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

   9. metadata-eval N/A

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

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

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

   11. lower-+.f32 N/A

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

   12. neg-sub0 N/A

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

   13. associate-+l- N/A

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

   14. neg-sub0 N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 57.1%

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

5. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 48.7

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

7. Applied rewrites 48.7%

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

8. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. distribute-lft-out-- N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower--.f32 N/A

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

   10. mul-1-neg N/A

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

   11. unsub-neg N/A

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

   12. lower--.f32 N/A

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

   13. sub-neg N/A

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

   14. metadata-eval N/A

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

   15. lower-+.f32 N/A

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

10. Applied rewrites 79.2%

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

11. Final simplification 79.2%

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

Alternative 10: 75.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{if}\;\left(1 - ux\right) + ux \cdot maxCos \leq 0.9997699856758118:\\ \;\;\;\;t\_0 \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy (PI)))))
   (if (<= (+ (- 1.0 ux) (* ux maxCos)) 0.9997699856758118)
     (* t_0 (sqrt (+ 1.0 (* (- 1.0 ux) (- (+ ux -1.0) (* ux maxCos))))))
     (* t_0 (sqrt (* ux (+ 2.0 (* maxCos -2.0))))))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.999769986

   1. Initial program 89.4%

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

   3. Taylor expanded in maxCos around 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 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 71.7

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

   8. Applied rewrites 71.7%

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

   9. Taylor expanded in maxCos around 0

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

       1. lower--.f32 70.0

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

   11. Applied rewrites 70.0%

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

   if 0.999769986 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 35.3%

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

      1. lift-*.f32 N/A

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

      2. associate-+l- N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. lower-/.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower-+.f32 N/A

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

      8. lower--.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower-+.f32 N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. lower-*.f32 N/A

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

   4. Applied rewrites 35.3%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 32.6

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

   7. Applied rewrites 32.6%

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

   8. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

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

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

      3. lower-+.f32 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f32 76.9

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

   10. Applied rewrites 76.9%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - ux\right) + ux \cdot maxCos \leq 0.9997699856758118:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(\left(ux + -1\right) - ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.1% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.1%

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

   1. lift-*.f32 N/A

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

   2. associate-+l- N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-+.f32 N/A

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

   8. lower--.f32 N/A

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

   9. metadata-eval N/A

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

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

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

   11. lower-+.f32 N/A

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

   12. neg-sub0 N/A

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

   13. associate-+l- N/A

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

   14. neg-sub0 N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 57.1%

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

5. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. distribute-lft-out-- N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f32 N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. lower-+.f32 N/A

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-sqrt.f32 N/A

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

   8. associate--r+ N/A

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

   9. lower--.f32 N/A

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

10. Applied rewrites 79.1%

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

11. Final simplification 79.1%

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

Alternative 12: 75.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{if}\;\left(1 - ux\right) + ux \cdot maxCos \leq 0.9997699856758118:\\ \;\;\;\;t\_0 \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* 2.0 (* uy (PI)))))
   (if (<= (+ (- 1.0 ux) (* ux maxCos)) 0.9997699856758118)
     (* t_0 (sqrt (+ 1.0 (* (- 1.0 ux) (+ ux -1.0)))))
     (* t_0 (sqrt (* ux (+ 2.0 (* maxCos -2.0))))))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.999769986

   1. Initial program 89.4%

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

   3. Taylor expanded in maxCos around 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 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 71.7

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

   8. Applied rewrites 71.7%

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

   9. Taylor expanded in maxCos around 0

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

       1. lower--.f32 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f32 N/A

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

       4. lower--.f32 N/A

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

       5. lower--.f32 69.7

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

   11. Applied rewrites 69.7%

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

   if 0.999769986 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 35.3%

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

      1. lift-*.f32 N/A

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

      2. associate-+l- N/A

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

      3. flip-- N/A

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

      4. clear-num N/A

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

      5. lower-/.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower-+.f32 N/A

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

      8. lower--.f32 N/A

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

      9. metadata-eval N/A

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

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

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

      11. lower-+.f32 N/A

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

      12. neg-sub0 N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. lower-*.f32 N/A

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

   4. Applied rewrites 35.3%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 32.6

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

   7. Applied rewrites 32.6%

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

   8. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

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

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

      3. lower-+.f32 N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f32 76.9

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

   10. Applied rewrites 76.9%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - ux\right) + ux \cdot maxCos \leq 0.9997699856758118:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux \cdot \left(2 + maxCos \cdot -2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.7% accurate, 4.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.1%

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

   1. lift-*.f32 N/A

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

   2. associate-+l- N/A

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

   3. flip-- N/A

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

   4. clear-num N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-+.f32 N/A

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

   8. lower--.f32 N/A

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

   9. metadata-eval N/A

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

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

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

   11. lower-+.f32 N/A

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

   12. neg-sub0 N/A

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

   13. associate-+l- N/A

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

   14. neg-sub0 N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 57.1%

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

5. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 48.7

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

7. Applied rewrites 48.7%

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

8. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

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

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

   3. lower-+.f32 N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f32 64.2

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

10. Applied rewrites 64.2%

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

11. Final simplification 64.2%

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

Alternative 14: 63.0% accurate, 4.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)} \]
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 30.4%

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

6. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. distribute-lft-in N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f32 62.5

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

8. Applied rewrites -0.0%

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

Alternative 15: 30.8% accurate, 6.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in ux around 0

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

5. Applied rewrites 26.2%

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

6. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

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

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

   7. *-lft-identity N/A

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

   8. mul-1-neg N/A

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

   9. distribute-rgt-in N/A

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

   10. mul-1-neg N/A

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

   11. sub-neg N/A

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

   12. sub-neg N/A

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

   13. mul-1-neg N/A

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

   14. lower-sqrt.f32 N/A

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

   15. +-commutative N/A

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

   16. distribute-lft-in N/A

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

   17. *-rgt-identity N/A

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

   18. lower-fma.f32 N/A

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

   19. mul-1-neg N/A

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

   20. lower-neg.f32 30.4

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

8. Applied rewrites -0.0%

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

9. Taylor expanded in maxCos around 0

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

    1. lower-sqrt.f32 30.5

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

11. Applied rewrites 30.5%

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

    1. lift-PI.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. lift-sqrt.f32 N/A

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

    5. *-commutative N/A

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

    6. lift-*.f32 N/A

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

    7. *-commutative N/A

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

    8. lift-*.f32 N/A

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

    9. associate-*l* N/A

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

    10. *-commutative N/A

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

    11. associate-*r* N/A

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

    12. lower-*.f32 N/A

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

    13. lower-*.f32 N/A

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

    14. lower-*.f32 30.5

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

13. Applied rewrites 30.5%

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

14. Final simplification 30.5%

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

Alternative 16: 30.8% accurate, 6.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in ux around 0

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

5. Applied rewrites 26.2%

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

6. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

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

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

   7. *-lft-identity N/A

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

   8. mul-1-neg N/A

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

   9. distribute-rgt-in N/A

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

   10. mul-1-neg N/A

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

   11. sub-neg N/A

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

   12. sub-neg N/A

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

   13. mul-1-neg N/A

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

   14. lower-sqrt.f32 N/A

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

   15. +-commutative N/A

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

   16. distribute-lft-in N/A

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

   17. *-rgt-identity N/A

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

   18. lower-fma.f32 N/A

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

   19. mul-1-neg N/A

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

   20. lower-neg.f32 30.4

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

8. Applied rewrites -0.0%

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

9. Taylor expanded in maxCos around 0

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

    1. lower-sqrt.f32 30.5

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

11. Applied rewrites 30.5%

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

Reproduce

herbie shell --seed 2024216 
(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)))))))