UniformSampleCone, x

Percentage Accurate: 57.4% → 99.0%

Time: 10.5s

Alternatives: 6

Speedup: 5.0×

Specification

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

Math

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

FPCore

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

Initial Program: 57.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.0% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 56.8%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

   4. metadata-eval N/A

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

   5. div-sub N/A

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

   6. associate-+l- N/A

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

   7. lower--.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f32 56.8

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f32 56.8

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

4. Applied rewrites 56.8%

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

5. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. associate--l+ N/A

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. associate-+r+ N/A

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

   8. mul-1-neg N/A

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

   9. unsub-neg N/A

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

   10. lower--.f32 N/A

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

   11. +-commutative N/A

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

   12. lower-fma.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lower-pow.f32 N/A

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

   16. lower--.f32 69.4

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

7. Applied rewrites 69.4%

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

9. Applied rewrites 99.0%

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

10. Final simplification 99.0%

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

Alternative 2: 91.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 56.8%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

   4. metadata-eval N/A

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

   5. div-sub N/A

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

   6. associate-+l- N/A

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

   7. lower--.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f32 56.8

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f32 56.8

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

4. Applied rewrites 56.8%

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

5. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. associate--l+ N/A

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. associate-+r+ N/A

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

   8. mul-1-neg N/A

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

   9. unsub-neg N/A

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

   10. lower--.f32 N/A

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

   11. +-commutative N/A

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

   12. lower-fma.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lower-pow.f32 N/A

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

   16. lower--.f32 69.4

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

7. Applied rewrites 69.4%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 91.7%

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

11. Final simplification 92.7%

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

Alternative 3: 92.9% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 56.8%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. flip-- N/A

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

   4. metadata-eval N/A

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

   5. div-sub N/A

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

   6. associate-+l- N/A

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

   7. lower--.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f32 56.8

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f32 56.8

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

4. Applied rewrites 56.8%

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

5. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. associate--l+ N/A

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

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

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. associate-+r+ N/A

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

   8. mul-1-neg N/A

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

   9. unsub-neg N/A

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

   10. lower--.f32 N/A

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

   11. +-commutative N/A

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

   12. lower-fma.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lower-pow.f32 N/A

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

   16. lower--.f32 69.4

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

7. Applied rewrites 69.4%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 92.7%

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

11. Final simplification 92.7%

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

Alternative 4: 62.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{4 \cdot ux} \cdot \sqrt{0.5} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (* (sqrt (* 4.0 ux)) (sqrt 0.5)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((4.0f * ux)) * sqrtf(0.5f);
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((4.0e0 * ux)) * sqrt(0.5e0)
end function

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32(4.0) * ux)) * sqrt(Float32(0.5)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((single(4.0) * ux)) * sqrt(single(0.5));
end

Derivation

1. Initial program 56.8%

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

   1. lift-*.f32 N/A

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

   2. lift-sqrt.f32 N/A

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

   3. lift--.f32 N/A

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

   4. flip-- N/A

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

   5. sqrt-div N/A

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

   6. associate-*r/ N/A

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

4. Applied rewrites 41.9%

   \[\leadsto \color{blue}{\frac{\cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right)}^{4}}}{\sqrt{{\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right)}^{2} + 1}}} \]

5. Taylor expanded in ux around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)}} \]

   2. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)}} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

   4. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

   5. lower-sqrt.f32 N/A

      \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{2}}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

   6. lower-cos.f32 N/A

      \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

   7. *-commutative N/A

      \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

   8. lower-*.f32 N/A

      \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

   9. *-commutative N/A

      \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)} \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

   10. lower-*.f32 N/A

       \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)} \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

   11. lower-PI.f32 N/A

       \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

   12. lower-sqrt.f32 N/A

       \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \color{blue}{\sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)}} \]

   13. *-commutative N/A

       \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(4 - 4 \cdot maxCos\right) \cdot ux}} \]

   14. lower-*.f32 N/A

       \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(4 - 4 \cdot maxCos\right) \cdot ux}} \]

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

       \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(4 + \left(\mathsf{neg}\left(4\right)\right) \cdot maxCos\right)} \cdot ux} \]

   16. metadata-eval N/A

       \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(4 + \color{blue}{-4} \cdot maxCos\right) \cdot ux} \]

   17. +-commutative N/A

       \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(-4 \cdot maxCos + 4\right)} \cdot ux} \]

   18. lower-fma.f32 56.7

       \[\leadsto \left(\sqrt{0.5} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-4, maxCos, 4\right)} \cdot ux} \]

7. Applied rewrites 57.2%

   \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\mathsf{fma}\left(-4, maxCos, 4\right) \cdot ux}} \]

8. Taylor expanded in maxCos around 0

   \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{4 \cdot ux} \]
9. Step-by-step derivation

10. Applied rewrites 74.1%

    \[\leadsto \left(\sqrt{0.5} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{4 \cdot ux} \]

11. Taylor expanded in uy around 0

    \[\leadsto \sqrt{\frac{1}{2}} \cdot \sqrt{\color{blue}{4 \cdot ux}} \]
12. Step-by-step derivation

13. Applied rewrites 63.8%

    \[\leadsto \sqrt{0.5} \cdot \sqrt{\color{blue}{4 \cdot ux}} \]

14. Final simplification 63.8%

    \[\leadsto \sqrt{4 \cdot ux} \cdot \sqrt{0.5} \]
15. Add Preprocessing

Alternative 5: 20.0% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{1 - \left(ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)} \end{array} \]

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((1.0f - ((ux * ux) * (maxCos * maxCos))));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((1.0e0 - ((ux * ux) * (maxcos * maxcos))))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(Float32(1.0) - Float32(Float32(ux * ux) * Float32(maxCos * maxCos))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((single(1.0) - ((ux * ux) * (maxCos * maxCos))));
end

Derivation

1. Initial program 56.8%

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

3. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. lower--.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 49.3

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

5. Applied rewrites 49.2%

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

6. Taylor expanded in maxCos around inf

   \[\leadsto \sqrt{1 - {maxCos}^{2} \cdot {ux}^{2}} \]
7. Step-by-step derivation

8. Applied rewrites 19.9%

   \[\leadsto \sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(ux \cdot ux\right)} \]

9. Final simplification 19.9%

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

Alternative 6: -0.0% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{-1} \cdot ux\right) \cdot maxCos \end{array} \]

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return (sqrtf(-1.0f) * ux) * maxCos;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = (sqrt((-1.0e0)) * ux) * maxcos
end function

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(sqrt(Float32(-1.0)) * ux) * maxCos)
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = (sqrt(single(-1.0)) * ux) * maxCos;
end

Derivation

1. Initial program 56.8%

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

3. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. lower--.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 49.3

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

5. Applied rewrites 49.2%

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

6. Taylor expanded in maxCos around inf

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

8. Applied rewrites -0.0%

   \[\leadsto \left(\sqrt{-1} \cdot ux\right) \cdot \color{blue}{maxCos} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024284 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :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)))
  (* (cos (* (* uy 2.0) (PI))) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))