UniformSampleCone, y

Percentage Accurate: 56.8% → 98.3%

Time: 8.0s

Alternatives: 11

Speedup: 4.2×

Specification

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

Math

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

FPCore

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

Initial Program: 56.8% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

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

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

   6. associate--l- N/A

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

   7. +-commutative N/A

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

   8. lower--.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-pow.f32 N/A

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

   13. lower--.f32 N/A

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

   14. lower-*.f32 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 98.4%

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

10. Applied rewrites 98.5%

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

    1. lift-*.f32 N/A

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

    2. *-commutative N/A

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

    3. count-2-rev N/A

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

    4. lower-+.f32 98.5

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

12. Applied rewrites 98.5%

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

Alternative 2: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

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

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

   6. associate--l- N/A

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

   7. +-commutative N/A

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

   8. lower--.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-pow.f32 N/A

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

   13. lower--.f32 N/A

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

   14. lower-*.f32 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 98.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 98.4

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

10. Applied rewrites 98.4%

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

Alternative 3: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

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

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

   6. associate--l- N/A

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

   7. +-commutative N/A

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

   8. lower--.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-pow.f32 N/A

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

   13. lower--.f32 N/A

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

   14. lower-*.f32 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 97.7%

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

Alternative 4: 95.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.500000053056283 \cdot 10^{-7}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(\left(\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - 1\right) - \left(\mathsf{fma}\left(maxCos, maxCos, -maxCos\right) - maxCos\right)\right) \cdot ux\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.500000053056283e-7)
   (* (sin (* (* uy 2.0) (PI))) (sqrt (* (- 2.0 ux) ux)))
   (*
    (* (* 2.0 uy) (PI))
    (sqrt
     (*
      (*
       (-
        (- (/ (fma -2.0 maxCos 2.0) ux) 1.0)
        (- (fma maxCos maxCos (- maxCos)) maxCos))
       ux)
      ux)))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.5000001e-7

   1. Initial program 52.8%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

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

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

      6. associate--l- N/A

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

      7. +-commutative N/A

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

      8. lower--.f32 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-pow.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower-*.f32 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 98.5%

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

   if 1.5000001e-7 < maxCos

   1. Initial program 55.5%

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

   3. Taylor expanded in 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. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 49.5

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

   5. Applied rewrites 49.5%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate--r+ N/A

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

      6. lower--.f32 N/A

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

      7. lower--.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lift-+.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

   7. Applied rewrites 48.8%

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

   8. Taylor expanded in ux around -inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. associate--r+ N/A

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

      5. lower--.f32 N/A

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

      6. +-commutative N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. div-add N/A

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

      11. lower--.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f32 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   10. Applied rewrites 83.0%

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

   12. Applied rewrites 83.1%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 1.500000053056283 \cdot 10^{-7}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(\left(\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - 1\right) - \left(\mathsf{fma}\left(maxCos, maxCos, -maxCos\right) - maxCos\right)\right) \cdot ux\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.9% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

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

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

   6. associate--l- N/A

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

   7. +-commutative N/A

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

   8. lower--.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-pow.f32 N/A

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

   13. lower--.f32 N/A

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

   14. lower-*.f32 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 98.4%

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

9. Taylor expanded in ux around 0

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

11. Applied rewrites 97.0%

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

Alternative 6: 76.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := \left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9985799789428711:\\ \;\;\;\;t\_1 \cdot \sqrt{1 - t\_0 \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))) (t_1 (* (* 2.0 uy) (PI))))
   (if (<= (* t_0 t_0) 0.9985799789428711)
     (* t_1 (sqrt (- 1.0 (* t_0 (- 1.0 ux)))))
     (* t_1 (sqrt (* (fma -2.0 maxCos 2.0) ux))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.99857998

   1. Initial program 90.3%

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

   3. 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. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 81.3

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

   5. Applied rewrites 81.3%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower--.f32 78.2

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

   8. Applied rewrites 78.2%

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

   if 0.99857998 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 39.1%

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

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 36.6

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

   5. Applied rewrites 36.6%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate--r+ N/A

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

      6. lower--.f32 N/A

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

      7. lower--.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lift-+.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

   7. Applied rewrites 34.3%

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

   8. Taylor expanded in ux around -inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. associate--r+ N/A

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

      5. lower--.f32 N/A

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

      6. +-commutative N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. div-add N/A

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

      11. lower--.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f32 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

   10. Applied rewrites 82.8%

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

   11. Taylor expanded in ux around 0

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

   13. Applied rewrites 78.3%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) \leq 0.9985799789428711:\\ \;\;\;\;\left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.0% accurate, 3.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

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

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

   6. associate--l- N/A

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

   7. +-commutative N/A

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

   8. lower--.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-pow.f32 N/A

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

   13. lower--.f32 N/A

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

   14. lower-*.f32 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 98.4%

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

10. Applied rewrites 98.5%

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

11. Taylor expanded in uy around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f32 N/A

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

    5. lower-PI.f32 84.0

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

13. Applied rewrites 84.0%

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

Alternative 8: 82.0% accurate, 3.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 53.3%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

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

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

   6. associate--l- N/A

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

   7. +-commutative N/A

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

   8. lower--.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-pow.f32 N/A

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

   13. lower--.f32 N/A

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

   14. lower-*.f32 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 98.4%

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

9. Taylor expanded in uy around 0

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

    1. lower-*.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower-PI.f32 84.0

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

11. Applied rewrites 84.0%

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

Alternative 9: 66.8% accurate, 4.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 53.3%

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

3. 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. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 49.0

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

5. Applied rewrites 49.0%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. distribute-rgt-in N/A

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

   5. associate--r+ N/A

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

   6. lower--.f32 N/A

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

   7. lower--.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-+.f32 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

7. Applied rewrites 47.4%

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

8. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. associate--r+ N/A

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

   5. lower--.f32 N/A

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

   6. +-commutative N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. div-add N/A

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

   11. lower--.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. +-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. mul-1-neg N/A

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

   16. +-commutative N/A

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

10. Applied rewrites 83.8%

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

11. Taylor expanded in ux around 0

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

13. Applied rewrites 70.1%

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

14. Final simplification 70.1%

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

Alternative 10: 41.0% accurate, 4.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 53.3%

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

3. 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. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 49.0

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

5. Applied rewrites 49.0%

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

6. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. metadata-eval N/A

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

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

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f32 41.7

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

8. Applied rewrites 41.7%

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

9. Taylor expanded in maxCos around 0

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

11. Applied rewrites 40.9%

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

Alternative 11: 7.1% accurate, 5.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 53.3%

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

3. 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. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 49.0

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

5. Applied rewrites 49.0%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 7.2%

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

Reproduce

herbie shell --seed 2025017 
(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)))))))