UniformSampleCone, x

Percentage Accurate: 56.7% → 99.1%

Time: 8.5s

Alternatives: 15

Speedup: 3.3×

Specification

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

Math

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

FPCore

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

Initial Program: 56.7% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 53.0%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

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

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

   6. associate--l- N/A

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

   7. +-commutative N/A

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

   8. lower--.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-pow.f32 N/A

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

   13. lower--.f32 N/A

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

   14. lower-*.f32 98.8

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

5. Applied rewrites 98.8%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. distribute-lft-neg-in N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. distribute-lft-neg-out N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lower-/.f32 99.1

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

7. Applied rewrites 99.1%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 99.1%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot -2\right)\right) \cdot \sqrt{\left(2 - \mathsf{fma}\left(\mathsf{fma}\left(ux, -2 + maxCos, 2\right), maxCos, ux\right)\right) \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (fma 0.5 (PI) (* (* (PI) uy) -2.0)))
  (sqrt (* (- 2.0 (fma (fma ux (+ -2.0 maxCos) 2.0) maxCos ux)) ux))))

Derivation

1. Initial program 53.0%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

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

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

   6. associate--l- N/A

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

   7. +-commutative N/A

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

   8. lower--.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-pow.f32 N/A

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

   13. lower--.f32 N/A

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

   14. lower-*.f32 98.8

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

5. Applied rewrites 98.8%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. distribute-lft-neg-in N/A

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

   7. lower-fma.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. distribute-lft-neg-out N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lower-/.f32 99.1

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

7. Applied rewrites 99.1%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 99.1%

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

11. Taylor expanded in uy around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f32 N/A

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

    3. lower-PI.f32 N/A

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

    4. *-commutative N/A

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

    5. lower-*.f32 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f32 N/A

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

    8. lower-PI.f32 98.8

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

13. Applied rewrites 98.8%

    \[\leadsto \sin \color{blue}{\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot -2\right)\right)} \cdot \sqrt{\left(2 - \mathsf{fma}\left(\mathsf{fma}\left(ux, -2 + maxCos, 2\right), maxCos, ux\right)\right) \cdot ux} \]
14. Add Preprocessing

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 53.0%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

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

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

   6. associate--l- N/A

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

   7. +-commutative N/A

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

   8. lower--.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-pow.f32 N/A

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

   13. lower--.f32 N/A

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

   14. lower-*.f32 98.8

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 98.8%

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

10. Applied rewrites 98.8%

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

Alternative 4: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 53.0%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

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

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

   6. associate--l- N/A

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

   7. +-commutative N/A

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

   8. lower--.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-pow.f32 N/A

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

   13. lower--.f32 N/A

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

   14. lower-*.f32 98.8

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 98.8%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 98.8

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

10. Applied rewrites 98.8%

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

Alternative 5: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 53.0%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

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

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

   6. associate--l- N/A

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

   7. +-commutative N/A

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

   8. lower--.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-pow.f32 N/A

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

   13. lower--.f32 N/A

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

   14. lower-*.f32 98.8

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 98.0%

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

Alternative 6: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.1000000199601345 \cdot 10^{-7}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\left(\left(2 - \left(\left(maxCos - 1\right) \cdot \left(-1 + maxCos\right)\right) \cdot ux\right) - 2 \cdot maxCos\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.1000000199601345e-7)
   (* (cos (* (* uy 2.0) (PI))) (sqrt (* (- 2.0 ux) ux)))
   (*
    (fma (* -2.0 (* uy uy)) (* (PI) (PI)) 1.0)
    (sqrt
     (*
      (- (- 2.0 (* (* (- maxCos 1.0) (+ -1.0 maxCos)) ux)) (* 2.0 maxCos))
      ux)))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.10000002e-7

   1. Initial program 52.5%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

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

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

      6. associate--l- N/A

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

      7. +-commutative N/A

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

      8. lower--.f32 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-pow.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower-*.f32 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 98.8%

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

   if 1.10000002e-7 < maxCos

   1. Initial program 54.9%

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

   3. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 50.7

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

   5. Applied rewrites 50.7%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. lift-+.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f32 N/A

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

      16. lower-*.f32 51.1

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

      17. lift-+.f32 N/A

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

      18. lift-*.f32 N/A

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

      19. *-commutative N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f32 51.1

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

   7. Applied rewrites 51.1%

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

   8. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   10. Applied rewrites 90.1%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 1.1000000199601345 \cdot 10^{-7}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\left(\left(2 - \left(\left(maxCos - 1\right) \cdot \left(-1 + maxCos\right)\right) \cdot ux\right) - 2 \cdot maxCos\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 53.0%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. mul-1-neg N/A

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

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

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

   6. associate--l- N/A

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

   7. +-commutative N/A

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

   8. lower--.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-pow.f32 N/A

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

   13. lower--.f32 N/A

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

   14. lower-*.f32 98.8

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 98.8%

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

9. Taylor expanded in ux around 0

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

11. Applied rewrites 97.3%

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

Alternative 8: 94.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.029999999329447746:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\left(\left(2 - \left(\left(maxCos - 1\right) \cdot \left(-1 + maxCos\right)\right) \cdot ux\right) - 2 \cdot maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.029999999329447746)
   (*
    (fma (* -2.0 (* uy uy)) (* (PI) (PI)) 1.0)
    (sqrt
     (*
      (- (- 2.0 (* (* (- maxCos 1.0) (+ -1.0 maxCos)) ux)) (* 2.0 maxCos))
      ux)))
   (* (cos (* (* uy 2.0) (PI))) (sqrt (* 2.0 ux)))))

Derivation

1. Split input into 2 regimes

2. if uy < 0.0299999993

   1. Initial program 53.8%

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

   3. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 53.4

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

   5. Applied rewrites 53.4%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. lift-+.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f32 N/A

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

      16. lower-*.f32 53.5

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

      17. lift-+.f32 N/A

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

      18. lift-*.f32 N/A

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

      19. *-commutative N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f32 53.5

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

   7. Applied rewrites 53.5%

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

   8. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   10. Applied rewrites 98.3%

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

   if 0.0299999993 < uy

   1. Initial program 47.8%

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

   3. Taylor expanded in ux around 0

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

      1. metadata-eval N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 82.8

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

   5. Applied rewrites 82.8%

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

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 74.6%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.029999999329447746:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\left(\left(2 - \left(\left(maxCos - 1\right) \cdot \left(-1 + maxCos\right)\right) \cdot ux\right) - 2 \cdot maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ t_1 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\sqrt{1 - t\_1 \cdot t\_1} \leq 0.037700001150369644:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{1 - t\_1 \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (fma (* -2.0 (* uy uy)) (* (PI) (PI)) 1.0))
        (t_1 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (sqrt (- 1.0 (* t_1 t_1))) 0.037700001150369644)
     (* t_0 (sqrt (* (fma -2.0 maxCos 2.0) ux)))
     (* t_0 (sqrt (- 1.0 (* t_1 (- 1.0 ux))))))))

Derivation

1. Split input into 2 regimes

2. if (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))) < 0.0377000012

   1. Initial program 38.9%

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

   3. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 36.6

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

   5. Applied rewrites 36.6%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. lift-+.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f32 N/A

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

      16. lower-*.f32 36.6

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

      17. lift-+.f32 N/A

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

      18. lift-*.f32 N/A

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

      19. *-commutative N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f32 36.6

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

   7. Applied rewrites 36.6%

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

   8. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} + \frac{-1}{2} \cdot \left(\sqrt{\frac{{ux}^{3}}{2 - 2 \cdot maxCos}} \cdot \left(-1 \cdot \left(maxCos - 1\right) + maxCos \cdot \left(maxCos - 1\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(\sqrt{\frac{{ux}^{3}}{2 - 2 \cdot maxCos}} \cdot \left(-1 \cdot \left(maxCos - 1\right) + maxCos \cdot \left(maxCos - 1\right)\right)\right) + \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \left(\color{blue}{\left(\sqrt{\frac{{ux}^{3}}{2 - 2 \cdot maxCos}} \cdot \left(-1 \cdot \left(maxCos - 1\right) + maxCos \cdot \left(maxCos - 1\right)\right)\right) \cdot \frac{-1}{2}} + \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{{ux}^{3}}{2 - 2 \cdot maxCos}} \cdot \left(-1 \cdot \left(maxCos - 1\right) + maxCos \cdot \left(maxCos - 1\right)\right), \frac{-1}{2}, \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)} \]

   10. Applied rewrites 90.5%

       \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\left(maxCos - 1\right) \cdot \left(-1 + maxCos\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right)} \]

   11. Taylor expanded in ux around 0

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

   13. Applied rewrites 84.4%

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

   if 0.0377000012 < (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))

   1. Initial program 89.9%

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

   3. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower--.f32 82.6

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

   8. Applied rewrites 82.6%

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

Alternative 10: 82.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := \mathsf{fma}\left(maxCos, ux, 1 - ux\right)\\ t_2 := \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9997000098228455:\\ \;\;\;\;t\_2 \cdot \sqrt{\mathsf{fma}\left(t\_1, -t\_1, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos)))
        (t_1 (fma maxCos ux (- 1.0 ux)))
        (t_2 (fma (* -2.0 (* uy uy)) (* (PI) (PI)) 1.0)))
   (if (<= (* t_0 t_0) 0.9997000098228455)
     (* t_2 (sqrt (fma t_1 (- t_1) 1.0)))
     (* t_2 (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))) < 0.99970001

   1. Initial program 88.0%

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

   3. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 83.8

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

   5. Applied rewrites 83.8%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. lift-+.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. +-commutative N/A

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

      11. lift-fma.f32 N/A

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

      12. lower-neg.f32 84.1

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

      13. lift-+.f32 N/A

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

      14. lift-*.f32 N/A

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

      15. *-commutative N/A

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

      16. +-commutative N/A

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

      17. lift-fma.f32 84.1

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

   7. Applied rewrites 84.1%

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

   if 0.99970001 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 36.6%

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

   3. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 34.3

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

   5. Applied rewrites 34.3%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. lift-+.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f32 N/A

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

      16. lower-*.f32 34.3

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

      17. lift-+.f32 N/A

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

      18. lift-*.f32 N/A

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

      19. *-commutative N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f32 34.3

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

   7. Applied rewrites 34.3%

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

   8. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} + \frac{-1}{2} \cdot \left(\sqrt{\frac{{ux}^{3}}{2 - 2 \cdot maxCos}} \cdot \left(-1 \cdot \left(maxCos - 1\right) + maxCos \cdot \left(maxCos - 1\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(\sqrt{\frac{{ux}^{3}}{2 - 2 \cdot maxCos}} \cdot \left(-1 \cdot \left(maxCos - 1\right) + maxCos \cdot \left(maxCos - 1\right)\right)\right) + \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \left(\color{blue}{\left(\sqrt{\frac{{ux}^{3}}{2 - 2 \cdot maxCos}} \cdot \left(-1 \cdot \left(maxCos - 1\right) + maxCos \cdot \left(maxCos - 1\right)\right)\right) \cdot \frac{-1}{2}} + \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{{ux}^{3}}{2 - 2 \cdot maxCos}} \cdot \left(-1 \cdot \left(maxCos - 1\right) + maxCos \cdot \left(maxCos - 1\right)\right), \frac{-1}{2}, \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)} \]

   10. Applied rewrites 90.3%

       \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\left(maxCos - 1\right) \cdot \left(-1 + maxCos\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right)} \]

   11. Taylor expanded in ux around 0

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

   13. Applied rewrites 85.4%

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

Alternative 11: 82.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ t_1 := \mathsf{fma}\left(maxCos, ux, 1\right) - ux\\ \mathbf{if}\;ux \leq 0.0001500000071246177:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{1 - t\_1 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (fma (* -2.0 (* uy uy)) (* (PI) (PI)) 1.0))
        (t_1 (- (fma maxCos ux 1.0) ux)))
   (if (<= ux 0.0001500000071246177)
     (* t_0 (sqrt (* (fma -2.0 maxCos 2.0) ux)))
     (* t_0 (sqrt (- 1.0 (* t_1 t_1)))))))

Derivation

1. Split input into 2 regimes

2. if ux < 1.50000007e-4

   1. Initial program 36.6%

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

   3. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 34.3

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

   5. Applied rewrites 34.3%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. lift-+.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f32 N/A

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

      16. lower-*.f32 34.3

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

      17. lift-+.f32 N/A

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

      18. lift-*.f32 N/A

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

      19. *-commutative N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f32 34.3

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

   7. Applied rewrites 34.3%

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

   8. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} + \frac{-1}{2} \cdot \left(\sqrt{\frac{{ux}^{3}}{2 - 2 \cdot maxCos}} \cdot \left(-1 \cdot \left(maxCos - 1\right) + maxCos \cdot \left(maxCos - 1\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(\sqrt{\frac{{ux}^{3}}{2 - 2 \cdot maxCos}} \cdot \left(-1 \cdot \left(maxCos - 1\right) + maxCos \cdot \left(maxCos - 1\right)\right)\right) + \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \left(\color{blue}{\left(\sqrt{\frac{{ux}^{3}}{2 - 2 \cdot maxCos}} \cdot \left(-1 \cdot \left(maxCos - 1\right) + maxCos \cdot \left(maxCos - 1\right)\right)\right) \cdot \frac{-1}{2}} + \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{{ux}^{3}}{2 - 2 \cdot maxCos}} \cdot \left(-1 \cdot \left(maxCos - 1\right) + maxCos \cdot \left(maxCos - 1\right)\right), \frac{-1}{2}, \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)} \]

   10. Applied rewrites 90.3%

       \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\left(maxCos - 1\right) \cdot \left(-1 + maxCos\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right)} \]

   11. Taylor expanded in ux around 0

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

   13. Applied rewrites 85.4%

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

   if 1.50000007e-4 < ux

   1. Initial program 88.0%

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

   3. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 83.8

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

   5. Applied rewrites 83.8%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. lift-+.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f32 N/A

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

      16. lower-*.f32 84.0

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

      17. lift-+.f32 N/A

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

      18. lift-*.f32 N/A

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

      19. *-commutative N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f32 84.0

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

   7. Applied rewrites 84.0%

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

      1. lift-fma.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. +-commutative N/A

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

      4. lift--.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate-*l* N/A

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

      9. lift-fma.f32 N/A

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

      10. +-commutative N/A

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

      11. lift--.f32 N/A

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

      12. *-commutative N/A

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

      13. *-commutative N/A

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

      14. distribute-lft-in N/A

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

      15. lift--.f32 N/A

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

      16. lower-*.f32 N/A

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

   9. Applied rewrites 83.8%

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

Alternative 12: 88.7% accurate, 2.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 53.0%

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 50.2

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

5. Applied rewrites 50.2%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lift-+.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. lift-fma.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

   15. lower-*.f32 N/A

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

   16. lower-*.f32 50.3

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

   17. lift-+.f32 N/A

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

   18. lift-*.f32 N/A

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

   19. *-commutative N/A

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

   20. +-commutative N/A

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

   21. lift-fma.f32 50.3

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

7. Applied rewrites 50.3%

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

8. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

10. Applied rewrites 91.2%

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

11. Final simplification 91.2%

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

Alternative 13: 80.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(maxCos, ux, 1\right) - ux\\ \mathbf{if}\;ux \leq 0.0002089000045089051:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \mathsf{fma}\left(t\_0 \cdot ux, maxCos, t\_0 \cdot \left(1 - ux\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- (fma maxCos ux 1.0) ux)))
   (if (<= ux 0.0002089000045089051)
     (*
      (fma (* -2.0 (* uy uy)) (* (PI) (PI)) 1.0)
      (sqrt (* (fma -2.0 maxCos 2.0) ux)))
     (sqrt (- 1.0 (fma (* t_0 ux) maxCos (* t_0 (- 1.0 ux))))))))

Derivation

1. Split input into 2 regimes

2. if ux < 2.08900005e-4

   1. Initial program 37.0%

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

   3. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 34.8

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

   5. Applied rewrites 34.8%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. lift-+.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f32 N/A

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

      16. lower-*.f32 34.8

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

      17. lift-+.f32 N/A

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

      18. lift-*.f32 N/A

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

      19. *-commutative N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f32 34.8

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

   7. Applied rewrites 34.8%

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

   8. Taylor expanded in ux around 0

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} + \frac{-1}{2} \cdot \left(\sqrt{\frac{{ux}^{3}}{2 - 2 \cdot maxCos}} \cdot \left(-1 \cdot \left(maxCos - 1\right) + maxCos \cdot \left(maxCos - 1\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(\sqrt{\frac{{ux}^{3}}{2 - 2 \cdot maxCos}} \cdot \left(-1 \cdot \left(maxCos - 1\right) + maxCos \cdot \left(maxCos - 1\right)\right)\right) + \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \left(\color{blue}{\left(\sqrt{\frac{{ux}^{3}}{2 - 2 \cdot maxCos}} \cdot \left(-1 \cdot \left(maxCos - 1\right) + maxCos \cdot \left(maxCos - 1\right)\right)\right) \cdot \frac{-1}{2}} + \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right) \]

      3. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{{ux}^{3}}{2 - 2 \cdot maxCos}} \cdot \left(-1 \cdot \left(maxCos - 1\right) + maxCos \cdot \left(maxCos - 1\right)\right), \frac{-1}{2}, \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)} \]

   10. Applied rewrites 90.3%

       \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\left(maxCos - 1\right) \cdot \left(-1 + maxCos\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right)} \]

   11. Taylor expanded in ux around 0

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

   13. Applied rewrites 85.3%

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

   if 2.08900005e-4 < ux

   1. Initial program 88.4%

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

   3. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-PI.f32 84.1

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

   5. Applied rewrites 84.1%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. distribute-rgt-in N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. lift-+.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lift-fma.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. associate-*r* N/A

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

      15. lower-*.f32 N/A

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

      16. lower-*.f32 84.3

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

      17. lift-+.f32 N/A

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

      18. lift-*.f32 N/A

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

      19. *-commutative N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f32 84.3

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

   7. Applied rewrites 84.3%

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

   8. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower--.f32 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f32 N/A

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

      15. lower--.f32 78.8

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

   10. Applied rewrites 78.8%

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

Alternative 14: 70.7% accurate, 3.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 53.0%

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 50.2

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

5. Applied rewrites 50.2%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lift-+.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. +-commutative N/A

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

   10. lift-fma.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. associate-*r* N/A

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

   15. lower-*.f32 N/A

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

   16. lower-*.f32 50.3

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

   17. lift-+.f32 N/A

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

   18. lift-*.f32 N/A

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

   19. *-commutative N/A

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

   20. +-commutative N/A

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

   21. lift-fma.f32 50.3

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

7. Applied rewrites 50.3%

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

8. Taylor expanded in ux around 0

   \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} + \frac{-1}{2} \cdot \left(\sqrt{\frac{{ux}^{3}}{2 - 2 \cdot maxCos}} \cdot \left(-1 \cdot \left(maxCos - 1\right) + maxCos \cdot \left(maxCos - 1\right)\right)\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(\sqrt{\frac{{ux}^{3}}{2 - 2 \cdot maxCos}} \cdot \left(-1 \cdot \left(maxCos - 1\right) + maxCos \cdot \left(maxCos - 1\right)\right)\right) + \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \left(\color{blue}{\left(\sqrt{\frac{{ux}^{3}}{2 - 2 \cdot maxCos}} \cdot \left(-1 \cdot \left(maxCos - 1\right) + maxCos \cdot \left(maxCos - 1\right)\right)\right) \cdot \frac{-1}{2}} + \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(\sqrt{\frac{{ux}^{3}}{2 - 2 \cdot maxCos}} \cdot \left(-1 \cdot \left(maxCos - 1\right) + maxCos \cdot \left(maxCos - 1\right)\right), \frac{-1}{2}, \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)} \]

10. Applied rewrites 85.7%

    \[\leadsto \mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\mathsf{fma}\left(\left(\left(maxCos - 1\right) \cdot \left(-1 + maxCos\right)\right) \cdot \sqrt{\frac{{ux}^{3}}{\mathsf{fma}\left(-2, maxCos, 2\right)}}, -0.5, \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\right)} \]

11. Taylor expanded in ux around 0

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

13. Applied rewrites 74.7%

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

Alternative 15: 6.6% accurate, 4.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 53.0%

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-PI.f32 50.2

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

5. Applied rewrites 50.2%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 6.6%

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

9. Taylor expanded in uy around inf

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

11. Applied rewrites 6.6%

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

Reproduce

herbie shell --seed 2025017 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (* (cos (* (* uy 2.0) (PI))) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))