UniformSampleCone, y

Percentage Accurate: 57.4% → 98.3%

Time: 6.2s

Alternatives: 14

Speedup: 5.0×

Specification

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

Math

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

FPCore

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

Initial Program: 57.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.4

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

5. Applied rewrites 98.4%

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

   1. lift--.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift--.f32 98.4

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

7. Applied rewrites 98.4%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 97.8

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

8. Applied rewrites 97.8%

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

Alternative 3: 96.9% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 97.1%

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

Alternative 4: 94.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.02500000037252903:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.02500000037252903)
   (*
    (*
     uy
     (fma
      -1.3333333333333333
      (* (* uy uy) (* (* (PI) (PI)) (PI)))
      (* 2.0 (PI))))
    (sqrt
     (*
      (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (* maxCos 2.0))
      ux)))
   (* (sin (* (* uy 2.0) (PI))) (sqrt (* (fma -2.0 maxCos 2.0) ux)))))

Derivation

1. Split input into 2 regimes

2. if uy < 0.0250000004

   1. Initial program 59.9%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 98.7

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

   5. Applied rewrites 98.7%

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

      1. lift--.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift--.f32 N/A

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

      6. lift--.f32 98.7

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

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \color{blue}{\left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

      2. lower-fma.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \color{blue}{{uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

      3. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, {uy}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{3}}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

      4. pow2 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

      5. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

      6. lower-pow.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{3}}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

      7. lift-PI.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

      8. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

      9. lift-PI.f32 98.4

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

   10. Applied rewrites 98.4%

       \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]
   11. Step-by-step derivation

       1. lift-PI.f32 N/A

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

       2. lift-pow.f32 N/A

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{3}}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

       3. unpow3 N/A

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

       4. unpow2 N/A

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

       5. lower-*.f32 N/A

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

       6. unpow2 N/A

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

       7. lower-*.f32 N/A

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

       8. lift-PI.f32 N/A

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

       9. lift-PI.f32 N/A

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

       10. lift-PI.f32 98.4

           \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

   12. Applied rewrites 98.4%

       \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

   if 0.0250000004 < uy

   1. Initial program 57.4%

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

   3. Taylor expanded in ux around 0

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

      1. metadata-eval N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 76.4

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

   5. Applied rewrites 76.4%

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

Alternative 5: 97.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.00800000037997961:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00800000037997961)
   (*
    (*
     uy
     (fma
      -1.3333333333333333
      (* (* uy uy) (* (* (PI) (PI)) (PI)))
      (* 2.0 (PI))))
    (sqrt
     (*
      (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (* maxCos 2.0))
      ux)))
   (* (sin (* (* uy 2.0) (PI))) (sqrt (* (- 2.0 ux) ux)))))

Derivation

1. Split input into 2 regimes

2. if uy < 0.00800000038

   1. Initial program 59.4%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 98.7

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

   5. Applied rewrites 98.7%

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

      1. lift--.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f32 N/A

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

      5. lift--.f32 N/A

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

      6. lift--.f32 98.7

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

   7. Applied rewrites 98.7%

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

   8. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \color{blue}{\left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

      2. lower-fma.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \color{blue}{{uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

      3. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, {uy}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{3}}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

      4. pow2 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

      5. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

      6. lower-pow.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{3}}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

      7. lift-PI.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

      8. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

      9. lift-PI.f32 98.7

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

   10. Applied rewrites 98.7%

       \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]
   11. Step-by-step derivation

       1. lift-PI.f32 N/A

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

       2. lift-pow.f32 N/A

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{3}}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

       3. unpow3 N/A

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

       4. unpow2 N/A

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

       5. lower-*.f32 N/A

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

       6. unpow2 N/A

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

       7. lower-*.f32 N/A

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

       8. lift-PI.f32 N/A

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

       9. lift-PI.f32 N/A

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

       10. lift-PI.f32 98.7

           \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

   12. Applied rewrites 98.7%

       \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

   if 0.00800000038 < uy

   1. Initial program 59.5%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 97.4

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

   5. Applied rewrites 97.4%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 89.0

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

   8. Applied rewrites 89.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.00800000037997961:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ \mathbf{if}\;t\_1 \leq 0.020500000566244125:\\ \;\;\;\;\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{maxCos \cdot \mathsf{fma}\left(-2, ux, 2 \cdot \frac{ux}{maxCos}\right)}\right)\right) \cdot uy\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))) (t_1 (sqrt (- 1.0 (* t_0 t_0)))))
   (if (<= t_1 0.020500000566244125)
     (*
      (* 2.0 (* (PI) (sqrt (* maxCos (fma -2.0 ux (* 2.0 (/ ux maxCos)))))))
      uy)
     (* (* (PI) (* 2.0 uy)) t_1))))

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))))) < 0.0205000006

   1. Initial program 38.4%

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

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{uy \cdot \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right) \cdot \color{blue}{uy} \]

      2. lower-*.f32 N/A

         \[\leadsto \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right) \cdot \color{blue}{uy} \]

   5. Applied rewrites 36.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{3} \cdot \left(uy \cdot uy\right)\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}^{2}}, -1.3333333333333333, \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}^{2}}\right) \cdot uy} \]

   6. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lower-sqrt.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-+.f32 N/A

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

      9. lower-*.f32 35.8

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

   8. Applied rewrites 35.8%

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

   9. Taylor expanded in ux around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 77.0

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

   11. Applied rewrites 77.0%

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

   12. Taylor expanded in maxCos around inf

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

       1. lower-*.f32 N/A

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

       2. lower-fma.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower-/.f32 77.1

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

   14. Applied rewrites 77.1%

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

   if 0.0205000006 < (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 90.1%

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

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 74.3

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

   5. Applied rewrites 74.3%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \leq 0.020500000566244125:\\ \;\;\;\;\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{maxCos \cdot \mathsf{fma}\left(-2, ux, 2 \cdot \frac{ux}{maxCos}\right)}\right)\right) \cdot uy\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \leq 0.019999999552965164:\\ \;\;\;\;\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{maxCos \cdot \mathsf{fma}\left(-2, ux, 2 \cdot \frac{ux}{maxCos}\right)}\right)\right) \cdot uy\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - t\_0 \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (sqrt (- 1.0 (* t_0 t_0))) 0.019999999552965164)
     (*
      (* 2.0 (* (PI) (sqrt (* maxCos (fma -2.0 ux (* 2.0 (/ ux maxCos)))))))
      uy)
     (*
      (* 2.0 (* uy (PI)))
      (sqrt (- 1.0 (* t_0 (- (fma maxCos ux 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))))) < 0.0199999996

   1. Initial program 38.2%

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

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{uy \cdot \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right) \cdot \color{blue}{uy} \]

      2. lower-*.f32 N/A

         \[\leadsto \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right) \cdot \color{blue}{uy} \]

   5. Applied rewrites 36.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{3} \cdot \left(uy \cdot uy\right)\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}^{2}}, -1.3333333333333333, \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}^{2}}\right) \cdot uy} \]

   6. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lower-sqrt.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-+.f32 N/A

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

      9. lower-*.f32 35.6

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

   8. Applied rewrites 35.6%

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

   9. Taylor expanded in ux around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 77.0

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

   11. Applied rewrites 77.0%

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

   12. Taylor expanded in maxCos around inf

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

       1. lower-*.f32 N/A

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

       2. lower-fma.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower-/.f32 77.1

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

   14. Applied rewrites 77.1%

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

   if 0.0199999996 < (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%

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

   3. Taylor expanded in maxCos around 0

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

      1. lower--.f32 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f32 89.9

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lift-PI.f32 74.1

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

   8. Applied rewrites 74.1%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \leq 0.019999999552965164:\\ \;\;\;\;\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{maxCos \cdot \mathsf{fma}\left(-2, ux, 2 \cdot \frac{ux}{maxCos}\right)}\right)\right) \cdot uy\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \leq 0.020500000566244125:\\ \;\;\;\;\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{maxCos \cdot \mathsf{fma}\left(-2, ux, 2 \cdot \frac{ux}{maxCos}\right)}\right)\right) \cdot uy\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - t\_0 \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (sqrt (- 1.0 (* t_0 t_0))) 0.020500000566244125)
     (*
      (* 2.0 (* (PI) (sqrt (* maxCos (fma -2.0 ux (* 2.0 (/ ux maxCos)))))))
      uy)
     (* (* (PI) (* 2.0 uy)) (sqrt (- 1.0 (* t_0 (- 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))))) < 0.0205000006

   1. Initial program 38.4%

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

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{uy \cdot \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right) \cdot \color{blue}{uy} \]

      2. lower-*.f32 N/A

         \[\leadsto \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right) \cdot \color{blue}{uy} \]

   5. Applied rewrites 36.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{3} \cdot \left(uy \cdot uy\right)\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}^{2}}, -1.3333333333333333, \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}^{2}}\right) \cdot uy} \]

   6. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lower-sqrt.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-+.f32 N/A

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

      9. lower-*.f32 35.8

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

   8. Applied rewrites 35.8%

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

   9. Taylor expanded in ux around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 77.0

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

   11. Applied rewrites 77.0%

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

   12. Taylor expanded in maxCos around inf

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

       1. lower-*.f32 N/A

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

       2. lower-fma.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower-/.f32 77.1

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

   14. Applied rewrites 77.1%

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

   if 0.0205000006 < (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 90.1%

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

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in maxCos around 0

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

      1. lift--.f32 71.9

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

   8. Applied rewrites 71.9%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \leq 0.020500000566244125:\\ \;\;\;\;\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{maxCos \cdot \mathsf{fma}\left(-2, ux, 2 \cdot \frac{ux}{maxCos}\right)}\right)\right) \cdot uy\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \leq 0.020500000566244125:\\ \;\;\;\;\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\right) \cdot uy\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - t\_0 \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (sqrt (- 1.0 (* t_0 t_0))) 0.020500000566244125)
     (* (* 2.0 (* (PI) (sqrt (* ux (- 2.0 (* 2.0 maxCos)))))) uy)
     (* (* (PI) (* 2.0 uy)) (sqrt (- 1.0 (* t_0 (- 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))))) < 0.0205000006

   1. Initial program 38.4%

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

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{uy \cdot \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right) \cdot \color{blue}{uy} \]

      2. lower-*.f32 N/A

         \[\leadsto \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right) \cdot \color{blue}{uy} \]

   5. Applied rewrites 36.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{3} \cdot \left(uy \cdot uy\right)\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}^{2}}, -1.3333333333333333, \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}^{2}}\right) \cdot uy} \]

   6. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lower-sqrt.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-+.f32 N/A

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

      9. lower-*.f32 35.8

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

   8. Applied rewrites 35.8%

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

   9. Taylor expanded in ux around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 77.0

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

   11. Applied rewrites 77.0%

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

   if 0.0205000006 < (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 90.1%

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

   3. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in maxCos around 0

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

      1. lift--.f32 71.9

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

   8. Applied rewrites 71.9%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \leq 0.020500000566244125:\\ \;\;\;\;\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\right) \cdot uy\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (*
   uy
   (fma -1.3333333333333333 (* (* uy uy) (* (* (PI) (PI)) (PI))) (* 2.0 (PI))))
  (sqrt
   (*
    (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (* maxCos 2.0))
    ux))))

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.4

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

5. Applied rewrites 98.4%

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

   1. lift--.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift--.f32 98.4

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \left(uy \cdot \color{blue}{\left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + 2 \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

   2. lower-fma.f32 N/A

      \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \color{blue}{{uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

   3. lower-*.f32 N/A

      \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, {uy}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{3}}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

   4. pow2 N/A

      \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

   5. lower-*.f32 N/A

      \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

   6. lower-pow.f32 N/A

      \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{3}}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

   7. lift-PI.f32 N/A

      \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

   8. lower-*.f32 N/A

      \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

   9. lift-PI.f32 87.8

      \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

10. Applied rewrites 87.8%

    \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]
11. Step-by-step derivation

    1. lift-PI.f32 N/A

       \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{3}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

    2. lift-pow.f32 N/A

       \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{3}}, 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

    3. unpow3 N/A

       \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

    4. unpow2 N/A

       \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

    5. lower-*.f32 N/A

       \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

    6. unpow2 N/A

       \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

    7. lower-*.f32 N/A

       \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

    8. lift-PI.f32 N/A

       \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

    9. lift-PI.f32 N/A

       \[\leadsto \left(uy \cdot \mathsf{fma}\left(\frac{-4}{3}, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

    10. lift-PI.f32 87.8

        \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]

12. Applied rewrites 87.8%

    \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right), 2 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]
13. Add Preprocessing

Alternative 11: 81.2% accurate, 2.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.4

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

5. Applied rewrites 98.4%

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

   1. lift--.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift--.f32 98.4

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lift-PI.f32 80.5

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

10. Applied rewrites 80.5%

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

Alternative 12: 66.0% accurate, 4.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{uy \cdot \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right) \cdot \color{blue}{uy} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right) \cdot \color{blue}{uy} \]

5. Applied rewrites 54.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{3} \cdot \left(uy \cdot uy\right)\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}^{2}}, -1.3333333333333333, \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}^{2}}\right) \cdot uy} \]

6. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-+.f32 N/A

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

   9. lower-*.f32 51.3

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

8. Applied rewrites 51.3%

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

9. Taylor expanded in ux around 0

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

    1. lower-*.f32 N/A

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

    2. lower--.f32 N/A

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

    3. lower-*.f32 64.6

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

11. Applied rewrites 64.6%

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

12. Final simplification 64.6%

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

Alternative 13: 63.2% accurate, 5.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{uy \cdot \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right) \cdot \color{blue}{uy} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right) \cdot \color{blue}{uy} \]

5. Applied rewrites 54.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{3} \cdot \left(uy \cdot uy\right)\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}^{2}}, -1.3333333333333333, \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}^{2}}\right) \cdot uy} \]

6. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-+.f32 N/A

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

   9. lower-*.f32 51.3

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

8. Applied rewrites 51.3%

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

9. Taylor expanded in ux around 0

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

    1. lower-*.f32 N/A

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

    2. lower--.f32 N/A

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

    3. lower-*.f32 64.6

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

11. Applied rewrites 64.6%

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

12. Taylor expanded in maxCos around 0

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

14. Applied rewrites 61.3%

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

15. Final simplification 61.3%

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

Alternative 14: 7.1% accurate, 5.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 51.5

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

5. Applied rewrites 51.5%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 7.1%

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

Reproduce

herbie shell --seed 2025037 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (* (sin (* (* uy 2.0) (PI))) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))