UniformSampleCone, y

Percentage Accurate: 57.4% → 98.4%

Time: 7.7s

Alternatives: 18

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: 98.4% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 61.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lift-pow.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 98.6

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

8. Applied rewrites 98.6%

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

   1. lift-*.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift-pow.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. distribute-rgt-in N/A

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

   8. lower-fma.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-pow.f32 N/A

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

   11. lift--.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. lift-*.f32 98.6

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

10. Applied rewrites 98.6%

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

11. Final simplification 98.6%

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

Alternative 2: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 61.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lift-pow.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 98.6

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

8. Applied rewrites 98.6%

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

Alternative 3: 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 61.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 98.6

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

5. Applied rewrites 98.6%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\mathsf{fma}\left(-ux, {\left(maxCos - 1\right)}^{2}, 2\right) - maxCos \cdot 2\right) \cdot ux}} \]
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 98.6

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

7. Applied rewrites 98.6%

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

Alternative 4: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 61.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 98.6

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

5. Applied rewrites 98.6%

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

6. Taylor expanded in maxCos around 0

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

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

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

   2. metadata-eval N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 97.8

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

8. Applied rewrites 97.8%

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

Alternative 5: 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 61.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 98.6

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

5. Applied rewrites 98.6%

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

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 6: 97.0% 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 61.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 98.6

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

5. Applied rewrites 98.6%

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

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 96.4%

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

Alternative 7: 90.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.003220000071451068:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.003220000071451068)
   (*
    (* 2.0 (* uy (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 < 0.00322000007

   1. Initial program 62.1%

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

   3. Taylor expanded in ux around 0

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

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]
   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. lift-*.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 96.1

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

   10. Applied rewrites 96.1%

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

   if 0.00322000007 < uy

   1. Initial program 60.5%

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

   3. Taylor expanded in ux around 0

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right)}} \]
   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 76.7

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

   5. Applied rewrites 76.7%

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

Alternative 8: 95.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.00019999999494757503:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00019999999494757503)
   (*
    (* 2.0 (* uy (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 < 1.99999995e-4

   1. Initial program 61.8%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      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

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux} \]
   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. lift-*.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 98.6

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

   10. Applied rewrites 98.6%

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

   if 1.99999995e-4 < uy

   1. Initial program 61.5%

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

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(2 + -1 \cdot ux\right) \cdot ux} \]
   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 91.1

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

   8. Applied rewrites 91.1%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.00019999999494757503:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos \cdot 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := uy \cdot \mathsf{PI}\left(\right)\\ t_1 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_1 \cdot t\_1 \leq 0.9996100068092346:\\ \;\;\;\;2 \cdot \left(t\_0 \cdot \sqrt{1 - \left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(1 + ux \cdot \left(maxCos - 1\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_0 \cdot \sqrt{maxCos \cdot \mathsf{fma}\left(-2, ux, 2 \cdot \frac{ux}{maxCos}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (PI))) (t_1 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_1 t_1) 0.9996100068092346)
     (*
      2.0
      (*
       t_0
       (sqrt
        (-
         1.0
         (* (- (+ 1.0 (* maxCos ux)) ux) (+ 1.0 (* ux (- maxCos 1.0))))))))
     (* 2.0 (* t_0 (sqrt (* maxCos (fma -2.0 ux (* 2.0 (/ ux maxCos))))))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.999610007

   1. Initial program 89.6%

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

      1. lift--.f32 N/A

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

      2. flip-- N/A

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

      3. lower-/.f32 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-+.f32 89.2

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

   7. Applied rewrites 76.3%

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

   8. Taylor expanded in ux around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lift--.f32 76.2

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

   10. Applied rewrites 76.2%

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

   if 0.999610007 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 41.9%

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

      1. lift--.f32 N/A

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

      2. flip-- N/A

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

      3. lower-/.f32 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-+.f32 42.0

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

   4. Applied rewrites 42.0%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

   7. Applied rewrites 39.3%

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

   8. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 78.0

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

   10. Applied rewrites 78.0%

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

   11. Taylor expanded in maxCos around inf

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

       1. lower-*.f32 N/A

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

       2. lower-fma.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower-/.f32 78.0

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

   13. Applied rewrites 78.0%

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

Alternative 10: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9996100068092346:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - t\_0 \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos \cdot ux, 2 \cdot ux\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_0 t_0) 0.9996100068092346)
     (* (* (PI) (* 2.0 uy)) (sqrt (- 1.0 (* t_0 (- 1.0 ux)))))
     (* 2.0 (* (* uy (PI)) (sqrt (fma -2.0 (* maxCos ux) (* 2.0 ux))))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.999610007

   1. Initial program 89.6%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 76.2

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

   5. Applied rewrites 76.2%

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

   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. flip-- N/A

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

      2. metadata-eval N/A

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

      3. lift--.f32 72.0

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

   8. Applied rewrites 72.0%

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

   if 0.999610007 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 41.9%

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

      1. lift--.f32 N/A

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

      2. flip-- N/A

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

      3. lower-/.f32 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-+.f32 42.0

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

   4. Applied rewrites 42.0%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

   7. Applied rewrites 39.3%

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

   8. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 78.0

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

   10. Applied rewrites 78.0%

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

   11. Taylor expanded in maxCos around 0

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

       1. lower-fma.f32 N/A

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

       2. lift-*.f32 N/A

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

       3. lower-*.f32 78.0

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

   13. Applied rewrites 78.0%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) \leq 0.9996100068092346:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos \cdot ux, 2 \cdot ux\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := uy \cdot \mathsf{PI}\left(\right)\\ t_1 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_1 \cdot t\_1 \leq 0.9996100068092346:\\ \;\;\;\;2 \cdot \left(t\_0 \cdot \sqrt{1 - \left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(1 - ux\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t\_0 \cdot \sqrt{\mathsf{fma}\left(-2, maxCos \cdot ux, 2 \cdot ux\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (PI))) (t_1 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_1 t_1) 0.9996100068092346)
     (* 2.0 (* t_0 (sqrt (- 1.0 (* (- (+ 1.0 (* maxCos ux)) ux) (- 1.0 ux))))))
     (* 2.0 (* t_0 (sqrt (fma -2.0 (* maxCos ux) (* 2.0 ux))))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.999610007

   1. Initial program 89.6%

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

      1. lift--.f32 N/A

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

      2. flip-- N/A

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

      3. lower-/.f32 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-+.f32 89.2

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

   7. Applied rewrites 76.3%

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

   8. Taylor expanded in maxCos around 0

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

      1. sub-div N/A

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

      2. metadata-eval N/A

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

      3. pow2 N/A

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

      4. flip-- N/A

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

      5. lower--.f32 72.0

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

   10. Applied rewrites 72.0%

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

   if 0.999610007 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 41.9%

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

      1. lift--.f32 N/A

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

      2. flip-- N/A

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

      3. lower-/.f32 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-+.f32 42.0

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

   4. Applied rewrites 42.0%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

   7. Applied rewrites 39.3%

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

   8. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 78.0

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

   10. Applied rewrites 78.0%

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

   11. Taylor expanded in maxCos around 0

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

       1. lower-fma.f32 N/A

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

       2. lift-*.f32 N/A

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

       3. lower-*.f32 78.0

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

   13. Applied rewrites 78.0%

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

Alternative 12: 76.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := uy \cdot \mathsf{PI}\left(\right)\\ t_1 := \mathsf{fma}\left(maxCos, ux, 1 - ux\right)\\ \mathbf{if}\;ux \leq 0.00019500000053085387:\\ \;\;\;\;2 \cdot \left(t\_0 \cdot \sqrt{maxCos \cdot \mathsf{fma}\left(-2, ux, 2 \cdot \frac{ux}{maxCos}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot t\_0\right) \cdot \sqrt{1 - t\_1 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* uy (PI))) (t_1 (fma maxCos ux (- 1.0 ux))))
   (if (<= ux 0.00019500000053085387)
     (* 2.0 (* t_0 (sqrt (* maxCos (fma -2.0 ux (* 2.0 (/ ux maxCos)))))))
     (* (* 2.0 t_0) (sqrt (- 1.0 (* t_1 t_1)))))))

Derivation

1. Split input into 2 regimes

2. if ux < 1.95000001e-4

   1. Initial program 41.9%

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

      1. lift--.f32 N/A

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

      2. flip-- N/A

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

      3. lower-/.f32 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-+.f32 42.0

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

   4. Applied rewrites 42.0%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

   7. Applied rewrites 39.3%

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

   8. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 78.0

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

   10. Applied rewrites 78.0%

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

   11. Taylor expanded in maxCos around inf

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

       1. lower-*.f32 N/A

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

       2. lower-fma.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower-/.f32 78.0

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

   13. Applied rewrites 78.0%

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

   if 1.95000001e-4 < ux

   1. Initial program 89.6%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. lift--.f32 89.6

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

      8. lift-+.f32 N/A

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

      9. lift--.f32 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lift--.f32 89.6

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

   4. Applied rewrites 89.6%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lift-PI.f32 76.2

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

   7. Applied rewrites 76.2%

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

Alternative 13: 76.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(maxCos, ux, 1 - ux\right)\\ t_1 := uy \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;ux \leq 0.00019500000053085387:\\ \;\;\;\;2 \cdot \left(t\_1 \cdot \sqrt{\mathsf{fma}\left(-2, maxCos \cdot ux, 2 \cdot ux\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot t\_1\right) \cdot \sqrt{1 - t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (fma maxCos ux (- 1.0 ux))) (t_1 (* uy (PI))))
   (if (<= ux 0.00019500000053085387)
     (* 2.0 (* t_1 (sqrt (fma -2.0 (* maxCos ux) (* 2.0 ux)))))
     (* (* 2.0 t_1) (sqrt (- 1.0 (* t_0 t_0)))))))

Derivation

1. Split input into 2 regimes

2. if ux < 1.95000001e-4

   1. Initial program 41.9%

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

      1. lift--.f32 N/A

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

      2. flip-- N/A

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

      3. lower-/.f32 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f32 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-+.f32 42.0

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

   4. Applied rewrites 42.0%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

   7. Applied rewrites 39.3%

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

   8. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 78.0

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

   10. Applied rewrites 78.0%

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

   11. Taylor expanded in maxCos around 0

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

       1. lower-fma.f32 N/A

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

       2. lift-*.f32 N/A

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

       3. lower-*.f32 78.0

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

   13. Applied rewrites 78.0%

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

   if 1.95000001e-4 < ux

   1. Initial program 89.6%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. lift--.f32 89.6

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

      8. lift-+.f32 N/A

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

      9. lift--.f32 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lift--.f32 89.6

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

   4. Applied rewrites 89.6%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lift-PI.f32 76.2

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

   7. Applied rewrites 76.2%

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

Alternative 14: 81.3% 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 61.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 98.6

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

5. Applied rewrites 98.6%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\mathsf{fma}\left(-ux, {\left(maxCos - 1\right)}^{2}, 2\right) - maxCos \cdot 2\right) \cdot ux}} \]
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 98.6

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

7. Applied rewrites 98.6%

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

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. lift-*.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 82.4

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

10. Applied rewrites 82.4%

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

Alternative 15: 66.0% accurate, 3.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 61.7%

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

   1. lift--.f32 N/A

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

   2. flip-- N/A

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

   3. lower-/.f32 N/A

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

   4. metadata-eval N/A

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

   5. unpow2 N/A

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

   6. lower--.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-+.f32 61.5

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

4. Applied rewrites 61.5%

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

5. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower--.f32 N/A

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

7. Applied rewrites 54.6%

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

8. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 64.9

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

10. Applied rewrites 64.9%

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

11. Taylor expanded in maxCos around 0

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

    1. lower-fma.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. lower-*.f32 64.9

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

13. Applied rewrites 64.9%

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

Alternative 16: 66.0% accurate, 4.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 61.7%

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

   1. lift--.f32 N/A

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

   2. flip-- N/A

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

   3. lower-/.f32 N/A

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

   4. metadata-eval N/A

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

   5. unpow2 N/A

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

   6. lower--.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-+.f32 61.5

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

4. Applied rewrites 61.5%

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

5. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower--.f32 N/A

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

7. Applied rewrites 54.6%

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

8. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 64.9

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

10. Applied rewrites 64.9%

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

Alternative 17: 63.5% accurate, 5.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 61.7%

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

   1. lift--.f32 N/A

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

   2. flip-- N/A

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

   3. lower-/.f32 N/A

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

   4. metadata-eval N/A

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

   5. unpow2 N/A

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

   6. lower--.f32 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-+.f32 61.5

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

4. Applied rewrites 61.5%

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

5. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower--.f32 N/A

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

7. Applied rewrites 54.6%

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

8. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 64.9

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

10. Applied rewrites 64.9%

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

11. Taylor expanded in maxCos around 0

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

13. Applied rewrites 62.8%

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

Alternative 18: 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 61.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 54.5

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

5. Applied rewrites 54.5%

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

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

   1. flip-- 7.1

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

   2. metadata-eval 7.1

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

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 2025051 
(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)))))))