UniformSampleCone, x

Percentage Accurate: 57.1% → 98.8%

Time: 16.4s

Alternatives: 16

Speedup: 6.5×

Specification

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

Math

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

FPCore

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

Initial Program: 57.1% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.8% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in maxCos around inf

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

   1. lower-*.f32 N/A

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

   2. associate--l+ N/A

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

   3. div-sub N/A

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

   4. lower-+.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower--.f32 57.8

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

5. Applied rewrites 57.8%

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

6. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f32 N/A

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

8. Applied rewrites 98.7%

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

   1. lift-/.f32 N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. div-inv N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*l* N/A

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

   9. lift-/.f32 N/A

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

   10. inv-pow N/A

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

   11. pow-plus N/A

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

   12. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{maxCos \cdot -2 + 2 \cdot {maxCos}^{\color{blue}{0}}}{ux} - \left(maxCos \cdot \left(1 - maxCos\right)\right) \cdot \left(\frac{1}{maxCos} + -1\right)\right)} \]

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-+.f32 N/A

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

   16. lower-*.f32 99.0

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

10. Applied rewrites 99.0%

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

    1. lift--.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. lift-/.f32 N/A

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

    4. distribute-rgt-in N/A

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

    5. +-commutative N/A

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

    6. lower-+.f32 N/A

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

    7. neg-mul-1 N/A

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

    8. lift-*.f32 N/A

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

    9. *-commutative N/A

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

    10. distribute-rgt-neg-in N/A

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

    11. lower-*.f32 N/A

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

    12. lower-neg.f32 N/A

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

    13. lift-*.f32 N/A

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

    14. associate-*r* N/A

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

    15. lift-/.f32 N/A

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

    16. inv-pow N/A

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

    17. pow-plus N/A

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

    18. metadata-eval N/A

        \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{maxCos \cdot -2 + 2}{ux} - \left(\left(1 - maxCos\right) \cdot \left(\mathsf{neg}\left(maxCos\right)\right) + {maxCos}^{\color{blue}{0}} \cdot \left(1 - maxCos\right)\right)\right)} \]

    19. metadata-eval N/A

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

    20. lower-*.f32 99.0

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

12. Applied rewrites 99.0%

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

13. Final simplification 99.0%

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

Alternative 2: 97.5% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in maxCos around inf

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

   1. lower-*.f32 N/A

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

   2. associate--l+ N/A

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

   3. div-sub N/A

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

   4. lower-+.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower--.f32 57.8

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

5. Applied rewrites 57.8%

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

6. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f32 N/A

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

8. Applied rewrites 98.7%

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

   1. lift-/.f32 N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. div-inv N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*l* N/A

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

   9. lift-/.f32 N/A

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

   10. inv-pow N/A

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

   11. pow-plus N/A

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

   12. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{maxCos \cdot -2 + 2 \cdot {maxCos}^{\color{blue}{0}}}{ux} - \left(maxCos \cdot \left(1 - maxCos\right)\right) \cdot \left(\frac{1}{maxCos} + -1\right)\right)} \]

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-+.f32 N/A

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

   16. lower-*.f32 99.0

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

10. Applied rewrites 99.0%

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

    1. lift--.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. lift-/.f32 N/A

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

    4. distribute-rgt-in N/A

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

    5. +-commutative N/A

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

    6. neg-mul-1 N/A

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

    7. lift-*.f32 N/A

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

    8. *-commutative N/A

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

    9. distribute-rgt-neg-in N/A

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

    10. lower-fma.f32 N/A

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

    11. lower-neg.f32 N/A

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

    12. lift-*.f32 N/A

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

    13. associate-*r* N/A

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

    14. lift-/.f32 N/A

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

    15. inv-pow N/A

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

    16. pow-plus N/A

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

    17. metadata-eval N/A

        \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{maxCos \cdot -2 + 2}{ux} - \mathsf{fma}\left(1 - maxCos, \mathsf{neg}\left(maxCos\right), {maxCos}^{\color{blue}{0}} \cdot \left(1 - maxCos\right)\right)\right)} \]

    18. metadata-eval N/A

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

    19. lower-*.f32 86.8

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

12. Applied rewrites 86.8%

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

13. Final simplification 86.8%

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

Alternative 3: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0008200000156648457:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{-2 \cdot \left(maxCos + -1\right)}{ux} + \left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(-1 + \frac{2}{ux}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0008200000156648457)
   (sqrt
    (*
     (* ux ux)
     (+ (/ (* -2.0 (+ maxCos -1.0)) ux) (* (- 1.0 maxCos) (+ maxCos -1.0)))))
   (* (cos (* (* uy 2.0) (PI))) (sqrt (* (* ux ux) (+ -1.0 (/ 2.0 ux)))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 8.20000016e-4

   1. Initial program 58.5%

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

   3. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. lower-fma.f32 N/A

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

   5. Applied rewrites 31.4%

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

   6. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f32 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f32 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f32 N/A

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. lower-+.f32 N/A

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

      19. sub-neg N/A

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

      20. metadata-eval N/A

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

      21. lower-+.f32 98.2

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

   8. Applied rewrites 98.2%

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

   if 8.20000016e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 55.9%

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

   3. Taylor expanded in maxCos around inf

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

      1. lower-*.f32 N/A

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

      2. associate--l+ N/A

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

      3. div-sub N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower--.f32 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in ux around -inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f32 N/A

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in maxCos around 0

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

       1. lower-*.f32 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f32 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-+.f32 N/A

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. lower-/.f32 94.2

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

   11. Applied rewrites 94.2%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0008200000156648457:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{-2 \cdot \left(maxCos + -1\right)}{ux} + \left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(-1 + \frac{2}{ux}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in maxCos around inf

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

   1. lower-*.f32 N/A

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

   2. associate--l+ N/A

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

   3. div-sub N/A

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

   4. lower-+.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower--.f32 57.8

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

5. Applied rewrites 57.8%

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

6. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f32 N/A

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

8. Applied rewrites 98.7%

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

   1. lift-/.f32 N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-in N/A

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

   4. *-commutative N/A

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

   5. lift-/.f32 N/A

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

   6. div-inv N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*l* N/A

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

   9. lift-/.f32 N/A

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

   10. inv-pow N/A

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

   11. pow-plus N/A

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

   12. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{maxCos \cdot -2 + 2 \cdot {maxCos}^{\color{blue}{0}}}{ux} - \left(maxCos \cdot \left(1 - maxCos\right)\right) \cdot \left(\frac{1}{maxCos} + -1\right)\right)} \]

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-+.f32 N/A

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

   16. lower-*.f32 99.0

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

10. Applied rewrites 99.0%

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

11. Taylor expanded in maxCos around 0

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

13. Applied rewrites 97.3%

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

14. Final simplification 97.3%

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

Alternative 5: 96.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0008200000156648457:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{-2 \cdot \left(maxCos + -1\right)}{ux} + \left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0008200000156648457)
   (sqrt
    (*
     (* ux ux)
     (+ (/ (* -2.0 (+ maxCos -1.0)) ux) (* (- 1.0 maxCos) (+ maxCos -1.0)))))
   (* (cos (* (* uy 2.0) (PI))) (sqrt (- (* 2.0 ux) (* ux ux))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 8.20000016e-4

   1. Initial program 58.5%

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

   3. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. lower-fma.f32 N/A

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

   5. Applied rewrites 31.4%

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

   6. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f32 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f32 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f32 N/A

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. lower-+.f32 N/A

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

      19. sub-neg N/A

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

      20. metadata-eval N/A

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

      21. lower-+.f32 98.2

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

   8. Applied rewrites 98.2%

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

   if 8.20000016e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 55.9%

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

   3. Taylor expanded in maxCos around inf

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

      1. lower-*.f32 N/A

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

      2. associate--l+ N/A

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

      3. div-sub N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower--.f32 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in ux around -inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f32 N/A

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in maxCos around 0

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

       1. lower-*.f32 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f32 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-+.f32 N/A

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. lower-/.f32 94.2

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

   11. Applied rewrites 94.2%

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

       1. lift-*.f32 N/A

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

       2. lift-/.f32 N/A

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

       3. distribute-rgt-in N/A

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

       4. +-commutative N/A

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

       5. lower-+.f32 N/A

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

       6. lift-*.f32 N/A

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

       7. associate-*r* N/A

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

       8. neg-mul-1 N/A

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

       9. lower-*.f32 N/A

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

       10. lower-neg.f32 N/A

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

       11. lift-/.f32 N/A

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

       12. div-inv N/A

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

       13. associate-*l* N/A

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

       14. inv-pow N/A

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

       15. lift-*.f32 N/A

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

       16. pow2 N/A

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

       17. pow-prod-up N/A

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

       18. metadata-eval N/A

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

       19. unpow1 N/A

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

       20. lower-*.f32 94.1

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

   13. Applied rewrites 94.1%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0008200000156648457:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{-2 \cdot \left(maxCos + -1\right)}{ux} + \left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0008200000156648457:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{-2 \cdot \left(maxCos + -1\right)}{ux} + \left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0008200000156648457)
   (sqrt
    (*
     (* ux ux)
     (+ (/ (* -2.0 (+ maxCos -1.0)) ux) (* (- 1.0 maxCos) (+ maxCos -1.0)))))
   (* (cos (* (* uy 2.0) (PI))) (sqrt (* ux (- 2.0 ux))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 8.20000016e-4

   1. Initial program 58.5%

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

   3. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. lower-fma.f32 N/A

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

   5. Applied rewrites 31.4%

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

   6. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f32 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f32 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f32 N/A

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. lower-+.f32 N/A

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

      19. sub-neg N/A

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

      20. metadata-eval N/A

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

      21. lower-+.f32 98.2

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

   8. Applied rewrites 98.2%

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

   if 8.20000016e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 55.9%

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

   3. Taylor expanded in maxCos around inf

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

      1. lower-*.f32 N/A

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

      2. associate--l+ N/A

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

      3. div-sub N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower--.f32 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in ux around -inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f32 N/A

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in maxCos around 0

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

       1. lower-*.f32 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f32 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-+.f32 N/A

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. lower-/.f32 94.2

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

   11. Applied rewrites 94.2%

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

   12. Taylor expanded in ux around 0

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

       1. lower-*.f32 N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

       4. lower--.f32 93.9

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

   14. Applied rewrites 93.9%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0008200000156648457:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{-2 \cdot \left(maxCos + -1\right)}{ux} + \left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.006000000052154064:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{-2 \cdot \left(maxCos + -1\right)}{ux} + \left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.006000000052154064)
   (sqrt
    (*
     (* ux ux)
     (+ (/ (* -2.0 (+ maxCos -1.0)) ux) (* (- 1.0 maxCos) (+ maxCos -1.0)))))
   (* (cos (* (* uy 2.0) (PI))) (sqrt (* 2.0 ux)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00600000005

   1. Initial program 59.5%

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

   3. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. lower-fma.f32 N/A

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

   5. Applied rewrites 31.4%

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

   6. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f32 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. sub-neg N/A

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

      12. metadata-eval N/A

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

      13. lower-+.f32 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f32 N/A

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. lower-+.f32 N/A

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

      19. sub-neg N/A

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

      20. metadata-eval N/A

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

      21. lower-+.f32 95.7

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

   8. Applied rewrites 95.7%

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

   if 0.00600000005 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 52.4%

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

   3. Taylor expanded in maxCos around inf

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

      1. lower-*.f32 N/A

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

      2. associate--l+ N/A

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

      3. div-sub N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower--.f32 52.4

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in ux around -inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. lower-+.f32 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f32 N/A

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

   8. Applied rewrites 98.5%

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

   9. Taylor expanded in maxCos around 0

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

       1. lower-*.f32 N/A

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

       2. unpow2 N/A

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

       3. lower-*.f32 N/A

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

       4. sub-neg N/A

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

       5. metadata-eval N/A

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

       6. lower-+.f32 N/A

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

       7. associate-*r/ N/A

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

       8. metadata-eval N/A

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

       9. lower-/.f32 93.4

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

   11. Applied rewrites 93.4%

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

   12. Taylor expanded in ux around 0

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

       1. lower-*.f32 78.5

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

   14. Applied rewrites 78.5%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.006000000052154064:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{-2 \cdot \left(maxCos + -1\right)}{ux} + \left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{-2 \cdot \left(maxCos + -1\right)}{ux} + \left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)} \end{array} \]

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(((ux * ux) * (((-2.0f * (maxCos + -1.0f)) / ux) + ((1.0f - maxCos) * (maxCos + -1.0f)))));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt(((ux * ux) * ((((-2.0e0) * (maxcos + (-1.0e0))) / ux) + ((1.0e0 - maxcos) * (maxcos + (-1.0e0))))))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(Float32(ux * ux) * Float32(Float32(Float32(Float32(-2.0) * Float32(maxCos + Float32(-1.0))) / ux) + Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt(((ux * ux) * (((single(-2.0) * (maxCos + single(-1.0))) / ux) + ((single(1.0) - maxCos) * (maxCos + single(-1.0))))));
end

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lower-fma.f32 N/A

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

5. Applied rewrites 30.7%

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

6. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. sub-neg N/A

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

   12. metadata-eval N/A

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

   13. lower-+.f32 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f32 N/A

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

   16. sub-neg N/A

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

   17. metadata-eval N/A

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

   18. lower-+.f32 N/A

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

   19. sub-neg N/A

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

   20. metadata-eval N/A

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

   21. lower-+.f32 81.4

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

8. Applied rewrites 81.4%

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

9. Final simplification 81.4%

   \[\leadsto \sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{-2 \cdot \left(maxCos + -1\right)}{ux} + \left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)} \]
10. Add Preprocessing

Alternative 9: 75.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot \left(maxCos \cdot \left(\left(-1 + \left(ux - \frac{ux}{maxCos}\right)\right) + \frac{2}{maxCos}\right)\right)} \end{array} \]

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * (maxCos * ((-1.0f + (ux - (ux / maxCos))) + (2.0f / maxCos)))));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * (maxcos * (((-1.0e0) + (ux - (ux / maxcos))) + (2.0e0 / maxcos)))))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(ux * Float32(maxCos * Float32(Float32(Float32(-1.0) + Float32(ux - Float32(ux / maxCos))) + Float32(Float32(2.0) / maxCos)))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((ux * (maxCos * ((single(-1.0) + (ux - (ux / maxCos))) + (single(2.0) / maxCos)))));
end

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in maxCos around inf

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

   1. lower-*.f32 N/A

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

   2. associate--l+ N/A

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

   3. div-sub N/A

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

   4. lower-+.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower--.f32 57.8

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

5. Applied rewrites 57.8%

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

6. Taylor expanded in maxCos around 0

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

   1. lower--.f32 55.8

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

8. Applied rewrites 55.8%

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

9. Taylor expanded in uy around 0

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

    1. lower-sqrt.f32 N/A

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

    2. lower--.f32 N/A

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

    3. associate-*r* N/A

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

    4. lower-*.f32 N/A

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

    5. lower-*.f32 N/A

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

    6. lower--.f32 N/A

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

    7. associate--l+ N/A

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

    8. div-sub N/A

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

    9. lower-+.f32 N/A

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

    10. lower-/.f32 N/A

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

    11. lower--.f32 49.5

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

11. Applied rewrites 49.5%

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

12. Taylor expanded in ux around 0

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

    1. lower-*.f32 N/A

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

    2. distribute-lft-out-- N/A

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

    3. lower-*.f32 N/A

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

    4. associate--r- N/A

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

    5. lower-+.f32 N/A

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

14. Applied rewrites 77.0%

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

15. Final simplification 77.0%

    \[\leadsto \sqrt{ux \cdot \left(maxCos \cdot \left(\left(-1 + \left(ux - \frac{ux}{maxCos}\right)\right) + \frac{2}{maxCos}\right)\right)} \]
16. Add Preprocessing

Alternative 10: 74.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - ux\right) + ux \cdot maxCos \leq 0.9997649788856506:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(ux \cdot \left(maxCos + -1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (+ (- 1.0 ux) (* ux maxCos)) 0.9997649788856506)
   (sqrt (+ 1.0 (* (- 1.0 ux) (+ ux -1.0))))
   (sqrt (* -2.0 (* ux (+ maxCos -1.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (((1.0f - ux) + (ux * maxCos)) <= 0.9997649788856506f) {
		tmp = sqrtf((1.0f + ((1.0f - ux) * (ux + -1.0f))));
	} else {
		tmp = sqrtf((-2.0f * (ux * (maxCos + -1.0f))));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: tmp
    if (((1.0e0 - ux) + (ux * maxcos)) <= 0.9997649788856506e0) then
        tmp = sqrt((1.0e0 + ((1.0e0 - ux) * (ux + (-1.0e0)))))
    else
        tmp = sqrt(((-2.0e0) * (ux * (maxcos + (-1.0e0)))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos)) <= Float32(0.9997649788856506))
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))));
	else
		tmp = sqrt(Float32(Float32(-2.0) * Float32(ux * Float32(maxCos + Float32(-1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (((single(1.0) - ux) + (ux * maxCos)) <= single(0.9997649788856506))
		tmp = sqrt((single(1.0) + ((single(1.0) - ux) * (ux + single(-1.0)))));
	else
		tmp = sqrt((single(-2.0) * (ux * (maxCos + single(-1.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.999764979

   1. Initial program 90.4%

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

   3. Taylor expanded in maxCos around inf

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

      1. lower-*.f32 N/A

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

      2. associate--l+ N/A

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

      3. div-sub N/A

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

      4. lower-+.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower--.f32 90.0

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower--.f32 85.4

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

   8. Applied rewrites 85.4%

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

   9. Taylor expanded in uy around 0

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

       1. lower-sqrt.f32 N/A

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

       2. lower--.f32 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. lower--.f32 N/A

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

       7. associate--l+ N/A

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

       8. div-sub N/A

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

       9. lower-+.f32 N/A

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

       10. lower-/.f32 N/A

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

       11. lower--.f32 75.7

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

   11. Applied rewrites 75.7%

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

   12. Taylor expanded in maxCos around 0

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

       1. unpow2 N/A

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

       2. lower-*.f32 N/A

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

       3. lower--.f32 N/A

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

       4. lower--.f32 75.8

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

   14. Applied rewrites 75.8%

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

   if 0.999764979 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 37.3%

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

   3. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. lower-fma.f32 N/A

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

   5. Applied rewrites 30.6%

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

   6. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

      5. lower-+.f32 75.0

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

   8. Applied rewrites 75.0%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - ux\right) + ux \cdot maxCos \leq 0.9997649788856506:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot \left(ux \cdot \left(maxCos + -1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.7% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(ux \cdot ux\right) \cdot \left(-1 + \frac{2}{ux}\right)} \end{array} \]

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(((ux * ux) * (-1.0f + (2.0f / ux))));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt(((ux * ux) * ((-1.0e0) + (2.0e0 / ux))))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(Float32(ux * ux) * Float32(Float32(-1.0) + Float32(Float32(2.0) / ux))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt(((ux * ux) * (single(-1.0) + (single(2.0) / ux))));
end

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in maxCos around inf

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

   1. lower-*.f32 N/A

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

   2. associate--l+ N/A

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

   3. div-sub N/A

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

   4. lower-+.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower--.f32 57.8

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

5. Applied rewrites 57.8%

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

6. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f32 N/A

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

8. Applied rewrites 98.7%

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

9. Taylor expanded in maxCos around 0

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

    1. lower-*.f32 N/A

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

    2. unpow2 N/A

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

    3. lower-*.f32 N/A

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

    4. sub-neg N/A

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

    5. metadata-eval N/A

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

    6. lower-+.f32 N/A

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

    7. associate-*r/ N/A

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

    8. metadata-eval N/A

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

    9. lower-/.f32 92.8

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

11. Applied rewrites 92.8%

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

12. Taylor expanded in uy around 0

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

14. Applied rewrites 76.9%

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

15. Final simplification 76.9%

    \[\leadsto \sqrt{\left(ux \cdot ux\right) \cdot \left(-1 + \frac{2}{ux}\right)} \]
16. Add Preprocessing

Alternative 12: 75.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ ux \cdot \sqrt{-1 + \frac{2}{ux}} \end{array} \]

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return ux * sqrtf((-1.0f + (2.0f / ux)));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = ux * sqrt(((-1.0e0) + (2.0e0 / ux)))
end function

Julia

function code(ux, uy, maxCos)
	return Float32(ux * sqrt(Float32(Float32(-1.0) + Float32(Float32(2.0) / ux))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = ux * sqrt((single(-1.0) + (single(2.0) / ux)));
end

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in maxCos around inf

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

   1. lower-*.f32 N/A

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

   2. associate--l+ N/A

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

   3. div-sub N/A

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

   4. lower-+.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower--.f32 57.8

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

5. Applied rewrites 57.8%

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

6. Taylor expanded in ux around -inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. sub-neg N/A

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

   8. metadata-eval N/A

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

   9. lower-+.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f32 N/A

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

8. Applied rewrites 98.7%

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

9. Taylor expanded in uy around 0

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

11. Applied rewrites 81.1%

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

12. Taylor expanded in maxCos around 0

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

    1. lower-*.f32 N/A

       \[\leadsto \color{blue}{ux \cdot \sqrt{2 \cdot \frac{1}{ux} - 1}} \]

    2. lower-sqrt.f32 N/A

       \[\leadsto ux \cdot \color{blue}{\sqrt{2 \cdot \frac{1}{ux} - 1}} \]

    3. sub-neg N/A

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

    4. metadata-eval N/A

       \[\leadsto ux \cdot \sqrt{2 \cdot \frac{1}{ux} + \color{blue}{-1}} \]

    5. +-commutative N/A

       \[\leadsto ux \cdot \sqrt{\color{blue}{-1 + 2 \cdot \frac{1}{ux}}} \]

    6. lower-+.f32 N/A

       \[\leadsto ux \cdot \sqrt{\color{blue}{-1 + 2 \cdot \frac{1}{ux}}} \]

    7. associate-*r/ N/A

       \[\leadsto ux \cdot \sqrt{-1 + \color{blue}{\frac{2 \cdot 1}{ux}}} \]

    8. metadata-eval N/A

       \[\leadsto ux \cdot \sqrt{-1 + \frac{\color{blue}{2}}{ux}} \]

    9. lower-/.f32 76.8

       \[\leadsto ux \cdot \sqrt{-1 + \color{blue}{\frac{2}{ux}}} \]

14. Applied rewrites 76.8%

    \[\leadsto \color{blue}{ux \cdot \sqrt{-1 + \frac{2}{ux}}} \]
15. Add Preprocessing

Alternative 13: 64.9% accurate, 6.5× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((-2.0f * (ux * (maxCos + -1.0f))));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt(((-2.0e0) * (ux * (maxcos + (-1.0e0)))))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(Float32(-2.0) * Float32(ux * Float32(maxCos + Float32(-1.0)))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((single(-2.0) * (ux * (maxCos + single(-1.0)))));
end

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lower-fma.f32 N/A

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

5. Applied rewrites 30.7%

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

6. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. sub-neg N/A

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

   4. metadata-eval N/A

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

   5. lower-+.f32 65.1

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

8. Applied rewrites 65.1%

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

Alternative 14: 19.9% accurate, 6.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lower-fma.f32 N/A

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

5. Applied rewrites 30.7%

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

6. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower--.f32 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-+.f32 19.9

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

8. Applied rewrites 20.0%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(1 - ux, ux + -1, 1\right)}} \]
9. Add Preprocessing

Alternative 15: 19.9% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(-1, 1, 1\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lower-fma.f32 N/A

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

5. Applied rewrites 30.7%

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

7. Applied rewrites 9.5%

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

8. Taylor expanded in ux around 0

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

10. Applied rewrites 20.0%

    \[\leadsto \sqrt{\mathsf{fma}\left(-1, \color{blue}{1}, 1\right)} \]
11. Add Preprocessing

Alternative 16: 6.6% accurate, 156.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 0.0)

C

float code(float ux, float uy, float maxCos) {
	return 0.0f;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 0.0e0
end function

Julia

function code(ux, uy, maxCos)
	return Float32(0.0)
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(0.0);
end

Derivation

1. Initial program 57.6%

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

3. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. distribute-rgt-neg-in N/A

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

   6. lower-fma.f32 N/A

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

5. Applied rewrites 30.7%

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

7. Applied rewrites 9.5%

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

8. Taylor expanded in ux around 0

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

10. Applied rewrites 19.9%

    \[\leadsto \sqrt{\mathsf{fma}\left(-1, \color{blue}{1}, 1\right)} \]
11. Step-by-step derivation

    1. lift-fma.f32 N/A

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

    2. pow1/2 N/A

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

    3. lift-fma.f32 N/A

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

    4. metadata-eval N/A

       \[\leadsto {\left(\color{blue}{-1} + 1\right)}^{\frac{1}{2}} \]

    5. metadata-eval N/A

       \[\leadsto {\color{blue}{0}}^{\frac{1}{2}} \]

    6. metadata-eval 6.6

       \[\leadsto \color{blue}{0} \]

12. Applied rewrites 6.6%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (* (cos (* (* uy 2.0) (PI))) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))