UniformSampleCone, x

Percentage Accurate: 56.8% → 99.0%

Time: 9.8s

Alternatives: 10

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: 56.8% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.0% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.7%

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

   1. lift-*.f32 N/A

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

   2. unpow1 N/A

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

   3. metadata-eval N/A

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

   4. sqrt-pow1 N/A

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

   5. pow2 N/A

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

   6. rem-sqrt-square-rev N/A

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

   7. lift-+.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-+l- N/A

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

   10. fabs-sub N/A

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

   11. unpow1 N/A

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

   12. metadata-eval N/A

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

   13. sqrt-pow1 N/A

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

   14. pow2 N/A

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

   15. rem-sqrt-square-rev N/A

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

   16. lift-+.f32 N/A

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

   17. lift--.f32 N/A

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

   18. associate-+l- N/A

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

   19. fabs-sub N/A

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

4. Applied rewrites 57.8%

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

5. Taylor expanded in maxCos around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate--r+ N/A

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

   4. lower--.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 58.0

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

7. Applied rewrites 58.0%

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

8. Taylor expanded in ux around 0

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

   1. metadata-eval N/A

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

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

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. mul-1-neg N/A

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

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

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

   9. lower--.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower--.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lower-pow.f32 N/A

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

   16. lower--.f32 99.0

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

10. Applied rewrites 99.0%

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

Alternative 2: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)\\ t_2 := \sqrt{1 - t\_0 \cdot t\_0}\\ \mathbf{if}\;t\_1 \cdot t\_2 \leq 0.025499999523162842:\\ \;\;\;\;t\_1 \cdot \sqrt{2 \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\left(\left(uy \cdot uy\right) \cdot -2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos)))
        (t_1 (cos (* (* uy 2.0) (PI))))
        (t_2 (sqrt (- 1.0 (* t_0 t_0)))))
   (if (<= (* t_1 t_2) 0.025499999523162842)
     (* t_1 (sqrt (* 2.0 ux)))
     (* (+ 1.0 (* (* (* (* uy uy) -2.0) (PI)) (PI))) t_2))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))) < 0.0255

   1. Initial program 39.9%

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

   3. Taylor expanded in ux around 0

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

      1. metadata-eval N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 65.4

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

   5. Applied rewrites 65.3%

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

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 87.2%

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

   if 0.0255 < (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))))

   1. Initial program 89.3%

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

   3. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 80.3

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

   5. Applied rewrites 80.3%

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

   7. Applied rewrites 85.8%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \leq 0.025499999523162842:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2 \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\left(\left(uy \cdot uy\right) \cdot -2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.7%

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

   1. lift-*.f32 N/A

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

   2. unpow1 N/A

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

   3. metadata-eval N/A

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

   4. sqrt-pow1 N/A

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

   5. pow2 N/A

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

   6. rem-sqrt-square-rev N/A

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

   7. lift-+.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-+l- N/A

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

   10. fabs-sub N/A

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

   11. unpow1 N/A

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

   12. metadata-eval N/A

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

   13. sqrt-pow1 N/A

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

   14. pow2 N/A

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

   15. rem-sqrt-square-rev N/A

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

   16. lift-+.f32 N/A

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

   17. lift--.f32 N/A

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

   18. associate-+l- N/A

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

   19. fabs-sub N/A

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

4. Applied rewrites 57.8%

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

5. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower--.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 98.8

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

7. Applied rewrites 98.8%

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

9. Applied rewrites 98.8%

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

10. Final simplification 98.8%

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

Alternative 4: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.7%

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

   1. lift-*.f32 N/A

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

   2. unpow1 N/A

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

   3. metadata-eval N/A

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

   4. sqrt-pow1 N/A

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

   5. pow2 N/A

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

   6. rem-sqrt-square-rev N/A

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

   7. lift-+.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-+l- N/A

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

   10. fabs-sub N/A

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

   11. unpow1 N/A

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

   12. metadata-eval N/A

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

   13. sqrt-pow1 N/A

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

   14. pow2 N/A

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

   15. rem-sqrt-square-rev N/A

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

   16. lift-+.f32 N/A

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

   17. lift--.f32 N/A

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

   18. associate-+l- N/A

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

   19. fabs-sub N/A

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

4. Applied rewrites 57.8%

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

5. Taylor expanded in maxCos around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate--r+ N/A

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

   4. lower--.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 58.0

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

7. Applied rewrites 58.0%

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

8. Taylor expanded in ux around 0

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

   1. metadata-eval N/A

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

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

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. mul-1-neg N/A

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

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

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

   9. lower--.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower--.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lower-pow.f32 N/A

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

   16. lower--.f32 99.0

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

10. Applied rewrites 99.0%

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

11. Taylor expanded in maxCos around 0

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

13. Applied rewrites 92.2%

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

Alternative 5: 97.3% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.7%

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

   1. lift-*.f32 N/A

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

   2. unpow1 N/A

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

   3. metadata-eval N/A

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

   4. sqrt-pow1 N/A

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

   5. pow2 N/A

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

   6. rem-sqrt-square-rev N/A

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

   7. lift-+.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-+l- N/A

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

   10. fabs-sub N/A

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

   11. unpow1 N/A

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

   12. metadata-eval N/A

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

   13. sqrt-pow1 N/A

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

   14. pow2 N/A

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

   15. rem-sqrt-square-rev N/A

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

   16. lift-+.f32 N/A

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

   17. lift--.f32 N/A

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

   18. associate-+l- N/A

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

   19. fabs-sub N/A

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

4. Applied rewrites 57.8%

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

5. Taylor expanded in maxCos around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate--r+ N/A

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

   4. lower--.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 58.0

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

7. Applied rewrites 58.0%

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

8. Taylor expanded in ux around 0

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

   1. metadata-eval N/A

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

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

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. mul-1-neg N/A

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

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

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

   9. lower--.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower--.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lower-pow.f32 N/A

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

   16. lower--.f32 99.0

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

10. Applied rewrites 99.0%

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

11. Taylor expanded in maxCos around 0

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

13. Applied rewrites 91.8%

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

Alternative 6: 96.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.00011700000322889537:\\ \;\;\;\;\sqrt{\left(\left(1 - maxCos\right) \cdot 2 - {\left(1 - maxCos\right)}^{2} \cdot ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00011700000322889537)
   (sqrt (* (- (* (- 1.0 maxCos) 2.0) (* (pow (- 1.0 maxCos) 2.0) ux)) ux))
   (* (cos (* (* uy 2.0) (PI))) (sqrt (* (- 2.0 ux) ux)))))

Derivation

1. Split input into 2 regimes

2. if uy < 1.17000003e-4

   1. Initial program 58.8%

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

      1. lift-*.f32 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. pow2 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. fabs-sub N/A

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

      11. unpow1 N/A

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

      12. metadata-eval N/A

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

      13. sqrt-pow1 N/A

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

      14. pow2 N/A

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

      15. rem-sqrt-square-rev N/A

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

      16. lift-+.f32 N/A

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

      17. lift--.f32 N/A

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

      18. associate-+l- N/A

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

      19. fabs-sub N/A

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

   4. Applied rewrites 59.0%

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

   5. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower--.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 55.8

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

   10. Applied rewrites 54.9%

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

   11. Taylor expanded in ux around 0

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

   13. Applied rewrites 99.4%

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

   if 1.17000003e-4 < uy

   1. Initial program 55.8%

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

      1. lift-*.f32 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. pow2 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. fabs-sub N/A

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

      11. unpow1 N/A

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

      12. metadata-eval N/A

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

      13. sqrt-pow1 N/A

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

      14. pow2 N/A

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

      15. rem-sqrt-square-rev N/A

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

      16. lift-+.f32 N/A

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

      17. lift--.f32 N/A

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

      18. associate-+l- N/A

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

      19. fabs-sub N/A

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

   4. Applied rewrites 55.8%

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

   5. Taylor expanded in maxCos around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate--r+ N/A

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

      4. lower--.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower-/.f32 54.7

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

   7. Applied rewrites 54.7%

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

   8. Taylor expanded in ux around 0

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

      1. metadata-eval N/A

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

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

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. associate-*r* N/A

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

      7. mul-1-neg N/A

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

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

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower--.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. lower-pow.f32 N/A

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

      16. lower--.f32 98.0

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

   10. Applied rewrites 98.0%

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

   11. Taylor expanded in maxCos around 0

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

   13. Applied rewrites 92.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.00011700000322889537:\\ \;\;\;\;\sqrt{\left(\left(1 - maxCos\right) \cdot 2 - {\left(1 - maxCos\right)}^{2} \cdot ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.6% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.7%

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

   1. lift-*.f32 N/A

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

   2. unpow1 N/A

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

   3. metadata-eval N/A

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

   4. sqrt-pow1 N/A

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

   5. pow2 N/A

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

   6. rem-sqrt-square-rev N/A

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

   7. lift-+.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-+l- N/A

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

   10. fabs-sub N/A

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

   11. unpow1 N/A

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

   12. metadata-eval N/A

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

   13. sqrt-pow1 N/A

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

   14. pow2 N/A

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

   15. rem-sqrt-square-rev N/A

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

   16. lift-+.f32 N/A

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

   17. lift--.f32 N/A

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

   18. associate-+l- N/A

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

   19. fabs-sub N/A

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

4. Applied rewrites 57.8%

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

5. Taylor expanded in maxCos around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate--r+ N/A

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

   4. lower--.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 58.0

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

7. Applied rewrites 58.0%

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

8. Taylor expanded in ux around 0

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

   1. metadata-eval N/A

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

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

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. associate-*r* N/A

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

   7. mul-1-neg N/A

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

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

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

   9. lower--.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower--.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lower-pow.f32 N/A

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

   16. lower--.f32 99.0

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

10. Applied rewrites 99.0%

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

11. Taylor expanded in maxCos around 0

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

13. Applied rewrites 92.7%

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

Alternative 8: 77.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;ux \leq 7.100000220816582 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(\left(1 - maxCos\right) \cdot ux\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\left(\left(uy \cdot uy\right) \cdot -2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= ux 7.100000220816582e-5)
     (sqrt (* (* (- 1.0 maxCos) ux) 2.0))
     (*
      (+ 1.0 (* (* (* (* uy uy) -2.0) (PI)) (PI)))
      (sqrt (- 1.0 (* t_0 t_0)))))))

Derivation

1. Split input into 2 regimes

2. if ux < 7.10000022e-5

   1. Initial program 36.2%

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

      1. lift-*.f32 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow1 N/A

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

      5. pow2 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. fabs-sub N/A

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

      11. unpow1 N/A

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

      12. metadata-eval N/A

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

      13. sqrt-pow1 N/A

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

      14. pow2 N/A

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

      15. rem-sqrt-square-rev N/A

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

      16. lift-+.f32 N/A

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

      17. lift--.f32 N/A

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

      18. associate-+l- N/A

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

      19. fabs-sub N/A

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-/.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower--.f32 N/A

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

      10. unpow2 N/A

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

      11. lower-*.f32 98.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 32.7

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

   10. Applied rewrites 31.1%

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

   11. Taylor expanded in ux around 0

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

   13. Applied rewrites 76.5%

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

   if 7.10000022e-5 < ux

   1. Initial program 87.1%

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

   3. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 75.7

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

   5. Applied rewrites 75.3%

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

   7. Applied rewrites 81.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 7.100000220816582 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(\left(1 - maxCos\right) \cdot ux\right) \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(\left(\left(uy \cdot uy\right) \cdot -2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.9% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

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((((1.0e0 - maxcos) * ux) * 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

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

   1. lift-*.f32 N/A

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

   2. unpow1 N/A

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

   3. metadata-eval N/A

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

   4. sqrt-pow1 N/A

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

   5. pow2 N/A

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

   6. rem-sqrt-square-rev N/A

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

   7. lift-+.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-+l- N/A

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

   10. fabs-sub N/A

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

   11. unpow1 N/A

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

   12. metadata-eval N/A

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

   13. sqrt-pow1 N/A

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

   14. pow2 N/A

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

   15. rem-sqrt-square-rev N/A

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

   16. lift-+.f32 N/A

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

   17. lift--.f32 N/A

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

   18. associate-+l- N/A

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

   19. fabs-sub N/A

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

4. Applied rewrites 57.8%

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

5. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower--.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 98.8

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. lower--.f32 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 48.6

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

10. Applied rewrites 43.4%

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

11. Taylor expanded in ux around 0

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

13. Applied rewrites 65.5%

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

14. Final simplification 65.5%

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

Alternative 10: -0.0% accurate, 7.4× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return (sqrtf(-1.0f) * ux) * 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((-1.0e0)) * ux) * maxcos
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

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

   1. lift-*.f32 N/A

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

   2. unpow1 N/A

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

   3. metadata-eval N/A

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

   4. sqrt-pow1 N/A

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

   5. pow2 N/A

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

   6. rem-sqrt-square-rev N/A

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

   7. lift-+.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-+l- N/A

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

   10. fabs-sub N/A

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

   11. unpow1 N/A

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

   12. metadata-eval N/A

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

   13. sqrt-pow1 N/A

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

   14. pow2 N/A

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

   15. rem-sqrt-square-rev N/A

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

   16. lift-+.f32 N/A

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

   17. lift--.f32 N/A

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

   18. associate-+l- N/A

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

   19. fabs-sub N/A

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

4. Applied rewrites 57.8%

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

5. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower--.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 98.8

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

7. Applied rewrites 98.8%

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

8. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-pow.f32 N/A

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

   4. lower--.f32 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 48.6

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

10. Applied rewrites 45.9%

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

11. Taylor expanded in maxCos around inf

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

13. Applied rewrites -0.0%

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

Reproduce

herbie shell --seed 2024343 
(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)))))))