UniformSampleCone, x

Percentage Accurate: 57.6% → 99.0%

Time: 12.5s

Alternatives: 13

Speedup: 5.8×

Specification

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

Math

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

FPCore

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

Initial Program: 57.6% 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

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

FPCore

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

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in maxCos around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate--l+ N/A

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

   4. div-sub N/A

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

   5. +-commutative N/A

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

   6. lower-+.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower--.f32 55.3

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

5. Applied rewrites 55.3%

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

7. Applied rewrites 31.2%

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

8. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-*.f32 98.6

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

10. Applied rewrites 98.6%

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

11. Final simplification 98.6%

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

Alternative 2: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot maxCos + \left(1 - ux\right)\\ t_1 := \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \cdot t\_1 \leq 0.0215000007301569:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - \left(ux - ux \cdot maxCos\right)\right) \cdot t\_0} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (* ux maxCos) (- 1.0 ux))) (t_1 (cos (* (PI) (* 2.0 uy)))))
   (if (<= (* (sqrt (- 1.0 (* t_0 t_0))) t_1) 0.0215000007301569)
     (* (sqrt (* (fma -2.0 maxCos 2.0) ux)) t_1)
     (* (sqrt (- 1.0 (* (- 1.0 (- ux (* ux maxCos))) t_0))) 1.0))))

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)))))) < 0.0215000007

   1. Initial program 37.9%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. 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. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. div-sub N/A

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

      6. associate-+l- N/A

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

      7. lower--.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f32 38.0

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f32 38.0

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

   4. Applied rewrites 38.0%

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

   5. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

      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 63.2

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

   7. Applied rewrites 63.1%

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

   if 0.0215000007 < (*.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 90.9%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 76.8%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f32 N/A

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

      5. lower--.f32 76.8

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

   7. Applied rewrites 76.8%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \leq 0.0215000007301569:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - \left(ux - ux \cdot maxCos\right)\right) \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot maxCos + \left(1 - ux\right)\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \leq 0.017480000853538513:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \sqrt{\left(6 - 6 \cdot maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{1 - \left(1 - ux\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (* ux maxCos) (- 1.0 ux))))
   (if (<=
        (* (sqrt (- 1.0 (* t_0 t_0))) (cos (* (PI) (* 2.0 uy))))
        0.017480000853538513)
     (* (sqrt 0.3333333333333333) (sqrt (* (- 6.0 (* 6.0 maxCos)) ux)))
     (* 1.0 (sqrt (- 1.0 (* (- 1.0 ux) t_0)))))))

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)))))) < 0.0174800009

   1. Initial program 36.8%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 29.6%

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

   7. Applied rewrites 29.2%

      \[\leadsto 1 \cdot \sqrt{\color{blue}{\frac{1 - {\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}^{6}}{\left({\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}^{4} + 1\right) + {\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}^{2}}}} \]

   8. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\frac{1}{3}}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\frac{1}{3}}\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)}} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\frac{1}{3}}\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{3}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{3}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      5. lower-sqrt.f32 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{3}}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      6. lower-cos.f32 N/A

         \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      8. lower-*.f32 N/A

         \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)} \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)} \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      11. lower-PI.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      12. lower-sqrt.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \color{blue}{\sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)}} \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right) \cdot ux}} \]

      14. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right) \cdot ux}} \]

      15. lower--.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \cdot ux} \]

      16. distribute-rgt-out N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(6 - \color{blue}{maxCos \cdot \left(2 + 4\right)}\right) \cdot ux} \]

      17. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(6 - maxCos \cdot \color{blue}{6}\right) \cdot ux} \]

      18. lower-*.f32 91.5

          \[\leadsto \left(\sqrt{0.3333333333333333} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(6 - \color{blue}{maxCos \cdot 6}\right) \cdot ux} \]

   10. Applied rewrites 91.5%

       \[\leadsto \color{blue}{\left(\sqrt{0.3333333333333333} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(6 - maxCos \cdot 6\right) \cdot ux}} \]

   11. Taylor expanded in uy around 0

       \[\leadsto \sqrt{\frac{1}{3}} \cdot \sqrt{\color{blue}{\left(6 - maxCos \cdot 6\right) \cdot ux}} \]
   12. Step-by-step derivation

   13. Applied rewrites 74.2%

       \[\leadsto \sqrt{0.3333333333333333} \cdot \sqrt{\color{blue}{\left(6 - maxCos \cdot 6\right) \cdot ux}} \]

   if 0.0174800009 < (*.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 89.9%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower--.f32 71.5

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

   8. Applied rewrites 71.5%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \leq 0.017480000853538513:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \sqrt{\left(6 - 6 \cdot maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot maxCos + \left(1 - ux\right)\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \leq 0.017480000853538513:\\ \;\;\;\;\sqrt{\left(\left(6 - 6 \cdot maxCos\right) \cdot ux\right) \cdot 0.3333333333333333} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{1 - \left(1 - ux\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (* ux maxCos) (- 1.0 ux))))
   (if (<=
        (* (sqrt (- 1.0 (* t_0 t_0))) (cos (* (PI) (* 2.0 uy))))
        0.017480000853538513)
     (* (sqrt (* (* (- 6.0 (* 6.0 maxCos)) ux) 0.3333333333333333)) 1.0)
     (* 1.0 (sqrt (- 1.0 (* (- 1.0 ux) t_0)))))))

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)))))) < 0.0174800009

   1. Initial program 36.8%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 29.6%

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

   7. Applied rewrites 29.3%

      \[\leadsto 1 \cdot \sqrt{\color{blue}{\frac{1 - {\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}^{6}}{\left({\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}^{4} + 1\right) + {\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}^{2}}}} \]

   8. Taylor expanded in ux around 0

      \[\leadsto 1 \cdot \sqrt{\color{blue}{\frac{1}{3} \cdot \left(ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\color{blue}{\frac{1}{3} \cdot \left(ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)\right)}} \]

      2. *-commutative N/A

         \[\leadsto 1 \cdot \sqrt{\frac{1}{3} \cdot \color{blue}{\left(\left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right) \cdot ux\right)}} \]

      3. lower-*.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\frac{1}{3} \cdot \color{blue}{\left(\left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right) \cdot ux\right)}} \]

      4. lower--.f32 N/A

         \[\leadsto 1 \cdot \sqrt{\frac{1}{3} \cdot \left(\color{blue}{\left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \cdot ux\right)} \]

      5. distribute-rgt-out N/A

         \[\leadsto 1 \cdot \sqrt{\frac{1}{3} \cdot \left(\left(6 - \color{blue}{maxCos \cdot \left(2 + 4\right)}\right) \cdot ux\right)} \]

      6. metadata-eval N/A

         \[\leadsto 1 \cdot \sqrt{\frac{1}{3} \cdot \left(\left(6 - maxCos \cdot \color{blue}{6}\right) \cdot ux\right)} \]

      7. lower-*.f32 74.2

         \[\leadsto 1 \cdot \sqrt{0.3333333333333333 \cdot \left(\left(6 - \color{blue}{maxCos \cdot 6}\right) \cdot ux\right)} \]

   10. Applied rewrites 74.2%

       \[\leadsto 1 \cdot \sqrt{\color{blue}{0.3333333333333333 \cdot \left(\left(6 - maxCos \cdot 6\right) \cdot ux\right)}} \]

   if 0.0174800009 < (*.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 89.9%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower--.f32 71.5

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

   8. Applied rewrites 71.5%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \leq 0.017480000853538513:\\ \;\;\;\;\sqrt{\left(\left(6 - 6 \cdot maxCos\right) \cdot ux\right) \cdot 0.3333333333333333} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ t_1 := ux \cdot maxCos + \left(1 - ux\right)\\ \mathbf{if}\;t\_1 \leq 0.9998499751091003:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot t\_1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* (PI) (* 2.0 uy)))) (t_1 (+ (* ux maxCos) (- 1.0 ux))))
   (if (<= t_1 0.9998499751091003)
     (* (sqrt (- 1.0 (* (- 1.0 ux) t_1))) t_0)
     (* (sqrt (* (fma -2.0 maxCos 2.0) ux)) t_0))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.999849975

   1. Initial program 89.7%

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

   3. Taylor expanded in maxCos around 0

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

      1. lower--.f32 83.0

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

   5. Applied rewrites 83.0%

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

   if 0.999849975 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 34.3%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. 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. lift--.f32 N/A

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

      3. flip-- N/A

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

      4. metadata-eval N/A

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

      5. div-sub N/A

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

      6. associate-+l- N/A

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

      7. lower--.f32 N/A

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

      8. lower-/.f32 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f32 34.5

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f32 34.5

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

   4. Applied rewrites 34.5%

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

   5. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

      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 66.5

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

   7. Applied rewrites 66.3%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \cdot maxCos + \left(1 - ux\right) \leq 0.9998499751091003:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 2.9000000722589903 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2}{ux} - 1\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(6 - 6 \cdot maxCos\right) \cdot ux} \cdot \left(\cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{0.3333333333333333}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 2.9000000722589903e-5)
   (* (sqrt (* (* ux ux) (- (/ 2.0 ux) 1.0))) (cos (* (PI) (* 2.0 uy))))
   (*
    (sqrt (* (- 6.0 (* 6.0 maxCos)) ux))
    (* (cos (* (* (PI) uy) 2.0)) (sqrt 0.3333333333333333)))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 2.90000007e-5

   1. Initial program 55.3%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate--r+ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower--.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f32 N/A

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

      15. lower-pow.f32 N/A

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

      16. lower--.f32 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f32 70.4

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

   5. Applied rewrites 70.4%

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

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 97.6%

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

   if 2.90000007e-5 < maxCos

   1. Initial program 55.2%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 46.2%

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

   7. Applied rewrites 32.9%

      \[\leadsto 1 \cdot \sqrt{\color{blue}{\frac{1 - {\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}^{6}}{\left({\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}^{4} + 1\right) + {\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}^{2}}}} \]

   8. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\frac{1}{3}}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\frac{1}{3}}\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)}} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\frac{1}{3}}\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{3}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{3}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      5. lower-sqrt.f32 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{3}}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      6. lower-cos.f32 N/A

         \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      8. lower-*.f32 N/A

         \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)} \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)} \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      11. lower-PI.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      12. lower-sqrt.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \color{blue}{\sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)}} \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right) \cdot ux}} \]

      14. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right) \cdot ux}} \]

      15. lower--.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \cdot ux} \]

      16. distribute-rgt-out N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(6 - \color{blue}{maxCos \cdot \left(2 + 4\right)}\right) \cdot ux} \]

      17. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(6 - maxCos \cdot \color{blue}{6}\right) \cdot ux} \]

      18. lower-*.f32 76.2

          \[\leadsto \left(\sqrt{0.3333333333333333} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(6 - \color{blue}{maxCos \cdot 6}\right) \cdot ux} \]

   10. Applied rewrites 76.2%

       \[\leadsto \color{blue}{\left(\sqrt{0.3333333333333333} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(6 - maxCos \cdot 6\right) \cdot ux}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 2.9000000722589903 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2}{ux} - 1\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(6 - 6 \cdot maxCos\right) \cdot ux} \cdot \left(\cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{0.3333333333333333}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.7% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate--r+ N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. div-sub N/A

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

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

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

   9. metadata-eval N/A

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

   10. lower--.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. +-commutative N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. lower-pow.f32 N/A

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

   16. lower--.f32 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f32 65.7

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

5. Applied rewrites 65.7%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 90.2%

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

9. Final simplification 90.2%

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

Alternative 8: 75.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot maxCos + \left(1 - ux\right)\\ \mathbf{if}\;t\_0 \leq 0.9998499751091003:\\ \;\;\;\;\sqrt{1 - \left(1 - \left(ux - ux \cdot maxCos\right)\right) \cdot t\_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \sqrt{\left(6 - 6 \cdot maxCos\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (* ux maxCos) (- 1.0 ux))))
   (if (<= t_0 0.9998499751091003)
     (* (sqrt (- 1.0 (* (- 1.0 (- ux (* ux maxCos))) t_0))) 1.0)
     (* (sqrt 0.3333333333333333) (sqrt (* (- 6.0 (* 6.0 maxCos)) ux))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (ux * maxCos) + (1.0f - ux);
	float tmp;
	if (t_0 <= 0.9998499751091003f) {
		tmp = sqrtf((1.0f - ((1.0f - (ux - (ux * maxCos))) * t_0))) * 1.0f;
	} else {
		tmp = sqrtf(0.3333333333333333f) * sqrtf(((6.0f - (6.0f * maxCos)) * ux));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (ux * maxcos) + (1.0e0 - ux)
    if (t_0 <= 0.9998499751091003e0) then
        tmp = sqrt((1.0e0 - ((1.0e0 - (ux - (ux * maxcos))) * t_0))) * 1.0e0
    else
        tmp = sqrt(0.3333333333333333e0) * sqrt(((6.0e0 - (6.0e0 * maxcos)) * ux))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(ux * maxCos) + Float32(Float32(1.0) - ux))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9998499751091003))
		tmp = Float32(sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - Float32(ux - Float32(ux * maxCos))) * t_0))) * Float32(1.0));
	else
		tmp = Float32(sqrt(Float32(0.3333333333333333)) * sqrt(Float32(Float32(Float32(6.0) - Float32(Float32(6.0) * maxCos)) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = (ux * maxCos) + (single(1.0) - ux);
	tmp = single(0.0);
	if (t_0 <= single(0.9998499751091003))
		tmp = sqrt((single(1.0) - ((single(1.0) - (ux - (ux * maxCos))) * t_0))) * single(1.0);
	else
		tmp = sqrt(single(0.3333333333333333)) * sqrt(((single(6.0) - (single(6.0) * maxCos)) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.999849975

   1. Initial program 89.7%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 71.0%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f32 N/A

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

      5. lower--.f32 71.1

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

   7. Applied rewrites 71.1%

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

   if 0.999849975 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 34.3%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 30.5%

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

   7. Applied rewrites 30.2%

      \[\leadsto 1 \cdot \sqrt{\color{blue}{\frac{1 - {\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}^{6}}{\left({\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}^{4} + 1\right) + {\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}^{2}}}} \]

   8. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\frac{1}{3}}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\frac{1}{3}}\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)}} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\frac{1}{3}}\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{3}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{3}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      5. lower-sqrt.f32 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{3}}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      6. lower-cos.f32 N/A

         \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      8. lower-*.f32 N/A

         \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)} \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)} \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      11. lower-PI.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      12. lower-sqrt.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \color{blue}{\sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)}} \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right) \cdot ux}} \]

      14. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right) \cdot ux}} \]

      15. lower--.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \cdot ux} \]

      16. distribute-rgt-out N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(6 - \color{blue}{maxCos \cdot \left(2 + 4\right)}\right) \cdot ux} \]

      17. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(6 - maxCos \cdot \color{blue}{6}\right) \cdot ux} \]

      18. lower-*.f32 93.5

          \[\leadsto \left(\sqrt{0.3333333333333333} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(6 - \color{blue}{maxCos \cdot 6}\right) \cdot ux} \]

   10. Applied rewrites 93.5%

       \[\leadsto \color{blue}{\left(\sqrt{0.3333333333333333} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(6 - maxCos \cdot 6\right) \cdot ux}} \]

   11. Taylor expanded in uy around 0

       \[\leadsto \sqrt{\frac{1}{3}} \cdot \sqrt{\color{blue}{\left(6 - maxCos \cdot 6\right) \cdot ux}} \]
   12. Step-by-step derivation

   13. Applied rewrites 77.4%

       \[\leadsto \sqrt{0.3333333333333333} \cdot \sqrt{\color{blue}{\left(6 - maxCos \cdot 6\right) \cdot ux}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \cdot maxCos + \left(1 - ux\right) \leq 0.9998499751091003:\\ \;\;\;\;\sqrt{1 - \left(1 - \left(ux - ux \cdot maxCos\right)\right) \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \sqrt{\left(6 - 6 \cdot maxCos\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot maxCos + \left(1 - ux\right)\\ \mathbf{if}\;t\_0 \leq 0.9998499751091003:\\ \;\;\;\;1 \cdot \sqrt{1 - t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \sqrt{\left(6 - 6 \cdot maxCos\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (* ux maxCos) (- 1.0 ux))))
   (if (<= t_0 0.9998499751091003)
     (* 1.0 (sqrt (- 1.0 (* t_0 t_0))))
     (* (sqrt 0.3333333333333333) (sqrt (* (- 6.0 (* 6.0 maxCos)) ux))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (ux * maxCos) + (1.0f - ux);
	float tmp;
	if (t_0 <= 0.9998499751091003f) {
		tmp = 1.0f * sqrtf((1.0f - (t_0 * t_0)));
	} else {
		tmp = sqrtf(0.3333333333333333f) * sqrtf(((6.0f - (6.0f * maxCos)) * ux));
	}
	return tmp;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (ux * maxcos) + (1.0e0 - ux)
    if (t_0 <= 0.9998499751091003e0) then
        tmp = 1.0e0 * sqrt((1.0e0 - (t_0 * t_0)))
    else
        tmp = sqrt(0.3333333333333333e0) * sqrt(((6.0e0 - (6.0e0 * maxcos)) * ux))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(ux * maxCos) + Float32(Float32(1.0) - ux))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9998499751091003))
		tmp = Float32(Float32(1.0) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))));
	else
		tmp = Float32(sqrt(Float32(0.3333333333333333)) * sqrt(Float32(Float32(Float32(6.0) - Float32(Float32(6.0) * maxCos)) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = (ux * maxCos) + (single(1.0) - ux);
	tmp = single(0.0);
	if (t_0 <= single(0.9998499751091003))
		tmp = single(1.0) * sqrt((single(1.0) - (t_0 * t_0)));
	else
		tmp = sqrt(single(0.3333333333333333)) * sqrt(((single(6.0) - (single(6.0) * maxCos)) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.999849975

   1. Initial program 89.7%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 71.0%

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

   if 0.999849975 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 34.3%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 30.5%

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

   7. Applied rewrites 30.2%

      \[\leadsto 1 \cdot \sqrt{\color{blue}{\frac{1 - {\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}^{6}}{\left({\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}^{4} + 1\right) + {\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}^{2}}}} \]

   8. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \cdot \left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\frac{1}{3}}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\frac{1}{3}}\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)}} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\frac{1}{3}}\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{3}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{3}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      5. lower-sqrt.f32 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{3}}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      6. lower-cos.f32 N/A

         \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      8. lower-*.f32 N/A

         \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)} \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)} \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      11. lower-PI.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \]

      12. lower-sqrt.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \color{blue}{\sqrt{ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)}} \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right) \cdot ux}} \]

      14. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right) \cdot ux}} \]

      15. lower--.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \cdot ux} \]

      16. distribute-rgt-out N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(6 - \color{blue}{maxCos \cdot \left(2 + 4\right)}\right) \cdot ux} \]

      17. metadata-eval N/A

          \[\leadsto \left(\sqrt{\frac{1}{3}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(6 - maxCos \cdot \color{blue}{6}\right) \cdot ux} \]

      18. lower-*.f32 93.5

          \[\leadsto \left(\sqrt{0.3333333333333333} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(6 - \color{blue}{maxCos \cdot 6}\right) \cdot ux} \]

   10. Applied rewrites 93.5%

       \[\leadsto \color{blue}{\left(\sqrt{0.3333333333333333} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(6 - maxCos \cdot 6\right) \cdot ux}} \]

   11. Taylor expanded in uy around 0

       \[\leadsto \sqrt{\frac{1}{3}} \cdot \sqrt{\color{blue}{\left(6 - maxCos \cdot 6\right) \cdot ux}} \]
   12. Step-by-step derivation

   13. Applied rewrites 77.4%

       \[\leadsto \sqrt{0.3333333333333333} \cdot \sqrt{\color{blue}{\left(6 - maxCos \cdot 6\right) \cdot ux}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \cdot maxCos + \left(1 - ux\right) \leq 0.9998499751091003:\\ \;\;\;\;1 \cdot \sqrt{1 - \left(ux \cdot maxCos + \left(1 - ux\right)\right) \cdot \left(ux \cdot maxCos + \left(1 - ux\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.3333333333333333} \cdot \sqrt{\left(6 - 6 \cdot maxCos\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.6% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(6 - 6 \cdot maxCos\right) \cdot ux\right) \cdot 0.3333333333333333} \cdot 1 \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (sqrt (* (* (- 6.0 (* 6.0 maxCos)) ux) 0.3333333333333333)) 1.0))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((((6.0f - (6.0f * maxCos)) * ux) * 0.3333333333333333f)) * 1.0f;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((((6.0e0 - (6.0e0 * maxcos)) * ux) * 0.3333333333333333e0)) * 1.0e0
end function

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32(Float32(Float32(6.0) - Float32(Float32(6.0) * maxCos)) * ux) * Float32(0.3333333333333333))) * Float32(1.0))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((((single(6.0) - (single(6.0) * maxCos)) * ux) * single(0.3333333333333333))) * single(1.0);
end

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in uy around 0

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

5. Applied rewrites 45.9%

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

7. Applied rewrites 43.7%

   \[\leadsto 1 \cdot \sqrt{\color{blue}{\frac{1 - {\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}^{6}}{\left({\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}^{4} + 1\right) + {\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right)}^{2}}}} \]

8. Taylor expanded in ux around 0

   \[\leadsto 1 \cdot \sqrt{\color{blue}{\frac{1}{3} \cdot \left(ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)\right)}} \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto 1 \cdot \sqrt{\color{blue}{\frac{1}{3} \cdot \left(ux \cdot \left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)\right)}} \]

   2. *-commutative N/A

      \[\leadsto 1 \cdot \sqrt{\frac{1}{3} \cdot \color{blue}{\left(\left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right) \cdot ux\right)}} \]

   3. lower-*.f32 N/A

      \[\leadsto 1 \cdot \sqrt{\frac{1}{3} \cdot \color{blue}{\left(\left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right) \cdot ux\right)}} \]

   4. lower--.f32 N/A

      \[\leadsto 1 \cdot \sqrt{\frac{1}{3} \cdot \left(\color{blue}{\left(6 - \left(2 \cdot maxCos + 4 \cdot maxCos\right)\right)} \cdot ux\right)} \]

   5. distribute-rgt-out N/A

      \[\leadsto 1 \cdot \sqrt{\frac{1}{3} \cdot \left(\left(6 - \color{blue}{maxCos \cdot \left(2 + 4\right)}\right) \cdot ux\right)} \]

   6. metadata-eval N/A

      \[\leadsto 1 \cdot \sqrt{\frac{1}{3} \cdot \left(\left(6 - maxCos \cdot \color{blue}{6}\right) \cdot ux\right)} \]

   7. lower-*.f32 65.6

      \[\leadsto 1 \cdot \sqrt{0.3333333333333333 \cdot \left(\left(6 - \color{blue}{maxCos \cdot 6}\right) \cdot ux\right)} \]

10. Applied rewrites 65.6%

    \[\leadsto 1 \cdot \sqrt{\color{blue}{0.3333333333333333 \cdot \left(\left(6 - maxCos \cdot 6\right) \cdot ux\right)}} \]

11. Final simplification 65.6%

    \[\leadsto \sqrt{\left(\left(6 - 6 \cdot maxCos\right) \cdot ux\right) \cdot 0.3333333333333333} \cdot 1 \]
12. Add Preprocessing

Alternative 11: 62.1% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in uy around 0

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

5. Applied rewrites 45.9%

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

6. Taylor expanded in ux around 0

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

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

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 62.0

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

8. Applied rewrites 61.8%

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

9. Final simplification 60.0%

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

Alternative 12: 17.7% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in uy around 0

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

5. Applied rewrites 45.9%

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

6. Taylor expanded in ux around 0

   \[\leadsto 1 \cdot \sqrt{1 - \color{blue}{1}} \]
7. Step-by-step derivation

8. Applied rewrites 6.6%

   \[\leadsto 1 \cdot \sqrt{1 - \color{blue}{1}} \]
9. Step-by-step derivation

   1. lift--.f32 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. neg-mul-1 N/A

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

   5. lower-fma.f32 19.3

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

10. Applied rewrites 19.8%

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

11. Final simplification 19.4%

    \[\leadsto \sqrt{\mathsf{fma}\left(-1, 1, 1\right)} \cdot 1 \]
12. Add Preprocessing

Alternative 13: 6.6% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{1 - 1} \cdot 1 \end{array} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in uy around 0

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

5. Applied rewrites 45.9%

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

6. Taylor expanded in ux around 0

   \[\leadsto 1 \cdot \sqrt{1 - \color{blue}{1}} \]
7. Step-by-step derivation

8. Applied rewrites 6.6%

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

9. Final simplification 6.6%

   \[\leadsto \sqrt{1 - 1} \cdot 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024271 
(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)))))))