UniformSampleCone, x

Percentage Accurate: 57.3% → 99.0%

Time: 11.8s

Alternatives: 12

Speedup: 3.7×

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: 57.3% 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

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

FPCore

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

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   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. cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

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

   8. unsub-neg N/A

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

   9. lower--.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-pow.f32 N/A

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

   13. lower--.f32 3.2

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

5. Applied rewrites 2.8%

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

7. Applied rewrites 98.9%

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

8. Final simplification 98.9%

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

Alternative 2: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{if}\;maxCos \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(maxCos \cdot maxCos\right) \cdot \left(\left(\frac{2 - \frac{2}{ux}}{maxCos} + \frac{\frac{\frac{2}{ux} - 1}{maxCos}}{maxCos}\right) \cdot \left(ux \cdot ux\right) - ux \cdot ux\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* (PI) (* 2.0 uy)))))
   (if (<= maxCos 9.99999993922529e-9)
     (* (sqrt (* (- 2.0 ux) ux)) t_0)
     (*
      (sqrt
       (*
        (* maxCos maxCos)
        (-
         (*
          (+
           (/ (- 2.0 (/ 2.0 ux)) maxCos)
           (/ (/ (- (/ 2.0 ux) 1.0) maxCos) maxCos))
          (* ux ux))
         (* ux ux))))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 9.99999994e-9

   1. Initial program 56.0%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      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. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-pow.f32 N/A

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

      13. lower--.f32 0.7

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

   5. Applied rewrites 2.2%

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

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 99.1%

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

   if 9.99999994e-9 < maxCos

   1. Initial program 61.4%

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

   3. Taylor expanded in 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. lower-fma.f32 N/A

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

      14. lower-pow.f32 N/A

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

      15. lower--.f32 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f32 47.0

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

   5. Applied rewrites 47.0%

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in maxCos around inf

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

   10. Applied rewrites 98.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 9.99999993922529 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(maxCos \cdot maxCos\right) \cdot \left(\left(\frac{2 - \frac{2}{ux}}{maxCos} + \frac{\frac{\frac{2}{ux} - 1}{maxCos}}{maxCos}\right) \cdot \left(ux \cdot ux\right) - ux \cdot ux\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{if}\;maxCos \leq 8.599999823388771 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{ux \cdot 2 - \left(2 - \frac{2 - ux}{maxCos}\right)}{maxCos} - ux\right) \cdot \left(maxCos \cdot maxCos\right)\right) \cdot ux} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* (PI) (* 2.0 uy)))))
   (if (<= maxCos 8.599999823388771e-9)
     (* (sqrt (* (- 2.0 ux) ux)) t_0)
     (*
      (sqrt
       (*
        (*
         (- (/ (- (* ux 2.0) (- 2.0 (/ (- 2.0 ux) maxCos))) maxCos) ux)
         (* maxCos maxCos))
        ux))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 8.59999982e-9

   1. Initial program 56.3%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      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. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-pow.f32 N/A

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

      13. lower--.f32 0.7

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

   5. Applied rewrites 1.2%

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

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 99.1%

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

   if 8.59999982e-9 < maxCos

   1. Initial program 60.4%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      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. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-pow.f32 N/A

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

      13. lower--.f32 7.6

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

   5. Applied rewrites 7.6%

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

   6. Taylor expanded in maxCos around -inf

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

   8. Applied rewrites 98.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 8.599999823388771 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{ux \cdot 2 - \left(2 - \frac{2 - ux}{maxCos}\right)}{maxCos} - ux\right) \cdot \left(maxCos \cdot maxCos\right)\right) \cdot ux} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{\left(\frac{2}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \cdot \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot 2} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.003000000026077032)
   (*
    (sqrt (* (- (/ 2.0 ux) 1.0) (* ux ux)))
    (fma (* (* uy uy) -2.0) (* (PI) (PI)) 1.0))
   (* (sqrt (* ux 2.0)) (cos (* (PI) (* 2.0 uy))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00300000003

   1. Initial program 56.5%

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

   3. Taylor expanded in 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. lower-fma.f32 N/A

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

      14. lower-pow.f32 N/A

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

      15. lower--.f32 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f32 92.4

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

   5. Applied rewrites 92.4%

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

   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 94.1%

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

   9. Taylor expanded in uy around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f32 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f32 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f32 N/A

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

       8. unpow2 N/A

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

       9. lower-*.f32 N/A

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

       10. lower-PI.f32 N/A

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

       11. lower-PI.f32 91.7

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

   11. Applied rewrites 91.1%

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

   if 0.00300000003 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 58.8%

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

   3. 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. lower-fma.f32 N/A

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

      14. lower-pow.f32 N/A

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

      15. lower--.f32 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f32 11.2

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

   5. Applied rewrites 11.2%

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

   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 92.9%

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

   9. Taylor expanded in ux around 0

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

   11. Applied rewrites 73.7%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{\left(\frac{2}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \cdot \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot 2} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.0% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   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. cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

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

   8. unsub-neg N/A

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

   9. lower--.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-pow.f32 N/A

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

   13. lower--.f32 3.7

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

5. Applied rewrites 2.6%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 93.9%

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

9. Final simplification 93.9%

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

Alternative 6: 73.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(uy \cdot uy\right) \cdot -2\\ t_1 := maxCos \cdot ux + \left(1 - ux\right)\\ \mathbf{if}\;1 - t\_1 \cdot t\_1 \leq 0.0007999999797903001:\\ \;\;\;\;\sqrt{\left(-2 \cdot maxCos + 2\right) \cdot ux} \cdot \mathsf{fma}\left(t\_0, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot \mathsf{fma}\left(t\_0 \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* uy uy) -2.0)) (t_1 (+ (* maxCos ux) (- 1.0 ux))))
   (if (<= (- 1.0 (* t_1 t_1)) 0.0007999999797903001)
     (* (sqrt (* (+ (* -2.0 maxCos) 2.0) ux)) (fma t_0 (* (PI) (PI)) 1.0))
     (*
      (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 (- ux (* maxCos ux))))))
      (fma (* t_0 (PI)) (PI) 1.0)))))

Derivation

1. Split input into 2 regimes

2. if (-.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)))) < 7.9999998e-4

   1. Initial program 37.6%

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

   3. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 32.2

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

   5. Applied rewrites 32.2%

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

   6. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 70.9

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

   8. Applied rewrites 70.9%

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

   10. Applied rewrites 73.3%

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

   if 7.9999998e-4 < (-.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 90.4%

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

   3. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 74.3

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 73.4%

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

   8. Taylor expanded in maxCos around 0

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

      1. lower--.f32 72.4

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

   10. Applied rewrites 71.4%

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

       1. lift-+.f32 N/A

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

       2. lift--.f32 N/A

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

       3. associate-+l- N/A

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

       4. lower--.f32 N/A

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

       5. lower--.f32 72.4

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

   12. Applied rewrites 71.4%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \left(maxCos \cdot ux + \left(1 - ux\right)\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right) \leq 0.0007999999797903001:\\ \;\;\;\;\sqrt{\left(-2 \cdot maxCos + 2\right) \cdot ux} \cdot \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot -2\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(uy \cdot uy\right) \cdot -2\\ t_1 := maxCos \cdot ux + \left(1 - ux\right)\\ \mathbf{if}\;1 - t\_1 \cdot t\_1 \leq 0.0007999999797903001:\\ \;\;\;\;\sqrt{\left(-2 \cdot maxCos + 2\right) \cdot ux} \cdot \mathsf{fma}\left(t\_0, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot t\_1} \cdot \mathsf{fma}\left(t\_0 \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* uy uy) -2.0)) (t_1 (+ (* maxCos ux) (- 1.0 ux))))
   (if (<= (- 1.0 (* t_1 t_1)) 0.0007999999797903001)
     (* (sqrt (* (+ (* -2.0 maxCos) 2.0) ux)) (fma t_0 (* (PI) (PI)) 1.0))
     (* (sqrt (- 1.0 (* (- 1.0 ux) t_1))) (fma (* t_0 (PI)) (PI) 1.0)))))

Derivation

1. Split input into 2 regimes

2. if (-.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)))) < 7.9999998e-4

   1. Initial program 37.6%

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

   3. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 32.2

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

   5. Applied rewrites 32.2%

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

   6. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 70.9

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

   8. Applied rewrites 70.9%

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

   10. Applied rewrites 73.3%

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

   if 7.9999998e-4 < (-.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 90.4%

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

   3. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 74.3

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 73.4%

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

   8. Taylor expanded in maxCos around 0

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

      1. lower--.f32 72.4

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

   10. Applied rewrites 71.4%

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

   12. Applied rewrites 71.4%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \left(maxCos \cdot ux + \left(1 - ux\right)\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right) \leq 0.0007999999797903001:\\ \;\;\;\;\sqrt{\left(-2 \cdot maxCos + 2\right) \cdot ux} \cdot \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot -2\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(uy \cdot uy\right) \cdot -2\\ t_1 := maxCos \cdot ux + \left(1 - ux\right)\\ \mathbf{if}\;1 - t\_1 \cdot t\_1 \leq 0.0007999999797903001:\\ \;\;\;\;\sqrt{\left(-2 \cdot maxCos + 2\right) \cdot ux} \cdot \mathsf{fma}\left(t\_0, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \mathsf{fma}\left(t\_0 \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* uy uy) -2.0)) (t_1 (+ (* maxCos ux) (- 1.0 ux))))
   (if (<= (- 1.0 (* t_1 t_1)) 0.0007999999797903001)
     (* (sqrt (* (+ (* -2.0 maxCos) 2.0) ux)) (fma t_0 (* (PI) (PI)) 1.0))
     (*
      (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux))))
      (fma (* t_0 (PI)) (PI) 1.0)))))

Derivation

1. Split input into 2 regimes

2. if (-.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)))) < 7.9999998e-4

   1. Initial program 37.6%

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

   3. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 32.2

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

   5. Applied rewrites 32.2%

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

   6. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 70.9

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

   8. Applied rewrites 70.3%

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

   10. Applied rewrites 72.7%

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

   if 7.9999998e-4 < (-.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 90.4%

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

   3. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 74.3

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

   5. Applied rewrites 73.4%

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

   7. Applied rewrites 73.4%

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

   8. Taylor expanded in maxCos around 0

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

      1. lower--.f32 72.4

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

   10. Applied rewrites 71.4%

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

   11. Taylor expanded in maxCos around 0

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

       1. lower--.f32 72.1

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

   13. Applied rewrites 72.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \left(maxCos \cdot ux + \left(1 - ux\right)\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right) \leq 0.0007999999797903001:\\ \;\;\;\;\sqrt{\left(-2 \cdot maxCos + 2\right) \cdot ux} \cdot \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot -2\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.2% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in 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. lower-fma.f32 N/A

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

   14. lower-pow.f32 N/A

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

   15. lower--.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 67.7

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

5. Applied rewrites 67.7%

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

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 93.8%

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

9. Taylor expanded in uy around 0

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

    3. lower-fma.f32 N/A

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

    4. *-commutative N/A

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

    5. lower-*.f32 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f32 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f32 N/A

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

    10. lower-PI.f32 N/A

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

    11. lower-PI.f32 75.1

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

11. Applied rewrites 74.8%

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

12. Final simplification 75.1%

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

Alternative 10: 64.4% accurate, 3.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-PI.f32 47.9

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

5. Applied rewrites 47.9%

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

6. Taylor expanded in ux around 0

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

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

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 61.5

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

8. Applied rewrites 61.5%

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

10. Applied rewrites 63.2%

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

11. Final simplification 63.3%

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

Alternative 11: 61.9% accurate, 3.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-PI.f32 47.9

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

5. Applied rewrites 47.5%

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

6. Taylor expanded in ux around 0

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

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

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 61.5

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

8. Applied rewrites 61.5%

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

9. Taylor expanded in maxCos around 0

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

11. Applied rewrites 61.4%

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

13. Applied rewrites 61.4%

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

14. Final simplification 61.5%

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

Alternative 12: 5.0% accurate, 3.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.2%

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-PI.f32 47.9

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

5. Applied rewrites 47.5%

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

6. Taylor expanded in ux around 0

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

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

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 61.5

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

8. Applied rewrites 61.4%

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

9. Taylor expanded in maxCos around 0

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

11. Applied rewrites 61.4%

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

12. Taylor expanded in uy around inf

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

14. Applied rewrites 5.3%

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

Reproduce

herbie shell --seed 2024332 
(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)))))))