UniformSampleCone, x

Percentage Accurate: 57.8% → 98.8%

Time: 11.0s

Alternatives: 10

Speedup: 3.3×

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: 57.8% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{if}\;maxCos \leq 5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(maxCos \cdot maxCos\right) \cdot \left(\left(\frac{\frac{\frac{2}{ux} - 1}{maxCos}}{maxCos} - \left(1 - \frac{2 - \frac{2}{ux}}{maxCos}\right)\right) \cdot \left(ux \cdot ux\right)\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* (PI) (* 2.0 uy)))))
   (if (<= maxCos 5.00000006675716e-11)
     (* (sqrt (* (- 2.0 ux) ux)) t_0)
     (*
      (sqrt
       (*
        (* maxCos maxCos)
        (*
         (-
          (/ (/ (- (/ 2.0 ux) 1.0) maxCos) maxCos)
          (- 1.0 (/ (- 2.0 (/ 2.0 ux)) maxCos)))
         (* ux ux))))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 5.00000007e-11

   1. Initial program 60.3%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate--r+ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower--.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. +-commutative N/A

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

      13. 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 71.9

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

   5. Applied rewrites 71.9%

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

   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 98.6%

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

   9. Taylor expanded in ux around 0

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

   11. Applied rewrites 98.8%

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

   if 5.00000007e-11 < maxCos

   1. Initial program 59.5%

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

   3. Taylor expanded in 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 53.6

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

   5. Applied rewrites 53.6%

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

   7. Applied rewrites 98.7%

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

   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 98.8%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(maxCos \cdot maxCos\right) \cdot \left(\left(\frac{\frac{\frac{2}{ux} - 1}{maxCos}}{maxCos} - \left(1 - \frac{2 - \frac{2}{ux}}{maxCos}\right)\right) \cdot \left(ux \cdot ux\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 60.1%

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

3. Taylor expanded in 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 67.9

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

5. Applied rewrites 67.9%

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

7. Applied rewrites 98.6%

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

8. Final simplification 98.6%

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

Alternative 3: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{if}\;maxCos \leq 5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{\frac{2}{ux} - 1}{maxCos \cdot maxCos} + \frac{2 - \frac{2}{ux}}{maxCos}\right) \cdot \left(ux \cdot ux\right) - ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* (PI) (* 2.0 uy)))))
   (if (<= maxCos 5.00000006675716e-11)
     (* (sqrt (* (- 2.0 ux) ux)) t_0)
     (*
      (sqrt
       (*
        (-
         (*
          (+
           (/ (- (/ 2.0 ux) 1.0) (* maxCos maxCos))
           (/ (- 2.0 (/ 2.0 ux)) maxCos))
          (* ux ux))
         (* ux ux))
        (* maxCos maxCos)))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 5.00000007e-11

   1. Initial program 60.3%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate--r+ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower--.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. +-commutative N/A

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

      13. 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 71.9

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

   5. Applied rewrites 71.9%

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

   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 98.6%

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

   9. Taylor expanded in ux around 0

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

   11. Applied rewrites 98.8%

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

   if 5.00000007e-11 < maxCos

   1. Initial program 59.5%

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

   3. Taylor expanded in 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 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in maxCos around inf

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

   8. Applied rewrites 98.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 5.00000006675716 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{\frac{2}{ux} - 1}{maxCos \cdot maxCos} + \frac{2 - \frac{2}{ux}}{maxCos}\right) \cdot \left(ux \cdot ux\right) - ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{\left(\frac{2}{ux} - \left(\frac{maxCos}{ux} - \left(maxCos - 1\right)\right)\right) \cdot \left(ux \cdot ux\right) - \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot maxCos\right) \cdot ux} \cdot \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot ux} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.0024999999441206455)
   (*
    (sqrt
     (-
      (* (- (/ 2.0 ux) (- (/ maxCos ux) (- maxCos 1.0))) (* ux ux))
      (* (* (fma maxCos ux (- 1.0 ux)) maxCos) ux)))
    (fma (* (* uy uy) -2.0) (* (PI) (PI)) 1.0))
   (* (sqrt (* 2.0 ux)) (cos (* (PI) (* 2.0 uy))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00249999994

   1. Initial program 59.8%

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

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 58.9

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

   5. Applied rewrites 58.4%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. distribute-lft-in N/A

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

      5. associate--r+ N/A

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

      6. *-commutative N/A

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

      7. lower--.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lift-+.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f32 N/A

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

      15. *-commutative N/A

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

   7. Applied rewrites 26.2%

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

   8. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f32 N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. lower--.f32 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f32 96.2

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

   10. Applied rewrites 96.2%

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

   if 0.00249999994 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 60.8%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 18.2

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

   5. Applied rewrites 18.9%

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

   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 90.2%

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

   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 70.3%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0024999999441206455:\\ \;\;\;\;\sqrt{\left(\frac{2}{ux} - \left(\frac{maxCos}{ux} - \left(maxCos - 1\right)\right)\right) \cdot \left(ux \cdot ux\right) - \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot maxCos\right) \cdot ux} \cdot \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot ux} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.2% 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 60.1%

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

3. Taylor expanded in 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 67.9

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

5. Applied rewrites 67.9%

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

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 92.3%

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

9. Taylor expanded in ux around 0

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

11. Applied rewrites 92.5%

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

12. Final simplification 92.5%

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

Alternative 6: 79.1% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 60.1%

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-PI.f32 49.9

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

5. Applied rewrites 49.7%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. distribute-lft-in N/A

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

   5. associate--r+ N/A

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

   6. *-commutative N/A

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

   7. lower--.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-+.f32 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. *-commutative N/A

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

7. Applied rewrites 22.2%

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

8. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f32 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. unsub-neg N/A

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

   10. lower--.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower--.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 75.6

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

10. Applied rewrites 78.0%

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

11. Final simplification 76.9%

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

Alternative 7: 71.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := maxCos \cdot ux + \left(1 - ux\right)\\ t_1 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;1 - t\_0 \cdot t\_0 \leq 0.00013000000035390258:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot \mathsf{fma}\left(uy, \left(t\_1 \cdot -2\right) \cdot uy, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot t\_0} \cdot \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, t\_1, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (* maxCos ux) (- 1.0 ux))) (t_1 (* (PI) (PI))))
   (if (<= (- 1.0 (* t_0 t_0)) 0.00013000000035390258)
     (* (sqrt (* (fma -2.0 maxCos 2.0) ux)) (fma uy (* (* t_1 -2.0) uy) 1.0))
     (* (sqrt (- 1.0 (* (- 1.0 ux) t_0))) (fma (* (* uy uy) -2.0) t_1 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)))) < 1.3e-4

   1. Initial program 34.5%

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

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 31.2

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

   5. Applied rewrites 30.9%

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

   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 67.8

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

   8. Applied rewrites 71.6%

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

   10. Applied rewrites 69.9%

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

   if 1.3e-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 88.2%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 70.6

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower--.f32 63.8

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

   8. Applied rewrites 68.0%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \left(maxCos \cdot ux + \left(1 - ux\right)\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right) \leq 0.00013000000035390258:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot \mathsf{fma}\left(uy, \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -2\right) \cdot uy, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot \mathsf{fma}\left(\left(uy \cdot uy\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.5% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 60.1%

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

3. Taylor expanded in 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 67.9

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

5. Applied rewrites 67.9%

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

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 92.3%

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

9. Taylor expanded in uy around 0

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

    1. +-commutative N/A

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

    2. associate-*r* N/A

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

    3. lower-fma.f32 N/A

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

    4. *-commutative N/A

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

    5. lower-*.f32 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f32 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f32 N/A

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

    10. lower-PI.f32 N/A

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

    11. lower-PI.f32 74.4

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

11. Applied rewrites 74.1%

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

12. Final simplification 74.4%

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

Alternative 9: 61.9% accurate, 3.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 60.1%

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-PI.f32 49.9

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

5. Applied rewrites 49.7%

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

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 58.4

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

8. Applied rewrites 59.7%

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

10. Applied rewrites 59.9%

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

11. Final simplification 58.0%

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

Alternative 10: 6.6% accurate, 3.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 60.1%

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

3. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-PI.f32 49.9

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

5. Applied rewrites 49.9%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 25.5%

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

9. Taylor expanded in ux around 0

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

11. Applied rewrites 6.6%

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

12. Final simplification 6.6%

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

Reproduce

herbie shell --seed 2024285 
(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)))))))