UniformSampleCone, x

Percentage Accurate: 57.3% → 98.8%

Time: 11.9s

Alternatives: 14

Speedup: 5.8×

Specification

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

Math

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

FPCore

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

Initial Program: 57.3% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\left(1 - ux\right) \cdot ux + ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(maxCos \cdot maxCos\right) \cdot \left(\frac{\left(\frac{\frac{2}{maxCos}}{ux} + 2\right) - \left(\frac{2}{ux} - \frac{-1}{maxCos}\right)}{maxCos} \cdot \left(ux \cdot ux\right) - ux \cdot ux\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.999999987845058e-8)
   (* (cos (* (* (PI) uy) 2.0)) (sqrt (+ (* (- 1.0 ux) ux) ux)))
   (*
    (sqrt
     (*
      (* maxCos maxCos)
      (-
       (*
        (/
         (- (+ (/ (/ 2.0 maxCos) ux) 2.0) (- (/ 2.0 ux) (/ -1.0 maxCos)))
         maxCos)
        (* ux ux))
       (* ux ux))))
    (cos (* (PI) (* 2.0 uy))))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999999e-8

   1. Initial program 60.5%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 52.4%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

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

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

      4. lift-+.f32 N/A

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

      5. lift--.f32 N/A

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

      6. sub-neg N/A

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

      7. lift-neg.f32 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. associate-+r+ N/A

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

      13. lift-+.f32 N/A

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

      14. lift-+.f32 N/A

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

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

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

      16. lift-+.f32 N/A

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

      17. distribute-rgt-in N/A

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

   7. Applied rewrites 52.3%

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

   8. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. cancel-sign-sub N/A

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

      7. lower-+.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-cos.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 98.8

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

   10. Applied rewrites 98.8%

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

   if 1.99999999e-8 < maxCos

   1. Initial program 58.1%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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. *-commutative N/A

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

      14. lower-fma.f32 N/A

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

      15. lower-pow.f32 N/A

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

      16. lower--.f32 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f32 56.3

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

   5. Applied rewrites 56.3%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(\frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux} - {\left(maxCos - 1\right)}^{2}\right) \cdot \left(ux \cdot ux\right)}} \]
   6. 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. 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 98.5%

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

   11. Taylor expanded in maxCos around inf

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

   13. Applied rewrites 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 1.999999987845058 \cdot 10^{-8}:\\ \;\;\;\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\left(1 - ux\right) \cdot ux + ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(maxCos \cdot maxCos\right) \cdot \left(\frac{\left(\frac{\frac{2}{maxCos}}{ux} + 2\right) - \left(\frac{2}{ux} - \frac{-1}{maxCos}\right)}{maxCos} \cdot \left(ux \cdot ux\right) - ux \cdot ux\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \leq 0.9999997019767761:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{\left(\left(\frac{-2 \cdot maxCos + 2}{ux} - {\left(maxCos - 1\right)}^{2}\right) \cdot ux\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* (PI) (* 2.0 uy))) 0.9999997019767761)
   (* (sqrt (* (- 2.0 ux) ux)) (cos (* (* (PI) uy) 2.0)))
   (*
    1.0
    (sqrt
     (*
      (* (- (/ (+ (* -2.0 maxCos) 2.0) ux) (pow (- maxCos 1.0) 2.0)) ux)
      ux)))))

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.999999702

   1. Initial program 61.4%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 42.9%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

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

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

      4. lift-+.f32 N/A

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

      5. lift--.f32 N/A

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

      6. sub-neg N/A

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

      7. lift-neg.f32 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. associate-+r+ N/A

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

      13. lift-+.f32 N/A

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

      14. lift-+.f32 N/A

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

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

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

      16. lift-+.f32 N/A

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

      17. distribute-rgt-in N/A

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

   7. Applied rewrites 41.0%

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

   8. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. cancel-sign-sub N/A

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

      7. lower-+.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-cos.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 93.9

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

   10. Applied rewrites 93.9%

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

   11. Taylor expanded in ux around 0

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

   13. Applied rewrites 93.9%

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

   if 0.999999702 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 59.2%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate--r+ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower--.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f32 N/A

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

      15. lower-pow.f32 N/A

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

      16. lower--.f32 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f32 93.2

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

   5. Applied rewrites 93.2%

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

   7. Applied rewrites 93.3%

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

   9. Applied rewrites 99.3%

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

   10. Taylor expanded in uy around 0

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

   12. Applied rewrites 99.2%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \leq 0.9999997019767761:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{\left(\left(\frac{-2 \cdot maxCos + 2}{ux} - {\left(maxCos - 1\right)}^{2}\right) \cdot ux\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt
   (* (* (- (/ (+ (* -2.0 maxCos) 2.0) ux) (pow (- maxCos 1.0) 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. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. lower-pow.f32 N/A

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

   16. lower--.f32 N/A

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

   17. unpow2 N/A

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

   18. lower-*.f32 70.4

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

5. Applied rewrites 70.4%

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

7. Applied rewrites 70.3%

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Alternative 4: 89.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \leq 0.9999480247497559:\\ \;\;\;\;\sqrt{ux \cdot 2} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{2}{ux} - \left(\frac{maxCos}{ux} - \left(maxCos - 1\right)\right)\right) \cdot \left(ux \cdot ux\right) - \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot maxCos\right) \cdot ux} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (cos (* (PI) (* 2.0 uy))) 0.9999480247497559)
   (* (sqrt (* ux 2.0)) (cos (* (* (PI) uy) 2.0)))
   (*
    (sqrt
     (-
      (* (- (/ 2.0 ux) (- (/ maxCos ux) (- maxCos 1.0))) (* ux ux))
      (* (* (fma maxCos ux (- 1.0 ux)) maxCos) ux)))
    1.0)))

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.999948025

   1. Initial program 55.4%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 30.4%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

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

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

      4. lift-+.f32 N/A

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

      5. lift--.f32 N/A

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

      6. sub-neg N/A

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

      7. lift-neg.f32 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. associate-+r+ N/A

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

      13. lift-+.f32 N/A

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

      14. lift-+.f32 N/A

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

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

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

      16. lift-+.f32 N/A

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

      17. distribute-rgt-in N/A

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

   7. Applied rewrites 29.8%

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

   8. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. cancel-sign-sub N/A

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

      7. lower-+.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-cos.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 93.2

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

   10. Applied rewrites 93.2%

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

   11. Taylor expanded in ux around 0

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

   13. Applied rewrites 74.8%

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

   if 0.999948025 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 61.8%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 60.5%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate--r+ N/A

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

      6. lower--.f32 N/A

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

      7. lower--.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lift-+.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lift-fma.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f32 N/A

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

   7. Applied rewrites 37.7%

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

   8. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f32 N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. lower--.f32 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f32 94.6

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

   10. Applied rewrites 94.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \leq 0.9999480247497559:\\ \;\;\;\;\sqrt{ux \cdot 2} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{2}{ux} - \left(\frac{maxCos}{ux} - \left(maxCos - 1\right)\right)\right) \cdot \left(ux \cdot ux\right) - \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot maxCos\right) \cdot ux} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\left(ux - ux \cdot ux\right) + ux} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{\frac{\left(\frac{2}{ux} - 1\right) \cdot ux}{maxCos} - \left(\frac{2}{ux} - 2\right) \cdot ux}{maxCos} - ux\right) \cdot \left(maxCos \cdot maxCos\right)\right) \cdot ux} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 9.999999717180685e-10)
   (* (sqrt (+ (- ux (* ux ux)) ux)) (cos (* (* (PI) uy) 2.0)))
   (*
    (sqrt
     (*
      (*
       (-
        (/
         (- (/ (* (- (/ 2.0 ux) 1.0) ux) maxCos) (* (- (/ 2.0 ux) 2.0) ux))
         maxCos)
        ux)
       (* maxCos maxCos))
      ux))
    (cos (* (PI) (* 2.0 uy))))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 9.99999972e-10

   1. Initial program 59.6%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 51.6%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

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

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

      4. lift-+.f32 N/A

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

      5. lift--.f32 N/A

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

      6. sub-neg N/A

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

      7. lift-neg.f32 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. associate-+r+ N/A

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

      13. lift-+.f32 N/A

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

      14. lift-+.f32 N/A

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

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

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

      16. lift-+.f32 N/A

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

      17. distribute-rgt-in N/A

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

   7. Applied rewrites 51.4%

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

   8. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. cancel-sign-sub N/A

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

      7. lower-+.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-cos.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 98.8

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

   10. Applied rewrites 98.8%

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

   12. Applied rewrites 98.8%

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

   if 9.99999972e-10 < maxCos

   1. Initial program 61.8%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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. *-commutative N/A

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

      14. lower-fma.f32 N/A

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

      15. lower-pow.f32 N/A

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

      16. lower--.f32 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f32 59.2

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

   5. Applied rewrites 59.3%

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

   7. Applied rewrites 59.3%

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

   8. Taylor expanded in maxCos around -inf

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

   10. Applied rewrites 98.9%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999717180685 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\left(ux - ux \cdot ux\right) + ux} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{\frac{\left(\frac{2}{ux} - 1\right) \cdot ux}{maxCos} - \left(\frac{2}{ux} - 2\right) \cdot ux}{maxCos} - ux\right) \cdot \left(maxCos \cdot maxCos\right)\right) \cdot ux} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 2.2000000043931323 \cdot 10^{-8}:\\ \;\;\;\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\left(1 - ux\right) \cdot ux + ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\frac{\frac{2}{ux} - 1}{maxCos \cdot maxCos} - \frac{\frac{2}{ux} - 2}{maxCos}\right) \cdot ux - ux\right) \cdot \left(maxCos \cdot maxCos\right)\right) \cdot ux} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 2.2000000043931323e-8)
   (* (cos (* (* (PI) uy) 2.0)) (sqrt (+ (* (- 1.0 ux) ux) ux)))
   (*
    (sqrt
     (*
      (*
       (-
        (*
         (-
          (/ (- (/ 2.0 ux) 1.0) (* maxCos maxCos))
          (/ (- (/ 2.0 ux) 2.0) maxCos))
         ux)
        ux)
       (* maxCos maxCos))
      ux))
    (cos (* (PI) (* 2.0 uy))))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 2.2e-8

   1. Initial program 60.5%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 52.4%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

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

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

      4. lift-+.f32 N/A

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

      5. lift--.f32 N/A

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

      6. sub-neg N/A

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

      7. lift-neg.f32 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. associate-+r+ N/A

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

      13. lift-+.f32 N/A

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

      14. lift-+.f32 N/A

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

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

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

      16. lift-+.f32 N/A

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

      17. distribute-rgt-in N/A

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

   7. Applied rewrites 52.2%

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

   8. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. cancel-sign-sub N/A

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

      7. lower-+.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-cos.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 98.8

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

   10. Applied rewrites 98.8%

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

   if 2.2e-8 < maxCos

   1. Initial program 58.4%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate--r+ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower--.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f32 N/A

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

      15. lower-pow.f32 N/A

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

      16. lower--.f32 N/A

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

      17. unpow2 N/A

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

      18. lower-*.f32 55.3

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

   5. Applied rewrites 55.3%

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

   7. Applied rewrites 55.2%

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

   8. Taylor expanded in maxCos around inf

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

   10. Applied rewrites 98.7%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 2.2000000043931323 \cdot 10^{-8}:\\ \;\;\;\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\left(1 - ux\right) \cdot ux + ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(\frac{\frac{2}{ux} - 1}{maxCos \cdot maxCos} - \frac{\frac{2}{ux} - 2}{maxCos}\right) \cdot ux - ux\right) \cdot \left(maxCos \cdot maxCos\right)\right) \cdot ux} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\left(1 - ux\right) \cdot ux + ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{\frac{2}{ux} - 1}{maxCos \cdot maxCos} - \frac{\frac{2}{ux} - 2}{maxCos}\right) \cdot \left(ux \cdot ux\right) - ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 9.999999747378752e-6)
   (* (cos (* (* (PI) uy) 2.0)) (sqrt (+ (* (- 1.0 ux) ux) ux)))
   (*
    (sqrt
     (*
      (-
       (*
        (-
         (/ (- (/ 2.0 ux) 1.0) (* maxCos maxCos))
         (/ (- (/ 2.0 ux) 2.0) maxCos))
        (* ux ux))
       (* ux ux))
      (* maxCos maxCos)))
    1.0)))

Derivation

1. Split input into 2 regimes

2. if maxCos < 9.99999975e-6

   1. Initial program 60.5%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 52.4%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

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

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

      4. lift-+.f32 N/A

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

      5. lift--.f32 N/A

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

      6. sub-neg N/A

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

      7. lift-neg.f32 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. associate-+r+ N/A

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

      13. lift-+.f32 N/A

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

      14. lift-+.f32 N/A

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

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

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

      16. lift-+.f32 N/A

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

      17. distribute-rgt-in N/A

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

   7. Applied rewrites 52.1%

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

   8. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. cancel-sign-sub N/A

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

      7. lower-+.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-cos.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 98.4

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

   10. Applied rewrites 98.4%

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

   if 9.99999975e-6 < maxCos

   1. Initial program 56.7%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate--r+ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower--.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-pow.f32 N/A

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

      15. lower--.f32 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f32 52.3

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

   8. Applied rewrites 51.5%

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

   9. Taylor expanded in maxCos around inf

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

   11. Applied rewrites 84.3%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\left(1 - ux\right) \cdot ux + ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{\frac{2}{ux} - 1}{maxCos \cdot maxCos} - \frac{\frac{2}{ux} - 2}{maxCos}\right) \cdot \left(ux \cdot ux\right) - ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{\frac{2}{ux} - 1}{maxCos \cdot maxCos} - \frac{\frac{2}{ux} - 2}{maxCos}\right) \cdot \left(ux \cdot ux\right) - ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 9.999999747378752e-6)
   (* (sqrt (* (- 2.0 ux) ux)) (cos (* (* (PI) uy) 2.0)))
   (*
    (sqrt
     (*
      (-
       (*
        (-
         (/ (- (/ 2.0 ux) 1.0) (* maxCos maxCos))
         (/ (- (/ 2.0 ux) 2.0) maxCos))
        (* ux ux))
       (* ux ux))
      (* maxCos maxCos)))
    1.0)))

Derivation

1. Split input into 2 regimes

2. if maxCos < 9.99999975e-6

   1. Initial program 60.5%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 52.4%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

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

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

      4. lift-+.f32 N/A

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

      5. lift--.f32 N/A

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

      6. sub-neg N/A

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

      7. lift-neg.f32 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. associate-+r+ N/A

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

      13. lift-+.f32 N/A

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

      14. lift-+.f32 N/A

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

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

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

      16. lift-+.f32 N/A

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

      17. distribute-rgt-in N/A

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

   7. Applied rewrites 52.2%

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

   8. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. cancel-sign-sub N/A

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

      7. lower-+.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-cos.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 98.4

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

   10. Applied rewrites 98.4%

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

   11. Taylor expanded in ux around 0

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

   13. Applied rewrites 98.3%

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

   if 9.99999975e-6 < maxCos

   1. Initial program 56.7%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 51.9%

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

   6. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate--r+ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

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

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

      9. metadata-eval N/A

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

      10. lower--.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-pow.f32 N/A

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

      15. lower--.f32 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f32 52.3

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

   8. Applied rewrites 51.5%

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

   9. Taylor expanded in maxCos around inf

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

   11. Applied rewrites 84.3%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 9.999999747378752 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{\frac{2}{ux} - 1}{maxCos \cdot maxCos} - \frac{\frac{2}{ux} - 2}{maxCos}\right) \cdot \left(ux \cdot ux\right) - ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.3% accurate, 2.0× speedup

Math

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

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)))
  1.0))

C

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

Julia

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

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

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

5. Applied rewrites 52.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. distribute-rgt-in N/A

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

   5. associate--r+ N/A

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

   6. lower--.f32 N/A

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

   7. lower--.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-+.f32 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-fma.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

7. Applied rewrites 30.0%

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

8. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f32 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. unsub-neg N/A

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

   10. lower--.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower--.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 79.8

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

10. Applied rewrites 79.8%

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

11. Final simplification 79.8%

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

Alternative 10: 79.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

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

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

5. Applied rewrites 52.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. distribute-rgt-in N/A

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

   5. associate--r+ N/A

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

   6. lower--.f32 N/A

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

   7. lower--.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-+.f32 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-fma.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

7. Applied rewrites 30.5%

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

8. Taylor expanded in ux around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. div-sub N/A

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

   9. sub-neg N/A

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

   10. mul-1-neg N/A

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

   11. lower--.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. mul-1-neg N/A

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

   14. sub-neg N/A

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

   15. lower--.f32 N/A

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

   16. mul-1-neg N/A

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

   17. unsub-neg N/A

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

   18. lower--.f32 N/A

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

   19. unpow2 N/A

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

   20. lower-*.f32 79.8

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

10. Applied rewrites 79.8%

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

11. Final simplification 79.8%

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

Alternative 11: 75.5% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

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

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

5. Applied rewrites 52.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

   4. lift-+.f32 N/A

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

   5. lift--.f32 N/A

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

   6. sub-neg N/A

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

   7. lift-neg.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. associate-+r+ N/A

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

   13. lift-+.f32 N/A

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

   14. lift-+.f32 N/A

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

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

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

   16. lift-+.f32 N/A

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

   17. distribute-rgt-in N/A

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

7. Applied rewrites 50.7%

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

8. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. cancel-sign-sub N/A

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

   4. lower-+.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 76.7

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

10. Applied rewrites 76.7%

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

11. Final simplification 76.7%

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

Alternative 12: 61.8% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 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

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

5. Applied rewrites 52.4%

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

6. Taylor expanded in ux around 0

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

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

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 60.9

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

8. Applied rewrites 60.5%

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

9. Final simplification 60.6%

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

Alternative 13: 30.4% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

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

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

5. Applied rewrites 52.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

   4. lift-+.f32 N/A

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

   5. lift--.f32 N/A

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

   6. sub-neg N/A

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

   7. lift-neg.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. associate-+r+ N/A

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

   13. lift-+.f32 N/A

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

   14. lift-+.f32 N/A

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

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

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

   16. lift-+.f32 N/A

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

   17. distribute-rgt-in N/A

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

7. Applied rewrites 50.7%

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

8. Taylor expanded in maxCos around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. associate-*r* N/A

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

   5. mul-1-neg N/A

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

   6. cancel-sign-sub N/A

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

   7. lower-+.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-cos.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. lower-PI.f32 93.6

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

10. Applied rewrites 93.6%

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

11. Taylor expanded in uy around 0

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

13. Applied rewrites 30.6%

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

Alternative 14: 6.6% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 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

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

5. Applied rewrites 52.4%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 6.6%

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

9. Final simplification 6.6%

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

Reproduce

herbie shell --seed 2024268 
(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)))))))