UniformSampleCone, y

Percentage Accurate: 57.0% → 98.1%

Time: 13.9s

Alternatives: 21

Speedup: 4.5×

Specification

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

Math

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

FPCore

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

Initial Program: 57.0% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{ux} - 1\\ t_1 := \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{if}\;maxCos \leq 4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;t\_1 \cdot \sqrt{t\_0 \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\left(\left(ux \cdot ux\right) \cdot \left(\frac{\frac{t\_0}{maxCos}}{maxCos} + \frac{2 - \frac{2}{ux}}{maxCos}\right) - ux \cdot ux\right) \cdot \left(maxCos \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- (/ 2.0 ux) 1.0)) (t_1 (sin (* (* uy 2.0) (PI)))))
   (if (<= maxCos 4.99999991225835e-14)
     (* t_1 (sqrt (* t_0 (* ux ux))))
     (*
      t_1
      (sqrt
       (*
        (-
         (*
          (* ux ux)
          (+ (/ (/ t_0 maxCos) maxCos) (/ (- 2.0 (/ 2.0 ux)) maxCos)))
         (* ux ux))
        (* maxCos maxCos)))))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.99999991e-14

   1. Initial program 63.8%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate--r+ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. metadata-eval N/A

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

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

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

      10. lower--.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-pow.f32 N/A

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

      15. lower--.f32 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f32 20.4

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

   5. Applied rewrites 23.4%

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in maxCos around 0

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

   10. Applied rewrites 98.2%

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

   if 4.99999991e-14 < maxCos

   1. Initial program 55.3%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate--r+ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. metadata-eval N/A

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

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

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

      10. lower--.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-pow.f32 N/A

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

      15. lower--.f32 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f32 18.9

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

   5. Applied rewrites 20.6%

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

   6. Taylor expanded in maxCos around inf

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

   8. Applied rewrites 98.1%

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

Alternative 2: 98.2% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate--r+ N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. div-sub N/A

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

   8. metadata-eval N/A

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

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

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

   10. lower--.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. lower-pow.f32 N/A

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

   15. lower--.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 20.6

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

5. Applied rewrites 21.2%

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

7. Applied rewrites 98.2%

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

Alternative 3: 98.2% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate--r+ N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. div-sub N/A

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

   8. metadata-eval N/A

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

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

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

   10. lower--.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. lower-pow.f32 N/A

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

   15. lower--.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 20.9

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

5. Applied rewrites 19.0%

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

7. Applied rewrites 98.2%

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

9. Applied rewrites 98.2%

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

Alternative 4: 97.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{ux} - 1\\ t_1 := \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{if}\;maxCos \leq 2.8000000525086298 \cdot 10^{-14}:\\ \;\;\;\;t\_1 \cdot \sqrt{t\_0 \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(-ux, ux, \left(ux \cdot ux\right) \cdot \left(\frac{t\_0}{maxCos \cdot maxCos} + \frac{2 - \frac{2}{ux}}{maxCos}\right)\right) \cdot \left(maxCos \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- (/ 2.0 ux) 1.0)) (t_1 (sin (* (* uy 2.0) (PI)))))
   (if (<= maxCos 2.8000000525086298e-14)
     (* t_1 (sqrt (* t_0 (* ux ux))))
     (*
      t_1
      (sqrt
       (*
        (fma
         (- ux)
         ux
         (*
          (* ux ux)
          (+ (/ t_0 (* maxCos maxCos)) (/ (- 2.0 (/ 2.0 ux)) maxCos))))
        (* maxCos maxCos)))))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 2.80000005e-14

   1. Initial program 63.7%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate--r+ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. metadata-eval N/A

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

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

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

      10. lower--.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-pow.f32 N/A

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

      15. lower--.f32 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f32 19.7

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

   5. Applied rewrites 19.5%

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in maxCos around 0

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

   10. Applied rewrites 98.2%

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

   if 2.80000005e-14 < maxCos

   1. Initial program 55.7%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate--r+ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. metadata-eval N/A

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

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

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

      10. lower--.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-pow.f32 N/A

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

      15. lower--.f32 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f32 19.2

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

   5. Applied rewrites 18.7%

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

   7. Applied rewrites 98.1%

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

   8. Taylor expanded in maxCos around inf

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

   10. Applied rewrites 47.5%

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

Alternative 5: 89.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9996500015258789:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(ux - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux - ux \cdot maxCos}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_0 t_0) 0.9996500015258789)
     (*
      (sin (* (* uy 2.0) (PI)))
      (sqrt (- 1.0 (* (- (- ux (* maxCos ux)) 1.0) (- ux 1.0)))))
     (*
      (* (sqrt 2.0) (sin (* (* (PI) uy) 2.0)))
      (sqrt (- ux (* ux maxCos)))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.99965

   1. Initial program 90.5%

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

      1. lift-*.f32 N/A

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

      2. sqr-abs-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. associate-+l- N/A

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

      6. fabs-sub N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. fabs-sub N/A

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

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 90.6

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 90.6

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

   4. Applied rewrites 90.6%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower--.f32 87.0

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

   7. Applied rewrites 87.0%

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

   if 0.99965 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 40.6%

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

      1. lift-*.f32 N/A

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

      2. sqr-abs-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. associate-+l- N/A

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

      6. fabs-sub N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. fabs-sub N/A

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

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 40.6

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 40.6

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

   4. Applied rewrites 40.6%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-lft-identity N/A

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

      14. metadata-eval N/A

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

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

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

      16. distribute-rgt-in N/A

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

      17. *-lft-identity N/A

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

      18. mul-1-neg N/A

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

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

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

      20. lower--.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 90.5

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

   7. Applied rewrites 90.5%

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

Alternative 6: 89.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9996500015258789:\\ \;\;\;\;t\_1 \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(ux - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\left(ux - ux \cdot maxCos\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))) (t_1 (sin (* (* uy 2.0) (PI)))))
   (if (<= (* t_0 t_0) 0.9996500015258789)
     (* t_1 (sqrt (- 1.0 (* (- (- ux (* maxCos ux)) 1.0) (- ux 1.0)))))
     (* t_1 (sqrt (* (- ux (* ux maxCos)) 2.0))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.99965

   1. Initial program 90.5%

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

      1. lift-*.f32 N/A

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

      2. sqr-abs-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. associate-+l- N/A

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

      6. fabs-sub N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. fabs-sub N/A

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

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 90.6

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 90.6

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

   4. Applied rewrites 90.6%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower--.f32 87.0

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

   7. Applied rewrites 87.0%

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

   if 0.99965 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 40.6%

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

      1. lift-*.f32 N/A

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

      2. sqr-abs-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. associate-+l- N/A

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

      6. fabs-sub N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. fabs-sub N/A

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

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 40.6

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 40.6

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

   4. Applied rewrites 40.6%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. mul-1-neg N/A

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

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

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 90.2

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

   7. Applied rewrites 90.2%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) \leq 0.9996500015258789:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(ux - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux - ux \cdot maxCos\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9996500015258789:\\ \;\;\;\;t\_1 \cdot \sqrt{1 - t\_0 \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\left(ux - ux \cdot maxCos\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))) (t_1 (sin (* (* uy 2.0) (PI)))))
   (if (<= (* t_0 t_0) 0.9996500015258789)
     (* t_1 (sqrt (- 1.0 (* t_0 (- 1.0 ux)))))
     (* t_1 (sqrt (* (- ux (* ux maxCos)) 2.0))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.99965

   1. Initial program 90.5%

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

   3. Taylor expanded in maxCos around 0

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

      1. lower--.f32 87.0

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

   5. Applied rewrites 87.0%

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

   if 0.99965 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 40.6%

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

      1. lift-*.f32 N/A

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

      2. sqr-abs-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. associate-+l- N/A

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

      6. fabs-sub N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. fabs-sub N/A

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

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 40.6

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 40.6

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

   4. Applied rewrites 40.6%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. mul-1-neg N/A

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

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

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 90.2

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

   7. Applied rewrites 90.2%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) \leq 0.9996500015258789:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux - ux \cdot maxCos\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9980000257492065:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux - ux \cdot maxCos\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_0 t_0) 0.9980000257492065)
     (*
      (* (* (PI) 2.0) uy)
      (sqrt (- 1.0 (* (- 1.0 (- ux (* maxCos ux))) t_0))))
     (* (sin (* (* uy 2.0) (PI))) (sqrt (* (- ux (* ux maxCos)) 2.0))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.998000026

   1. Initial program 92.8%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 77.6

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

   5. Applied rewrites 77.6%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f32 N/A

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

      5. lower--.f32 77.6

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 77.6

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

   7. Applied rewrites 77.6%

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

   if 0.998000026 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 43.8%

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

      1. lift-*.f32 N/A

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

      2. sqr-abs-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. associate-+l- N/A

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

      6. fabs-sub N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. fabs-sub N/A

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

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 43.8

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 43.8

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

   4. Applied rewrites 43.8%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

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

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

      6. distribute-rgt-in N/A

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

      7. *-lft-identity N/A

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

      8. mul-1-neg N/A

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

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

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 88.1

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

   7. Applied rewrites 88.1%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) \leq 0.9980000257492065:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(ux - ux \cdot maxCos\right) \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9993500113487244:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_0 t_0) 0.9993500113487244)
     (*
      (* (* (PI) 2.0) uy)
      (sqrt (- 1.0 (* (- 1.0 (- ux (* maxCos ux))) t_0))))
     (* (sin (* (* uy 2.0) (PI))) (sqrt (* (fma -2.0 maxCos 2.0) ux))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.999350011

   1. Initial program 91.3%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 76.3

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

   5. Applied rewrites 76.3%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f32 N/A

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

      5. lower--.f32 76.4

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 76.4

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

   7. Applied rewrites 76.4%

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

   if 0.999350011 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 41.7%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate--r+ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. metadata-eval N/A

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

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

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

      10. lower--.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-pow.f32 N/A

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

      15. lower--.f32 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f32 20.5

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

   5. Applied rewrites 20.5%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 38.9%

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

Alternative 10: 92.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate--r+ N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. div-sub N/A

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

   8. metadata-eval N/A

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

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

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

   10. lower--.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. lower-pow.f32 N/A

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

   15. lower--.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 20.4

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

5. Applied rewrites 22.0%

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

7. Applied rewrites 98.2%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 93.2%

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

Alternative 11: 92.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate--r+ N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. div-sub N/A

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

   8. metadata-eval N/A

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

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

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

   10. lower--.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. lower-pow.f32 N/A

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

   15. lower--.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 19.3

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

5. Applied rewrites 20.1%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 93.2%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(\left(\frac{2}{ux} - 1\right) \cdot ux\right) \cdot \color{blue}{ux}} \]
9. Add Preprocessing

Alternative 12: 80.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 6.000000233523199 \cdot 10^{-16}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - \mathsf{fma}\left(\frac{maxCos}{ux}, 2, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right) \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\left(\left(\left(\frac{2}{maxCos} - \frac{-2}{\left(maxCos \cdot maxCos\right) \cdot ux}\right) - \left(\left(1 + \frac{1}{maxCos \cdot maxCos}\right) + \frac{\frac{2}{maxCos}}{ux}\right)\right) \cdot \left(maxCos \cdot maxCos\right)\right) \cdot \left(ux \cdot ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 6.000000233523199e-16)
   (*
    (* (* (PI) 2.0) uy)
    (sqrt
     (*
      (- (/ 2.0 ux) (fma (/ maxCos ux) 2.0 (* (- maxCos 1.0) (- maxCos 1.0))))
      (* ux ux))))
   (*
    (* (* (PI) uy) 2.0)
    (sqrt
     (*
      (*
       (-
        (- (/ 2.0 maxCos) (/ -2.0 (* (* maxCos maxCos) ux)))
        (+ (+ 1.0 (/ 1.0 (* maxCos maxCos))) (/ (/ 2.0 maxCos) ux)))
       (* maxCos maxCos))
      (* ux ux))))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 6.0000002e-16

   1. Initial program 63.4%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 54.1

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

   5. Applied rewrites 54.1%

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

      1. *-lft-identity N/A

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

      2. lift-*.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. distribute-rgt-in N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lift-+.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f32 N/A

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

   7. Applied rewrites 19.0%

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

   8. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   10. Applied rewrites 80.0%

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

   if 6.0000002e-16 < maxCos

   1. Initial program 57.2%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate--r+ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. div-sub N/A

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

      8. metadata-eval N/A

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

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

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

      10. lower--.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lower-pow.f32 N/A

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

      15. lower--.f32 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f32 22.1

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

   5. Applied rewrites 21.2%

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

   7. Applied rewrites 98.1%

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

   8. Taylor expanded in maxCos around inf

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

   10. Applied rewrites 97.8%

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

   11. Taylor expanded in uy around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f32 N/A

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

       5. lower-PI.f32 82.0

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

   13. Applied rewrites 82.0%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 6.000000233523199 \cdot 10^{-16}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - \mathsf{fma}\left(\frac{maxCos}{ux}, 2, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right) \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\left(\left(\left(\frac{2}{maxCos} - \frac{-2}{\left(maxCos \cdot maxCos\right) \cdot ux}\right) - \left(\left(1 + \frac{1}{maxCos \cdot maxCos}\right) + \frac{\frac{2}{maxCos}}{ux}\right)\right) \cdot \left(maxCos \cdot maxCos\right)\right) \cdot \left(ux \cdot ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux - ux \cdot maxCos\\ t_1 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\sqrt{1 - t\_1 \cdot t\_1} \leq 0.03500000014901161:\\ \;\;\;\;\left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - t\_0\right) \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- ux (* ux maxCos))) (t_1 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (sqrt (- 1.0 (* t_1 t_1))) 0.03500000014901161)
     (* (* (* (* (sqrt 2.0) (PI)) uy) 2.0) (sqrt t_0))
     (* (* (* (PI) 2.0) uy) (sqrt (- 1.0 (* (- 1.0 t_0) (- 1.0 ux))))))))

Derivation

1. Split input into 2 regimes

2. if (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))) < 0.0350000001

   1. Initial program 42.3%

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

      1. lift-*.f32 N/A

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

      2. sqr-abs-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. associate-+l- N/A

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

      6. fabs-sub N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. fabs-sub N/A

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

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 42.4

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 42.4

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

   4. Applied rewrites 42.4%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-lft-identity N/A

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

      14. metadata-eval N/A

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

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

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

      16. distribute-rgt-in N/A

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

      17. *-lft-identity N/A

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

      18. mul-1-neg N/A

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

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

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

      20. lower--.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 89.5

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

   7. Applied rewrites 89.5%

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

   8. Taylor expanded in uy around 0

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

   10. Applied rewrites 76.2%

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

   if 0.0350000001 < (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))

   1. Initial program 91.7%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower--.f32 74.6

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

   8. Applied rewrites 74.6%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lower--.f32 74.6

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f32 74.6

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

   10. Applied rewrites 74.6%

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

Alternative 14: 73.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \leq 0.01860000006854534:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - ux \cdot maxCos\right)\right) \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (sqrt (- 1.0 (* t_0 t_0))) 0.01860000006854534)
     (* (* (+ (PI) (PI)) uy) (sqrt (* (fma -2.0 maxCos 2.0) ux)))
     (*
      (* (* (PI) 2.0) uy)
      (sqrt (- 1.0 (* (- 1.0 (- ux (* ux maxCos))) (- 1.0 ux))))))))

Derivation

1. Split input into 2 regimes

2. if (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))) < 0.0186000001

   1. Initial program 40.6%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 37.3

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

   5. Applied rewrites 37.3%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower--.f32 36.7

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

   8. Applied rewrites 36.7%

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

   9. Taylor expanded in ux around 0

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

       1. metadata-eval N/A

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

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

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

       3. *-commutative N/A

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

       4. lower-*.f32 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f32 74.1

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

   11. Applied rewrites 73.7%

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

   13. Applied rewrites 63.3%

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

   if 0.0186000001 < (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))

   1. Initial program 90.5%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 74.8

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower--.f32 72.9

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

   8. Applied rewrites 72.9%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lower--.f32 72.9

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f32 72.9

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

   10. Applied rewrites 72.9%

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

Alternative 15: 73.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \leq 0.01860000006854534:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - t\_0 \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (sqrt (- 1.0 (* t_0 t_0))) 0.01860000006854534)
     (* (* (+ (PI) (PI)) uy) (sqrt (* (fma -2.0 maxCos 2.0) ux)))
     (* (* (* (PI) 2.0) uy) (sqrt (- 1.0 (* t_0 (- 1.0 ux))))))))

Derivation

1. Split input into 2 regimes

2. if (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))) < 0.0186000001

   1. Initial program 40.6%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 37.3

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

   5. Applied rewrites 37.3%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower--.f32 36.7

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

   8. Applied rewrites 36.7%

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

   9. Taylor expanded in ux around 0

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

       1. metadata-eval N/A

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

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

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

       3. *-commutative N/A

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

       4. lower-*.f32 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f32 74.1

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

   11. Applied rewrites 73.7%

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

   13. Applied rewrites 74.1%

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

   if 0.0186000001 < (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))

   1. Initial program 90.5%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 74.8

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower--.f32 72.9

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

   8. Applied rewrites 72.9%

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

Alternative 16: 73.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \leq 0.01860000006854534:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (sqrt (- 1.0 (* t_0 t_0))) 0.01860000006854534)
     (* (* (+ (PI) (PI)) uy) (sqrt (* (fma -2.0 maxCos 2.0) ux)))
     (* (* (* (PI) 2.0) uy) (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux))))))))

Derivation

1. Split input into 2 regimes

2. if (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))) < 0.0186000001

   1. Initial program 40.6%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 37.3

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

   5. Applied rewrites 37.3%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower--.f32 36.7

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

   8. Applied rewrites 36.7%

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

   9. Taylor expanded in ux around 0

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

       1. metadata-eval N/A

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

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

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

       3. *-commutative N/A

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

       4. lower-*.f32 N/A

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

       5. +-commutative N/A

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

       6. lower-fma.f32 74.1

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

   11. Applied rewrites 73.7%

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

   13. Applied rewrites 63.3%

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

   if 0.0186000001 < (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))

   1. Initial program 90.5%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 74.8

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower--.f32 72.9

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

   8. Applied rewrites 72.9%

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

   9. Taylor expanded in maxCos around 0

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

       1. lower--.f32 72.7

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

   11. Applied rewrites 72.7%

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

Alternative 17: 77.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\\ \mathbf{if}\;ux \leq 0.0035000001080334187:\\ \;\;\;\;t\_0 \cdot \sqrt{\left(\frac{2}{ux} - \mathsf{fma}\left(\frac{maxCos}{ux}, 2, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right) \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (PI) 2.0) uy)))
   (if (<= ux 0.0035000001080334187)
     (*
      t_0
      (sqrt
       (*
        (-
         (/ 2.0 ux)
         (fma (/ maxCos ux) 2.0 (* (- maxCos 1.0) (- maxCos 1.0))))
        (* ux ux))))
     (*
      t_0
      (sqrt
       (-
        1.0
        (* (- 1.0 (- ux (* maxCos ux))) (+ (- 1.0 ux) (* ux maxCos)))))))))

Derivation

1. Split input into 2 regimes

2. if ux < 0.00350000011

   1. Initial program 47.5%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 42.0

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

   5. Applied rewrites 42.0%

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

      1. *-lft-identity N/A

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

      2. lift-*.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. distribute-rgt-in N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lift-+.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. +-commutative N/A

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

      13. lift-fma.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. associate-*r* N/A

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

      19. lower-*.f32 N/A

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

   7. Applied rewrites 17.2%

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

   8. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   10. Applied rewrites 77.4%

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

   if 0.00350000011 < ux

   1. Initial program 94.9%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 79.6

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

   5. Applied rewrites 79.6%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f32 N/A

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

      5. lower--.f32 79.7

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 79.7

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

   7. Applied rewrites 79.7%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.0035000001080334187:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - \mathsf{fma}\left(\frac{maxCos}{ux}, 2, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right) \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 76.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00011999999696854502:\\ \;\;\;\;\left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{ux - ux \cdot maxCos}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00011999999696854502)
   (* (* (* (* (sqrt 2.0) (PI)) uy) 2.0) (sqrt (- ux (* ux maxCos))))
   (*
    (* (* (PI) 2.0) uy)
    (sqrt
     (- 1.0 (* (- 1.0 (- ux (* maxCos ux))) (+ (- 1.0 ux) (* ux maxCos))))))))

Derivation

1. Split input into 2 regimes

2. if ux < 1.19999997e-4

   1. Initial program 39.2%

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

      1. lift-*.f32 N/A

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

      2. sqr-abs-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. associate-+l- N/A

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

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux - ux \cdot maxCos\right) - 1\right) \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)}} \]

      12. lower-*.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux - ux \cdot maxCos\right) - 1\right) \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)}} \]

      13. lower--.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux - ux \cdot maxCos\right) - 1\right)} \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)} \]

      14. lower--.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\color{blue}{\left(ux - ux \cdot maxCos\right)} - 1\right) \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)} \]

      15. lift-*.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - \color{blue}{ux \cdot maxCos}\right) - 1\right) \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)} \]

      16. *-commutative N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - \color{blue}{maxCos \cdot ux}\right) - 1\right) \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)} \]

      17. lower-*.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - \color{blue}{maxCos \cdot ux}\right) - 1\right) \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)} \]

      18. lower--.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \color{blue}{\left(\left(ux - ux \cdot maxCos\right) - 1\right)}} \]

      19. lower--.f32 39.2

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(\color{blue}{\left(ux - ux \cdot maxCos\right)} - 1\right)} \]

      20. lift-*.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(\left(ux - \color{blue}{ux \cdot maxCos}\right) - 1\right)} \]

      21. *-commutative N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(\left(ux - \color{blue}{maxCos \cdot ux}\right) - 1\right)} \]

      22. lower-*.f32 39.2

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(\left(ux - \color{blue}{maxCos \cdot ux}\right) - 1\right)} \]

   4. Applied rewrites 39.2%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 1\right)}} \]

   5. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(1 - maxCos\right)} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{ux \cdot \left(1 - maxCos\right)}} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{ux \cdot \left(1 - maxCos\right)}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \left(1 - maxCos\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \left(1 - maxCos\right)} \]

      5. lower-sqrt.f32 N/A

         \[\leadsto \left(\color{blue}{\sqrt{2}} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux \cdot \left(1 - maxCos\right)} \]

      6. lower-sin.f32 N/A

         \[\leadsto \left(\sqrt{2} \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \sqrt{ux \cdot \left(1 - maxCos\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\sqrt{2} \cdot \sin \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}\right) \cdot \sqrt{ux \cdot \left(1 - maxCos\right)} \]

      8. lower-*.f32 N/A

         \[\leadsto \left(\sqrt{2} \cdot \sin \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}\right) \cdot \sqrt{ux \cdot \left(1 - maxCos\right)} \]

      9. *-commutative N/A

         \[\leadsto \left(\sqrt{2} \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)} \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(1 - maxCos\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)} \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(1 - maxCos\right)} \]

      11. lower-PI.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(1 - maxCos\right)} \]

      12. lower-sqrt.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \color{blue}{\sqrt{ux \cdot \left(1 - maxCos\right)}} \]

      13. *-lft-identity N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(1 - \color{blue}{1 \cdot maxCos}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot maxCos\right)} \]

      15. fp-cancel-sign-sub-inv N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(1 + -1 \cdot maxCos\right)}} \]

      16. distribute-rgt-in N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{1 \cdot ux + \left(-1 \cdot maxCos\right) \cdot ux}} \]

      17. *-lft-identity N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{ux} + \left(-1 \cdot maxCos\right) \cdot ux} \]

      18. mul-1-neg N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux + \color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)} \cdot ux} \]

      19. fp-cancel-sub-sign N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{ux - maxCos \cdot ux}} \]

      20. lower--.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{ux - maxCos \cdot ux}} \]

      21. *-commutative N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux - \color{blue}{ux \cdot maxCos}} \]

      22. lower-*.f32 91.2

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux - \color{blue}{ux \cdot maxCos}} \]

   7. Applied rewrites 91.2%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux - ux \cdot maxCos}} \]

   8. Taylor expanded in uy around 0

      \[\leadsto \left(2 \cdot \left(uy \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)\right) \cdot \sqrt{\color{blue}{ux - ux \cdot maxCos}} \]
   9. Step-by-step derivation

   10. Applied rewrites 78.1%

       \[\leadsto \left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\color{blue}{ux - ux \cdot maxCos}} \]

   if 1.19999997e-4 < ux

   1. Initial program 89.5%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lower-PI.f32 74.2

         \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. Applied rewrites 74.2%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   6. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. lift--.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. associate-+l- N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - ux \cdot maxCos\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. lower--.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - ux \cdot maxCos\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower--.f32 74.2

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \color{blue}{\left(ux - ux \cdot maxCos\right)}\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lift-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{ux \cdot maxCos}\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{maxCos \cdot ux}\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      8. lower-*.f32 74.2

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{maxCos \cdot ux}\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   7. Applied rewrites 74.2%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 76.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;ux \leq 0.00011999999696854502:\\ \;\;\;\;\left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{ux - ux \cdot maxCos}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= ux 0.00011999999696854502)
     (* (* (* (* (sqrt 2.0) (PI)) uy) 2.0) (sqrt (- ux (* ux maxCos))))
     (* (* (* (PI) 2.0) uy) (sqrt (- 1.0 (* t_0 t_0)))))))

Derivation

1. Split input into 2 regimes

2. if ux < 1.19999997e-4

   1. Initial program 39.2%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux - ux \cdot maxCos\right) - 1\right) \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)}} \]

      12. lower-*.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux - ux \cdot maxCos\right) - 1\right) \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)}} \]

      13. lower--.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux - ux \cdot maxCos\right) - 1\right)} \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)} \]

      14. lower--.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\color{blue}{\left(ux - ux \cdot maxCos\right)} - 1\right) \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)} \]

      15. lift-*.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - \color{blue}{ux \cdot maxCos}\right) - 1\right) \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)} \]

      16. *-commutative N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - \color{blue}{maxCos \cdot ux}\right) - 1\right) \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)} \]

      17. lower-*.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - \color{blue}{maxCos \cdot ux}\right) - 1\right) \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)} \]

      18. lower--.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \color{blue}{\left(\left(ux - ux \cdot maxCos\right) - 1\right)}} \]

      19. lower--.f32 39.2

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(\color{blue}{\left(ux - ux \cdot maxCos\right)} - 1\right)} \]

      20. lift-*.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(\left(ux - \color{blue}{ux \cdot maxCos}\right) - 1\right)} \]

      21. *-commutative N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(\left(ux - \color{blue}{maxCos \cdot ux}\right) - 1\right)} \]

      22. lower-*.f32 39.2

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(\left(ux - \color{blue}{maxCos \cdot ux}\right) - 1\right)} \]

   4. Applied rewrites 39.2%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 1\right)}} \]

   5. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(1 - maxCos\right)} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{ux \cdot \left(1 - maxCos\right)}} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right) \cdot \sqrt{ux \cdot \left(1 - maxCos\right)}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \left(1 - maxCos\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \left(1 - maxCos\right)} \]

      5. lower-sqrt.f32 N/A

         \[\leadsto \left(\color{blue}{\sqrt{2}} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux \cdot \left(1 - maxCos\right)} \]

      6. lower-sin.f32 N/A

         \[\leadsto \left(\sqrt{2} \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \sqrt{ux \cdot \left(1 - maxCos\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\sqrt{2} \cdot \sin \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}\right) \cdot \sqrt{ux \cdot \left(1 - maxCos\right)} \]

      8. lower-*.f32 N/A

         \[\leadsto \left(\sqrt{2} \cdot \sin \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}\right) \cdot \sqrt{ux \cdot \left(1 - maxCos\right)} \]

      9. *-commutative N/A

         \[\leadsto \left(\sqrt{2} \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)} \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(1 - maxCos\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)} \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(1 - maxCos\right)} \]

      11. lower-PI.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(1 - maxCos\right)} \]

      12. lower-sqrt.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \color{blue}{\sqrt{ux \cdot \left(1 - maxCos\right)}} \]

      13. *-lft-identity N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(1 - \color{blue}{1 \cdot maxCos}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot maxCos\right)} \]

      15. fp-cancel-sign-sub-inv N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(1 + -1 \cdot maxCos\right)}} \]

      16. distribute-rgt-in N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{1 \cdot ux + \left(-1 \cdot maxCos\right) \cdot ux}} \]

      17. *-lft-identity N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{ux} + \left(-1 \cdot maxCos\right) \cdot ux} \]

      18. mul-1-neg N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux + \color{blue}{\left(\mathsf{neg}\left(maxCos\right)\right)} \cdot ux} \]

      19. fp-cancel-sub-sign N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{ux - maxCos \cdot ux}} \]

      20. lower--.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{ux - maxCos \cdot ux}} \]

      21. *-commutative N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux - \color{blue}{ux \cdot maxCos}} \]

      22. lower-*.f32 91.2

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux - \color{blue}{ux \cdot maxCos}} \]

   7. Applied rewrites 91.2%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux - ux \cdot maxCos}} \]

   8. Taylor expanded in uy around 0

      \[\leadsto \left(2 \cdot \left(uy \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)\right) \cdot \sqrt{\color{blue}{ux - ux \cdot maxCos}} \]
   9. Step-by-step derivation

   10. Applied rewrites 78.1%

       \[\leadsto \left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\color{blue}{ux - ux \cdot maxCos}} \]

   if 1.19999997e-4 < ux

   1. Initial program 89.5%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lower-PI.f32 74.2

         \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. Applied rewrites 74.2%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 63.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (* (+ (PI) (PI)) uy) (sqrt (* (fma -2.0 maxCos 2.0) ux))))

Derivation

1. Initial program 61.0%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   3. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   4. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. lower-PI.f32 52.7

      \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

5. Applied rewrites 52.7%

   \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

6. Taylor expanded in maxCos around 0

   \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]
7. Step-by-step derivation

   1. lower--.f32 51.6

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

8. Applied rewrites 51.6%

   \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

9. Taylor expanded in ux around 0

   \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right)}} \]
10. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot maxCos\right)} \]

    2. fp-cancel-sign-sub-inv N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + -2 \cdot maxCos\right)}} \]

    3. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

    4. lower-*.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

    5. +-commutative N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(-2 \cdot maxCos + 2\right)} \cdot ux} \]

    6. lower-fma.f32 62.1

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right)} \cdot ux} \]

11. Applied rewrites 62.0%

    \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}} \]
12. Step-by-step derivation

13. Applied rewrites 62.1%

    \[\leadsto \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \]
14. Add Preprocessing

Alternative 21: 7.1% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - 1} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (* (* (PI) 2.0) uy) (sqrt (- 1.0 1.0))))

Derivation

1. Initial program 61.0%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   3. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   4. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. lower-PI.f32 52.7

      \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

5. Applied rewrites 52.7%

   \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
6. Step-by-step derivation

   1. *-lft-identity N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{1 \cdot \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} \]

   2. lift-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - 1 \cdot \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} \]

   3. lift-+.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - 1 \cdot \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}\right)} \]

   4. distribute-rgt-in N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - 1 \cdot \color{blue}{\left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} \]

   5. distribute-lft-in N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 \cdot \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + 1 \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)}} \]

   6. lower-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\mathsf{fma}\left(1, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right), 1 \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)}} \]

   7. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(1, \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)}, 1 \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   8. lower-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(1, \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)}, 1 \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   9. lift-+.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(1, \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)} \cdot \left(1 - ux\right), 1 \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   10. lift-*.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(1, \left(\left(1 - ux\right) + \color{blue}{ux \cdot maxCos}\right) \cdot \left(1 - ux\right), 1 \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   11. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(1, \left(\left(1 - ux\right) + \color{blue}{maxCos \cdot ux}\right) \cdot \left(1 - ux\right), 1 \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   12. +-commutative N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(1, \color{blue}{\left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot \left(1 - ux\right), 1 \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   13. lift-fma.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(1, \color{blue}{\mathsf{fma}\left(maxCos, ux, 1 - ux\right)} \cdot \left(1 - ux\right), 1 \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   14. lower-*.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(1, \mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right), \color{blue}{1 \cdot \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}\right)} \]

   15. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(1, \mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right), 1 \cdot \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(ux \cdot maxCos\right)\right)}\right)} \]

   16. lift-*.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(1, \mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right), 1 \cdot \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(ux \cdot maxCos\right)}\right)\right)} \]

   17. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(1, \mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right), 1 \cdot \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(maxCos \cdot ux\right)}\right)\right)} \]

   18. associate-*r* N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(1, \mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right), 1 \cdot \color{blue}{\left(\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot maxCos\right) \cdot ux\right)}\right)} \]

   19. lower-*.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(1, \mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right), 1 \cdot \color{blue}{\left(\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot maxCos\right) \cdot ux\right)}\right)} \]

7. Applied rewrites 20.0%

   \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\mathsf{fma}\left(1, \mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right), 1 \cdot \left(\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot maxCos\right) \cdot ux\right)\right)}} \]
8. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)\right) + 1 \cdot \left(\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot maxCos\right) \cdot ux\right)\right)}} \]

   2. +-commutative N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 \cdot \left(\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot maxCos\right) \cdot ux\right) + 1 \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)\right)\right)}} \]

   3. lift-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\color{blue}{1 \cdot \left(\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot maxCos\right) \cdot ux\right)} + 1 \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)\right)\right)} \]

   4. *-lft-identity N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\color{blue}{\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot maxCos\right) \cdot ux} + 1 \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)\right)\right)} \]

   5. lift-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\color{blue}{\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot maxCos\right) \cdot ux} + 1 \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)\right)\right)} \]

   6. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\color{blue}{ux \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot maxCos\right)} + 1 \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)\right)\right)} \]

   7. lower-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\mathsf{fma}\left(ux, \mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot maxCos, 1 \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)\right)\right)}} \]

   8. lift-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \color{blue}{\left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot maxCos, 1 \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)\right)\right)} \]

   9. lift-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \left(\color{blue}{maxCos \cdot ux} + \left(1 - ux\right)\right) \cdot maxCos, 1 \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)\right)\right)} \]

   10. lift--.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \left(maxCos \cdot ux + \color{blue}{\left(1 - ux\right)}\right) \cdot maxCos, 1 \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)\right)\right)} \]

   11. associate-+r- N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \color{blue}{\left(\left(maxCos \cdot ux + 1\right) - ux\right)} \cdot maxCos, 1 \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)\right)\right)} \]

   12. lower--.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \color{blue}{\left(\left(maxCos \cdot ux + 1\right) - ux\right)} \cdot maxCos, 1 \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)\right)\right)} \]

   13. lift-*.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \left(\left(\color{blue}{maxCos \cdot ux} + 1\right) - ux\right) \cdot maxCos, 1 \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)\right)\right)} \]

   14. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \left(\left(\color{blue}{ux \cdot maxCos} + 1\right) - ux\right) \cdot maxCos, 1 \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)\right)\right)} \]

   15. lower-fma.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \left(\color{blue}{\mathsf{fma}\left(ux, maxCos, 1\right)} - ux\right) \cdot maxCos, 1 \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)\right)\right)} \]

   16. *-lft-identity 51.4

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right) \cdot maxCos, \color{blue}{\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)}\right)} \]

   17. lift-*.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right) \cdot maxCos, \color{blue}{\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)}\right)} \]

   18. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right) \cdot maxCos, \color{blue}{\left(1 - ux\right) \cdot \mathsf{fma}\left(maxCos, ux, 1 - ux\right)}\right)} \]

   19. lift-fma.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right) \cdot maxCos, \left(1 - ux\right) \cdot \color{blue}{\left(maxCos \cdot ux + \left(1 - ux\right)\right)}\right)} \]

9. Applied rewrites 49.9%

   \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\mathsf{fma}\left(ux, \left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right) \cdot maxCos, \left(1 - ux\right) \cdot \left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right)\right)}} \]

10. Taylor expanded in ux around 0

    \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{1}} \]
11. Step-by-step derivation

12. Applied rewrites 7.1%

    \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{1}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :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)))
  (* (sin (* (* uy 2.0) (PI))) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))