UniformSampleCone, y

Percentage Accurate: 57.4% → 98.1%

Time: 8.0s

Alternatives: 14

Speedup: 5.0×

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: 57.4% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ t_1 := \frac{2}{ux} - 1\\ \mathbf{if}\;maxCos \leq 1.999999943436137 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot t\_1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(maxCos \cdot maxCos\right) \cdot \left(\frac{2 - \left(\frac{2}{ux} - \frac{t\_1}{maxCos}\right)}{maxCos} - 1\right)\right) \cdot \left(ux \cdot ux\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sin (* (PI) (* 2.0 uy)))) (t_1 (- (/ 2.0 ux) 1.0)))
   (if (<= maxCos 1.999999943436137e-9)
     (* (sqrt (* (* ux ux) t_1)) t_0)
     (*
      (sqrt
       (*
        (*
         (* maxCos maxCos)
         (- (/ (- 2.0 (- (/ 2.0 ux) (/ t_1 maxCos))) maxCos) 1.0))
        (* ux ux)))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999994e-9

   1. Initial program 56.2%

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

      1. lift--.f32 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} + 1} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) - 1\right)}} \]

      6. lower--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) - 1\right)}} \]

      7. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} - 1\right)} \]

      8. pow2 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}^{2}} - 1\right)} \]

      9. pow-to-exp N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{e^{\log \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot 2}} - 1\right)} \]

      10. rem-log-exp N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\color{blue}{\log \left(e^{\log \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot 2}\right)}} - 1\right)} \]

      11. pow-to-exp N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left({\left(\left(1 - ux\right) + ux \cdot maxCos\right)}^{2}\right)}} - 1\right)} \]

      12. pow2 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} - 1\right)} \]

      13. lift-*.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} - 1\right)} \]

      14. lower-expm1.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \color{blue}{\mathsf{expm1}\left(\log \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)}} \]

   4. Applied rewrites 45.1%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \mathsf{expm1}\left(\mathsf{log1p}\left(ux \cdot \left(-1 + maxCos\right)\right) \cdot 2\right)}} \]

   5. Taylor expanded in ux around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 98.3%

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

   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 98.3%

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

   if 1.99999994e-9 < maxCos

   1. Initial program 60.1%

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

      1. lift--.f32 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} + 1} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) - 1\right)}} \]

      6. lower--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) - 1\right)}} \]

      7. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} - 1\right)} \]

      8. pow2 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}^{2}} - 1\right)} \]

      9. pow-to-exp N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{e^{\log \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot 2}} - 1\right)} \]

      10. rem-log-exp N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\color{blue}{\log \left(e^{\log \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot 2}\right)}} - 1\right)} \]

      11. pow-to-exp N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left({\left(\left(1 - ux\right) + ux \cdot maxCos\right)}^{2}\right)}} - 1\right)} \]

      12. pow2 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} - 1\right)} \]

      13. lift-*.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} - 1\right)} \]

      14. lower-expm1.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \color{blue}{\mathsf{expm1}\left(\log \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)}} \]

   4. Applied rewrites 41.6%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \mathsf{expm1}\left(\mathsf{log1p}\left(ux \cdot \left(-1 + maxCos\right)\right) \cdot 2\right)}} \]

   5. Taylor expanded in ux around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in maxCos around -inf

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

   10. Applied rewrites 98.5%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 1.999999943436137 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2}{ux} - 1\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(maxCos \cdot maxCos\right) \cdot \left(\frac{2 - \left(\frac{2}{ux} - \frac{\frac{2}{ux} - 1}{maxCos}\right)}{maxCos} - 1\right)\right) \cdot \left(ux \cdot ux\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in maxCos around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate--l+ N/A

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

   4. div-sub N/A

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

   5. +-commutative N/A

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

   6. lower-+.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower--.f32 57.3

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

5. Applied rewrites 57.3%

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

6. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-*.f32 98.4

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

8. Applied rewrites 98.4%

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

9. Final simplification 98.4%

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

Alternative 3: 90.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\\ \mathbf{if}\;2 \cdot uy \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{\left(\left(2 - \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right) - maxCos \cdot 2\right) \cdot ux} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\frac{2}{maxCos} - 2\right) \cdot maxCos\right) \cdot ux} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (PI) 2.0) uy)))
   (if (<= (* 2.0 uy) 0.003000000026077032)
     (*
      (sqrt
       (*
        (- (- 2.0 (* (* (- maxCos 1.0) (- maxCos 1.0)) ux)) (* maxCos 2.0))
        ux))
      t_0)
     (* (sqrt (* (* (- (/ 2.0 maxCos) 2.0) maxCos) ux)) (sin t_0)))))

Derivation

1. Split input into 2 regimes

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

   1. Initial program 56.4%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 55.9

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

   5. Applied rewrites 55.9%

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

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

      5. lower-/.f32 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f32 N/A

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

      8. pow2 N/A

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

      9. lower-pow.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower-+.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f32 N/A

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

      18. lower--.f32 56.2

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

   7. Applied rewrites 56.2%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift--.f32 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f32 N/A

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

      6. unpow2 N/A

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

      7. lift-+.f32 N/A

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

      8. flip-- N/A

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

      9. lift--.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. associate-+l- N/A

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

      14. lift-*.f32 N/A

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

      15. lift-+.f32 N/A

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

      16. lift--.f32 N/A

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

      17. lift-*.f32 N/A

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

      18. *-lft-identity N/A

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

   9. Applied rewrites 19.8%

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

   10. Taylor expanded in ux around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. lower--.f32 N/A

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

       4. mul-1-neg N/A

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

       5. unsub-neg N/A

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

       6. lower--.f32 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f32 N/A

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

       9. distribute-rgt-out N/A

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

       10. lower-*.f32 N/A

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

       11. lower--.f32 N/A

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

       12. lower-+.f32 N/A

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

       13. lower-*.f32 96.3

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

   12. Applied rewrites 96.3%

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

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

   1. Initial program 58.7%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f32 N/A

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

      6. lower-/.f32 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-sin.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-sqrt.f32 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f32 11.1

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

   8. Applied rewrites 11.1%

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

   9. Taylor expanded in maxCos around inf

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

   11. Applied rewrites 76.2%

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

4. Final simplification 90.2%

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

Alternative 4: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\\ \mathbf{if}\;2 \cdot uy \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{\left(\left(2 - \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right) - maxCos \cdot 2\right) \cdot ux} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot maxCos + 2\right) \cdot ux} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (PI) 2.0) uy)))
   (if (<= (* 2.0 uy) 0.003000000026077032)
     (*
      (sqrt
       (*
        (- (- 2.0 (* (* (- maxCos 1.0) (- maxCos 1.0)) ux)) (* maxCos 2.0))
        ux))
      t_0)
     (* (sqrt (* (+ (* -2.0 maxCos) 2.0) ux)) (sin t_0)))))

Derivation

1. Split input into 2 regimes

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

   1. Initial program 56.4%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 55.9

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

   5. Applied rewrites 55.9%

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

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

      5. lower-/.f32 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f32 N/A

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

      8. pow2 N/A

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

      9. lower-pow.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower-+.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f32 N/A

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

      18. lower--.f32 56.2

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

   7. Applied rewrites 56.2%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift--.f32 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f32 N/A

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

      6. unpow2 N/A

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

      7. lift-+.f32 N/A

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

      8. flip-- N/A

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

      9. lift--.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. associate-+l- N/A

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

      14. lift-*.f32 N/A

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

      15. lift-+.f32 N/A

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

      16. lift--.f32 N/A

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

      17. lift-*.f32 N/A

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

      18. *-lft-identity N/A

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

   9. Applied rewrites 21.0%

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

   10. Taylor expanded in ux around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. lower--.f32 N/A

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

       4. mul-1-neg N/A

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

       5. unsub-neg N/A

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

       6. lower--.f32 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f32 N/A

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

       9. distribute-rgt-out N/A

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

       10. lower-*.f32 N/A

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

       11. lower--.f32 N/A

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

       12. lower-+.f32 N/A

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

       13. lower-*.f32 96.3

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

   12. Applied rewrites 96.3%

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

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

   1. Initial program 58.7%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f32 N/A

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

      6. lower-/.f32 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-sin.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-sqrt.f32 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f32 11.1

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

   8. Applied rewrites 11.1%

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

   10. Applied rewrites 76.2%

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

4. Final simplification 90.2%

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

Alternative 5: 92.3% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.1%

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

   1. lift--.f32 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. neg-sub0 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} + 1} \]

   5. associate-+l- N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) - 1\right)}} \]

   6. lower--.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) - 1\right)}} \]

   7. lift-*.f32 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} - 1\right)} \]

   8. pow2 N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}^{2}} - 1\right)} \]

   9. pow-to-exp N/A

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{e^{\log \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot 2}} - 1\right)} \]

   10. rem-log-exp N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\color{blue}{\log \left(e^{\log \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot 2}\right)}} - 1\right)} \]

   11. pow-to-exp N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left({\left(\left(1 - ux\right) + ux \cdot maxCos\right)}^{2}\right)}} - 1\right)} \]

   12. pow2 N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} - 1\right)} \]

   13. lift-*.f32 N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} - 1\right)} \]

   14. lower-expm1.f32 N/A

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \color{blue}{\mathsf{expm1}\left(\log \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)}} \]

4. Applied rewrites 46.9%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \mathsf{expm1}\left(\mathsf{log1p}\left(ux \cdot \left(-1 + maxCos\right)\right) \cdot 2\right)}} \]

5. Taylor expanded in ux around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

7. Applied rewrites 98.3%

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

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 93.4%

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

11. Final simplification 93.4%

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

Alternative 6: 89.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\\ \mathbf{if}\;2 \cdot uy \leq 0.003000000026077032:\\ \;\;\;\;\sqrt{\left(\left(2 - \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right) \cdot ux\right) - maxCos \cdot 2\right) \cdot ux} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot ux} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (PI) 2.0) uy)))
   (if (<= (* 2.0 uy) 0.003000000026077032)
     (*
      (sqrt
       (*
        (- (- 2.0 (* (* (- maxCos 1.0) (- maxCos 1.0)) ux)) (* maxCos 2.0))
        ux))
      t_0)
     (* (sqrt (* 2.0 ux)) (sin t_0)))))

Derivation

1. Split input into 2 regimes

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

   1. Initial program 56.4%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 55.9

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

   5. Applied rewrites 55.9%

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

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

      5. lower-/.f32 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f32 N/A

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

      8. pow2 N/A

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

      9. lower-pow.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower-+.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f32 N/A

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

      18. lower--.f32 56.2

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

   7. Applied rewrites 56.2%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift--.f32 N/A

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

      4. metadata-eval N/A

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

      5. lift-pow.f32 N/A

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

      6. unpow2 N/A

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

      7. lift-+.f32 N/A

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

      8. flip-- N/A

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

      9. lift--.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. associate-+l- N/A

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

      14. lift-*.f32 N/A

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

      15. lift-+.f32 N/A

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

      16. lift--.f32 N/A

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

      17. lift-*.f32 N/A

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

      18. *-lft-identity N/A

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

   9. Applied rewrites 20.4%

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

   10. Taylor expanded in ux around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. lower--.f32 N/A

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

       4. mul-1-neg N/A

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

       5. unsub-neg N/A

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

       6. lower--.f32 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f32 N/A

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

       9. distribute-rgt-out N/A

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

       10. lower-*.f32 N/A

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

       11. lower--.f32 N/A

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

       12. lower-+.f32 N/A

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

       13. lower-*.f32 96.3

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

   12. Applied rewrites 96.3%

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

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

   1. Initial program 58.7%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f32 N/A

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

      6. lower-/.f32 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-sin.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-sqrt.f32 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f32 11.1

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

   8. Applied rewrites 11.1%

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

   9. Taylor expanded in maxCos around 0

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

   11. Applied rewrites 73.7%

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

4. Final simplification 89.4%

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

Alternative 7: 76.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := maxCos \cdot ux + \left(1 - ux\right)\\ \mathbf{if}\;t\_0 \leq 0.9998199939727783:\\ \;\;\;\;\sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot t\_0} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot \left(\sqrt{\left(\left(\frac{2}{maxCos} - 2\right) \cdot maxCos\right) \cdot ux} \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (* maxCos ux) (- 1.0 ux))))
   (if (<= t_0 0.9998199939727783)
     (*
      (sqrt (- 1.0 (* (- 1.0 (- ux (* maxCos ux))) t_0)))
      (* (* (PI) 2.0) uy))
     (* (* (PI) uy) (* (sqrt (* (* (- (/ 2.0 maxCos) 2.0) maxCos) ux)) 2.0)))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.999819994

   1. Initial program 88.4%

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

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

      6. *-commutative N/A

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

      7. lift-*.f32 N/A

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

      8. lower--.f32 74.9

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

   7. Applied rewrites 74.9%

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

   if 0.999819994 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 35.0%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f32 N/A

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

      6. lower-/.f32 36.2

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

   5. Applied rewrites 36.2%

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

   6. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-sin.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-sqrt.f32 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f32 25.9

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

   8. Applied rewrites 28.8%

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

   9. Taylor expanded in uy around 0

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

   11. Applied rewrites 73.3%

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

   12. Taylor expanded in maxCos around inf

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

   14. Applied rewrites 76.3%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \cdot ux + \left(1 - ux\right) \leq 0.9998199939727783:\\ \;\;\;\;\sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot \left(\sqrt{\left(\left(\frac{2}{maxCos} - 2\right) \cdot maxCos\right) \cdot ux} \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := maxCos \cdot ux + \left(1 - ux\right)\\ \mathbf{if}\;t\_0 \leq 0.9998199939727783:\\ \;\;\;\;\sqrt{1 - t\_0 \cdot t\_0} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot \left(\sqrt{\left(\left(\frac{2}{maxCos} - 2\right) \cdot maxCos\right) \cdot ux} \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (* maxCos ux) (- 1.0 ux))))
   (if (<= t_0 0.9998199939727783)
     (* (sqrt (- 1.0 (* t_0 t_0))) (* (* (PI) 2.0) uy))
     (* (* (PI) uy) (* (sqrt (* (* (- (/ 2.0 maxCos) 2.0) maxCos) ux)) 2.0)))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.999819994

   1. Initial program 88.4%

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

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

   if 0.999819994 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 35.0%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f32 N/A

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

      6. lower-/.f32 36.2

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

   5. Applied rewrites 36.2%

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

   6. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-sin.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-sqrt.f32 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f32 28.6

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

   8. Applied rewrites 26.7%

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

   9. Taylor expanded in uy around 0

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

   11. Applied rewrites 73.3%

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

   12. Taylor expanded in maxCos around inf

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

   14. Applied rewrites 76.3%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \cdot ux + \left(1 - ux\right) \leq 0.9998199939727783:\\ \;\;\;\;\sqrt{1 - \left(maxCos \cdot ux + \left(1 - ux\right)\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot \left(\sqrt{\left(\left(\frac{2}{maxCos} - 2\right) \cdot maxCos\right) \cdot ux} \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := maxCos \cdot ux + \left(1 - ux\right)\\ \mathbf{if}\;t\_0 \leq 0.9995999932289124:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot t\_0} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot \left(\sqrt{\left(\left(\frac{2}{maxCos} - 2\right) \cdot maxCos\right) \cdot ux} \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (* maxCos ux) (- 1.0 ux))))
   (if (<= t_0 0.9995999932289124)
     (* (sqrt (- 1.0 (* (- 1.0 ux) t_0))) (* (* (PI) 2.0) uy))
     (* (* (PI) uy) (* (sqrt (* (* (- (/ 2.0 maxCos) 2.0) maxCos) ux)) 2.0)))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.999599993

   1. Initial program 89.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 76.0

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

   5. Applied rewrites 76.0%

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

   6. 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 73.8

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

   8. Applied rewrites 73.8%

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

   if 0.999599993 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 37.8%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f32 N/A

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

      6. lower-/.f32 38.8

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

   5. Applied rewrites 38.8%

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

   6. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-sin.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-sqrt.f32 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f32 26.8

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

   8. Applied rewrites 27.2%

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

   9. Taylor expanded in uy around 0

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

   11. Applied rewrites 72.1%

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

   12. Taylor expanded in maxCos around inf

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

   14. Applied rewrites 75.2%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \cdot ux + \left(1 - ux\right) \leq 0.9995999932289124:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot \left(\sqrt{\left(\left(\frac{2}{maxCos} - 2\right) \cdot maxCos\right) \cdot ux} \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := maxCos \cdot ux + \left(1 - ux\right)\\ \mathbf{if}\;t\_0 \leq 0.9995999932289124:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot t\_0} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(-2 \cdot maxCos + 2\right) \cdot ux} \cdot 2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot uy\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (* maxCos ux) (- 1.0 ux))))
   (if (<= t_0 0.9995999932289124)
     (* (sqrt (- 1.0 (* (- 1.0 ux) t_0))) (* (* (PI) 2.0) uy))
     (* (* (sqrt (* (+ (* -2.0 maxCos) 2.0) ux)) 2.0) (* (PI) uy)))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.999599993

   1. Initial program 89.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 76.0

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

   5. Applied rewrites 76.0%

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

   6. 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 73.8

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

   8. Applied rewrites 73.8%

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

   if 0.999599993 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 37.8%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f32 N/A

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

      6. lower-/.f32 38.8

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

   5. Applied rewrites 38.8%

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

   6. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-sin.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-sqrt.f32 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f32 26.8

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

   8. Applied rewrites 28.1%

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

   9. Taylor expanded in uy around 0

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

   11. Applied rewrites 72.1%

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

   13. Applied rewrites 75.2%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \cdot ux + \left(1 - ux\right) \leq 0.9995999932289124:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(-2 \cdot maxCos + 2\right) \cdot ux} \cdot 2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.0% accurate, 2.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.1%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 49.4

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

5. Applied rewrites 49.4%

   \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
6. 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. flip-- N/A

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

   5. lower-/.f32 N/A

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

   6. metadata-eval N/A

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

   7. lower--.f32 N/A

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

   8. pow2 N/A

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

   9. lower-pow.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lower--.f32 N/A

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

   14. lower-+.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. lower--.f32 49.7

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

7. Applied rewrites 49.7%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift--.f32 N/A

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

   4. metadata-eval N/A

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

   5. lift-pow.f32 N/A

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

   6. unpow2 N/A

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

   7. lift-+.f32 N/A

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

   8. flip-- N/A

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

   9. lift--.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. associate-+l- N/A

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

   14. lift-*.f32 N/A

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

   15. lift-+.f32 N/A

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

   16. lift--.f32 N/A

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

   17. lift-*.f32 N/A

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

   18. *-lft-identity N/A

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

9. Applied rewrites 19.9%

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

10. Taylor expanded in ux around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. lower--.f32 N/A

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

    4. mul-1-neg N/A

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

    5. unsub-neg N/A

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

    6. lower--.f32 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f32 N/A

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

    9. distribute-rgt-out N/A

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

    10. lower-*.f32 N/A

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

    11. lower--.f32 N/A

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

    12. lower-+.f32 N/A

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

    13. lower-*.f32 79.9

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

12. Applied rewrites 79.9%

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

13. Final simplification 79.9%

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

Alternative 12: 75.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \cdot ux + \left(1 - ux\right) \leq 0.9995999932289124:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(-2 \cdot maxCos + 2\right) \cdot ux} \cdot 2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot uy\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (+ (* maxCos ux) (- 1.0 ux)) 0.9995999932289124)
   (* (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))) (* (* (PI) 2.0) uy))
   (* (* (sqrt (* (+ (* -2.0 maxCos) 2.0) ux)) 2.0) (* (PI) uy))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.999599993

   1. Initial program 89.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 76.0

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

   5. Applied rewrites 76.0%

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

   6. 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 73.8

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

   8. Applied rewrites 73.8%

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

   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 73.6

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

   11. Applied rewrites 73.6%

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

   if 0.999599993 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 37.8%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f32 N/A

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

      6. lower-/.f32 38.8

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

   5. Applied rewrites 38.8%

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

   6. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-sin.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-sqrt.f32 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f32 27.8

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

   8. Applied rewrites 30.1%

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

   9. Taylor expanded in uy around 0

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

   11. Applied rewrites 72.1%

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

   13. Applied rewrites 75.2%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \cdot ux + \left(1 - ux\right) \leq 0.9995999932289124:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(-2 \cdot maxCos + 2\right) \cdot ux} \cdot 2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.8% accurate, 4.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f32 N/A

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

   6. lower-/.f32 58.1

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

5. Applied rewrites 58.1%

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

6. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-sin.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-sqrt.f32 N/A

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. +-commutative N/A

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

   16. lower-fma.f32 29.8

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

8. Applied rewrites 28.3%

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

9. Taylor expanded in uy around 0

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

11. Applied rewrites 58.2%

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

13. Applied rewrites 64.7%

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

14. Final simplification 64.7%

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

Alternative 14: 63.2% accurate, 5.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f32 N/A

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

   6. lower-/.f32 58.1

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

5. Applied rewrites 58.1%

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

6. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-sin.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. lower-sqrt.f32 N/A

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

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

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. +-commutative N/A

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

   16. lower-fma.f32 30.2

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

8. Applied rewrites 28.1%

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

9. Taylor expanded in uy around 0

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

11. Applied rewrites 62.6%

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

12. Taylor expanded in maxCos around 0

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

14. Applied rewrites 62.9%

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

15. Final simplification 62.9%

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

Reproduce

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