UniformSampleCone, x

Percentage Accurate: 57.4% → 98.7%

Time: 11.7s

Alternatives: 13

Speedup: 5.8×

Specification

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

Math

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

FPCore

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

Initial Program: 57.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.7% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 56.3%

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

3. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate--r+ N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. div-sub N/A

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

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

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

   9. metadata-eval N/A

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

   10. lower--.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. lower-pow.f32 N/A

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

   15. lower--.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 70.0

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

5. Applied rewrites 70.1%

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

7. Applied rewrites 70.2%

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

9. Applied rewrites 98.8%

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

10. Final simplification 98.8%

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

Alternative 2: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := maxCos \cdot ux + \left(1 - ux\right)\\ t_1 := \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \cdot t\_1 \leq 0.03200000151991844:\\ \;\;\;\;\sqrt{0.5 \cdot \left(4 \cdot ux\right)} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\left(\left(\frac{1}{ux} + maxCos\right) - 1\right) \cdot ux\right) \cdot t\_0} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (* maxCos ux) (- 1.0 ux))) (t_1 (cos (* (PI) (* 2.0 uy)))))
   (if (<= (* (sqrt (- 1.0 (* t_0 t_0))) t_1) 0.03200000151991844)
     (* (sqrt (* 0.5 (* 4.0 ux))) t_1)
     (* (sqrt (- 1.0 (* (* (- (+ (/ 1.0 ux) maxCos) 1.0) ux) t_0))) 1.0))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))) < 0.0320000015

   1. Initial program 39.3%

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

      1. lift--.f32 N/A

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

      2. flip-- N/A

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

      3. lower-/.f32 N/A

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

   4. Applied rewrites 25.4%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\frac{1 - {\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right)}^{4}}{{\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right)}^{2} + 1}}} \]

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)}} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

      5. lower-sqrt.f32 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{2}}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

      6. lower-cos.f32 N/A

         \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

      8. lower-*.f32 N/A

         \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

      9. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)} \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)} \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

      11. lower-PI.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

      12. lower-sqrt.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \color{blue}{\sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)}} \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(4 - 4 \cdot maxCos\right) \cdot ux}} \]

      14. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(4 - 4 \cdot maxCos\right) \cdot ux}} \]

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

          \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(4 + \left(\mathsf{neg}\left(4\right)\right) \cdot maxCos\right)} \cdot ux} \]

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 62.3

          \[\leadsto \left(\sqrt{0.5} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-4, maxCos, 4\right)} \cdot ux} \]

   7. Applied rewrites 62.8%

      \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\mathsf{fma}\left(-4, maxCos, 4\right) \cdot ux}} \]

   8. Taylor expanded in maxCos around 0

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

   10. Applied rewrites 85.7%

       \[\leadsto \left(\sqrt{0.5} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{4 \cdot ux} \]
   11. Step-by-step derivation

   12. Applied rewrites 85.9%

       \[\leadsto \color{blue}{\sqrt{\left(4 \cdot ux\right) \cdot 0.5} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right)} \]

   if 0.0320000015 < (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))))

   1. Initial program 92.2%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 82.5%

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

   6. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f32 N/A

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

      6. lower-/.f32 82.9

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

   8. Applied rewrites 82.9%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - \left(maxCos \cdot ux + \left(1 - ux\right)\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \leq 0.03200000151991844:\\ \;\;\;\;\sqrt{0.5 \cdot \left(4 \cdot ux\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\left(\left(\frac{1}{ux} + maxCos\right) - 1\right) \cdot ux\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 56.3%

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

3. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate--r+ N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. div-sub N/A

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

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

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

   9. metadata-eval N/A

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

   10. lower--.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. lower-pow.f32 N/A

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

   15. lower--.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 70.1

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

5. Applied rewrites 70.1%

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

7. Applied rewrites 98.8%

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

8. Final simplification 98.8%

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

Alternative 4: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{if}\;t\_0 \leq 0.9999849796295166:\\ \;\;\;\;\sqrt{0.5 \cdot \left(4 \cdot ux\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(-2, maxCos, 2\right) - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (cos (* (PI) (* 2.0 uy)))))
   (if (<= t_0 0.9999849796295166)
     (* (sqrt (* 0.5 (* 4.0 ux))) t_0)
     (*
      (sqrt (* (- (fma -2.0 maxCos 2.0) (* (pow (- maxCos 1.0) 2.0) ux)) ux))
      1.0))))

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.99998498

   1. Initial program 54.1%

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

      1. lift--.f32 N/A

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

      2. flip-- N/A

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

      3. lower-/.f32 N/A

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

   4. Applied rewrites 9.2%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\frac{1 - {\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right)}^{4}}{{\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right)}^{2} + 1}}} \]

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)}} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{\frac{1}{2}}\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)}} \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{2}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

      5. lower-sqrt.f32 N/A

         \[\leadsto \left(\color{blue}{\sqrt{\frac{1}{2}}} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

      6. lower-cos.f32 N/A

         \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

      8. lower-*.f32 N/A

         \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \color{blue}{\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

      9. *-commutative N/A

         \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)} \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)} \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

      11. lower-PI.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)} \]

      12. lower-sqrt.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \color{blue}{\sqrt{ux \cdot \left(4 - 4 \cdot maxCos\right)}} \]

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(4 - 4 \cdot maxCos\right) \cdot ux}} \]

      14. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(4 - 4 \cdot maxCos\right) \cdot ux}} \]

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

          \[\leadsto \left(\sqrt{\frac{1}{2}} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(4 + \left(\mathsf{neg}\left(4\right)\right) \cdot maxCos\right)} \cdot ux} \]

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 20.0

          \[\leadsto \left(\sqrt{0.5} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-4, maxCos, 4\right)} \cdot ux} \]

   7. Applied rewrites 20.0%

      \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\mathsf{fma}\left(-4, maxCos, 4\right) \cdot ux}} \]

   8. Taylor expanded in maxCos around 0

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

   10. Applied rewrites 74.4%

       \[\leadsto \left(\sqrt{0.5} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{4 \cdot ux} \]
   11. Step-by-step derivation

   12. Applied rewrites 74.6%

       \[\leadsto \color{blue}{\sqrt{\left(4 \cdot ux\right) \cdot 0.5} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right)} \]

   if 0.99998498 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 57.3%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. associate--l+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. associate-+r+ N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f32 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. lower-pow.f32 N/A

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

      16. lower--.f32 94.0

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

   8. Applied rewrites 94.0%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \leq 0.9999849796295166:\\ \;\;\;\;\sqrt{0.5 \cdot \left(4 \cdot ux\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\mathsf{fma}\left(-2, maxCos, 2\right) - {\left(maxCos - 1\right)}^{2} \cdot ux\right) \cdot ux} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.8% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 56.3%

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

3. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate--r+ N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. div-sub N/A

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

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

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

   9. metadata-eval N/A

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

   10. lower--.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f32 N/A

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

   14. lower-pow.f32 N/A

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

   15. lower--.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 70.1

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

5. Applied rewrites 70.1%

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

7. Applied rewrites 70.2%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 95.1%

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

11. Final simplification 95.1%

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

Alternative 6: 72.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.0001500000071246177:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\left(\left(\frac{1}{ux} + maxCos\right) - 1\right) \cdot ux\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.0001500000071246177)
   (* (sqrt (* (fma -2.0 maxCos 2.0) ux)) 1.0)
   (*
    (sqrt
     (-
      1.0
      (* (* (- (+ (/ 1.0 ux) maxCos) 1.0) ux) (+ (* maxCos ux) (- 1.0 ux)))))
    1.0)))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.0001500000071246177f) {
		tmp = sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux)) * 1.0f;
	} else {
		tmp = sqrtf((1.0f - (((((1.0f / ux) + maxCos) - 1.0f) * ux) * ((maxCos * ux) + (1.0f - ux))))) * 1.0f;
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.0001500000071246177))
		tmp = Float32(sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)) * Float32(1.0));
	else
		tmp = Float32(sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(Float32(Float32(Float32(1.0) / ux) + maxCos) - Float32(1.0)) * ux) * Float32(Float32(maxCos * ux) + Float32(Float32(1.0) - ux))))) * Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.50000007e-4

   1. Initial program 34.3%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 31.9%

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

   6. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 73.6

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

   8. Applied rewrites 73.2%

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

   if 1.50000007e-4 < ux

   1. Initial program 89.9%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f32 N/A

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

      6. lower-/.f32 76.1

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

   8. Applied rewrites 76.1%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.0001500000071246177:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\left(\left(\frac{1}{ux} + maxCos\right) - 1\right) \cdot ux\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.0001500000071246177:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.0001500000071246177)
   (* (sqrt (* (fma -2.0 maxCos 2.0) ux)) 1.0)
   (*
    (sqrt
     (- 1.0 (* (- 1.0 (- ux (* maxCos ux))) (+ (* maxCos ux) (- 1.0 ux)))))
    1.0)))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.0001500000071246177f) {
		tmp = sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux)) * 1.0f;
	} else {
		tmp = sqrtf((1.0f - ((1.0f - (ux - (maxCos * ux))) * ((maxCos * ux) + (1.0f - ux))))) * 1.0f;
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.0001500000071246177))
		tmp = Float32(sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)) * Float32(1.0));
	else
		tmp = Float32(sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - Float32(ux - Float32(maxCos * ux))) * Float32(Float32(maxCos * ux) + Float32(Float32(1.0) - ux))))) * Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.50000007e-4

   1. Initial program 34.3%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 31.9%

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

   6. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 73.6

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

   8. Applied rewrites 73.2%

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

   if 1.50000007e-4 < ux

   1. Initial program 89.9%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 75.6%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f32 N/A

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

      5. lower--.f32 75.8

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 75.8

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

   7. Applied rewrites 75.8%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.0001500000071246177:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := maxCos \cdot ux + \left(1 - ux\right)\\ \mathbf{if}\;ux \leq 0.0001500000071246177:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - t\_0 \cdot t\_0} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (* maxCos ux) (- 1.0 ux))))
   (if (<= ux 0.0001500000071246177)
     (* (sqrt (* (fma -2.0 maxCos 2.0) ux)) 1.0)
     (* (sqrt (- 1.0 (* t_0 t_0))) 1.0))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (maxCos * ux) + (1.0f - ux);
	float tmp;
	if (ux <= 0.0001500000071246177f) {
		tmp = sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux)) * 1.0f;
	} else {
		tmp = sqrtf((1.0f - (t_0 * t_0))) * 1.0f;
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(maxCos * ux) + Float32(Float32(1.0) - ux))
	tmp = Float32(0.0)
	if (ux <= Float32(0.0001500000071246177))
		tmp = Float32(sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)) * Float32(1.0));
	else
		tmp = Float32(sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))) * Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.50000007e-4

   1. Initial program 34.3%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 31.9%

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

   6. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 68.1

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

   8. Applied rewrites 73.2%

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

   if 1.50000007e-4 < ux

   1. Initial program 89.9%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 75.6%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.0001500000071246177:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(maxCos \cdot ux + \left(1 - ux\right)\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.0001500000071246177:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right) \cdot \left(\left(maxCos - 1\right) \cdot ux + 1\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.0001500000071246177)
   (* (sqrt (* (fma -2.0 maxCos 2.0) ux)) 1.0)
   (*
    (sqrt (- 1.0 (* (- (fma ux maxCos 1.0) ux) (+ (* (- maxCos 1.0) ux) 1.0))))
    1.0)))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.0001500000071246177f) {
		tmp = sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux)) * 1.0f;
	} else {
		tmp = sqrtf((1.0f - ((fmaf(ux, maxCos, 1.0f) - ux) * (((maxCos - 1.0f) * ux) + 1.0f)))) * 1.0f;
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.0001500000071246177))
		tmp = Float32(sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)) * Float32(1.0));
	else
		tmp = Float32(sqrt(Float32(Float32(1.0) - Float32(Float32(fma(ux, maxCos, Float32(1.0)) - ux) * Float32(Float32(Float32(maxCos - Float32(1.0)) * ux) + Float32(1.0))))) * Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.50000007e-4

   1. Initial program 34.3%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 31.9%

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

   6. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 73.6

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

   8. Applied rewrites 73.2%

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

   if 1.50000007e-4 < ux

   1. Initial program 89.9%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f32 N/A

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

      6. lower-/.f32 76.1

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

   8. Applied rewrites 76.1%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 76.1

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

   10. Applied rewrites 73.5%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.0001500000071246177:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\mathsf{fma}\left(ux, maxCos, 1\right) - ux\right) \cdot \left(\left(maxCos - 1\right) \cdot ux + 1\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.0001500000071246177:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.0001500000071246177)
   (* (sqrt (* (fma -2.0 maxCos 2.0) ux)) 1.0)
   (* (sqrt (- 1.0 (* (- 1.0 ux) (+ (* maxCos ux) (- 1.0 ux))))) 1.0)))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.0001500000071246177f) {
		tmp = sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux)) * 1.0f;
	} else {
		tmp = sqrtf((1.0f - ((1.0f - ux) * ((maxCos * ux) + (1.0f - ux))))) * 1.0f;
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.0001500000071246177))
		tmp = Float32(sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)) * Float32(1.0));
	else
		tmp = Float32(sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(maxCos * ux) + Float32(Float32(1.0) - ux))))) * Float32(1.0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.50000007e-4

   1. Initial program 34.3%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 31.9%

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

   6. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 73.6

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

   8. Applied rewrites 73.2%

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

   if 1.50000007e-4 < ux

   1. Initial program 89.9%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower--.f32 74.2

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

   8. Applied rewrites 74.2%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.0001500000071246177:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.8% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.3%

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

3. Taylor expanded in uy around 0

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

5. Applied rewrites 49.2%

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

6. Taylor expanded in ux around 0

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

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

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 61.1

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

8. Applied rewrites 62.6%

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

9. Final simplification 59.4%

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

Alternative 12: 19.7% accurate, 7.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.3%

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

3. Taylor expanded in uy around 0

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

5. Applied rewrites 49.2%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 6.6%

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

   1. lift--.f32 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. neg-mul-1 N/A

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

   5. lower-fma.f32 19.4

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

10. Applied rewrites 20.0%

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

11. Final simplification 19.9%

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

Alternative 13: 6.6% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.3%

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

3. Taylor expanded in uy around 0

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

5. Applied rewrites 49.2%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 6.6%

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

9. Final simplification 6.6%

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

Reproduce

herbie shell --seed 2024276 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (* (cos (* (* uy 2.0) (PI))) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))