UniformSampleCone, x

Percentage Accurate: 57.2% → 99.1%

Time: 9.0s

Alternatives: 16

Speedup: 3.3×

Specification

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

Math

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

FPCore

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

Initial Program: 57.2% 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: 99.1% accurate, 0.6× speedup

Math

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

FPCore

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

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 0

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

5. Applied rewrites 99.0%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. distribute-lft-neg-in N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lower-/.f32 99.2

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

7. Applied rewrites 99.2%

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

8. Final simplification 99.2%

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

Alternative 2: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot -2\right)\right) \cdot \sqrt{\left(2 - \mathsf{fma}\left({\left(maxCos - 1\right)}^{2}, ux, 2 \cdot maxCos\right)\right) \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (fma 0.5 (PI) (* (* (PI) uy) -2.0)))
  (sqrt (* (- 2.0 (fma (pow (- maxCos 1.0) 2.0) ux (* 2.0 maxCos))) ux))))

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 0

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

5. Applied rewrites 99.0%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. distribute-lft-neg-in N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-neg.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lower-/.f32 99.2

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

7. Applied rewrites 99.2%

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

8. Taylor expanded in uy around 0

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

10. Applied rewrites 99.0%

    \[\leadsto \sin \color{blue}{\left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot -2\right)\right)} \cdot \sqrt{\left(2 - \mathsf{fma}\left({\left(maxCos - 1\right)}^{2}, ux, 2 \cdot maxCos\right)\right) \cdot ux} \]

11. Final simplification 99.0%

    \[\leadsto \sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), \left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot -2\right)\right) \cdot \sqrt{\left(2 - \mathsf{fma}\left({\left(maxCos - 1\right)}^{2}, ux, 2 \cdot maxCos\right)\right) \cdot ux} \]
12. Add Preprocessing

Alternative 3: 99.0% accurate, 0.6× speedup

Math

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

FPCore

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

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 0

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

5. Applied rewrites 99.0%

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

6. Final simplification 99.0%

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

Alternative 4: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

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 0

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

5. Applied rewrites 99.0%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 98.9%

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

9. Final simplification 98.9%

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

Alternative 5: 96.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 8.499999967170879 \cdot 10^{-5}:\\ \;\;\;\;1 \cdot \sqrt{\left(2 - \mathsf{fma}\left({\left(maxCos - 1\right)}^{2}, ux, 2 \cdot maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 8.499999967170879e-5)
   (*
    1.0
    (sqrt (* (- 2.0 (fma (pow (- maxCos 1.0) 2.0) ux (* 2.0 maxCos))) ux)))
   (* (cos (* (* uy 2.0) (PI))) (sqrt (* (- 2.0 ux) ux)))))

Derivation

1. Split input into 2 regimes

2. if uy < 8.49999997e-5

   1. Initial program 55.8%

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

   3. Taylor expanded in ux around 0

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 99.6%

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

   if 8.49999997e-5 < uy

   1. Initial program 57.0%

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

   3. Taylor expanded in ux around 0

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 93.2%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 8.499999967170879 \cdot 10^{-5}:\\ \;\;\;\;1 \cdot \sqrt{\left(2 - \mathsf{fma}\left({\left(maxCos - 1\right)}^{2}, ux, 2 \cdot maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

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 0

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

5. Applied rewrites 99.0%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 98.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 98.3

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

10. Applied rewrites 98.3%

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

11. Final simplification 98.3%

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

Alternative 7: 96.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 8.499999967170879 \cdot 10^{-5}:\\ \;\;\;\;1 \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(-1 + maxCos\right), 2\right) - 2 \cdot maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 8.499999967170879e-5)
   (*
    1.0
    (sqrt
     (*
      (- (fma (- ux) (* (- maxCos 1.0) (+ -1.0 maxCos)) 2.0) (* 2.0 maxCos))
      ux)))
   (* (cos (* (* uy 2.0) (PI))) (sqrt (* (- 2.0 ux) ux)))))

Derivation

1. Split input into 2 regimes

2. if uy < 8.49999997e-5

   1. Initial program 55.8%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 55.7%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate--r+ N/A

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

      6. lower--.f32 N/A

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

      7. lower--.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lift-+.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

   7. Applied rewrites 54.3%

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

   8. Taylor expanded in ux around 0

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

   10. Applied rewrites 99.5%

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

   if 8.49999997e-5 < uy

   1. Initial program 57.0%

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

   3. Taylor expanded in ux around 0

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 93.2%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 8.499999967170879 \cdot 10^{-5}:\\ \;\;\;\;1 \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(-1 + maxCos\right), 2\right) - 2 \cdot maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.4% accurate, 1.1× speedup

Math

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

FPCore

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

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 0

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

5. Applied rewrites 99.0%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 98.3%

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

9. Taylor expanded in ux around 0

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

11. Applied rewrites 97.4%

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

12. Final simplification 97.4%

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

Alternative 9: 75.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(maxCos, ux, 1 - ux\right)\\ t_1 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_1 \cdot t\_1 \leq 0.9995499849319458:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux - \mathsf{fma}\left(maxCos, ux, 1\right), t\_0, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{\left(2 - maxCos\right) \cdot ux - \left(t\_0 \cdot maxCos\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (fma maxCos ux (- 1.0 ux))) (t_1 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_1 t_1) 0.9995499849319458)
     (sqrt (fma (- ux (fma maxCos ux 1.0)) t_0 1.0))
     (* 1.0 (sqrt (- (* (- 2.0 maxCos) ux) (* (* t_0 maxCos) ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = fmaf(maxCos, ux, (1.0f - ux));
	float t_1 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((t_1 * t_1) <= 0.9995499849319458f) {
		tmp = sqrtf(fmaf((ux - fmaf(maxCos, ux, 1.0f)), t_0, 1.0f));
	} else {
		tmp = 1.0f * sqrtf((((2.0f - maxCos) * ux) - ((t_0 * maxCos) * ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = fma(maxCos, ux, Float32(Float32(1.0) - ux))
	t_1 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (Float32(t_1 * t_1) <= Float32(0.9995499849319458))
		tmp = sqrt(fma(Float32(ux - fma(maxCos, ux, Float32(1.0))), t_0, Float32(1.0)));
	else
		tmp = Float32(Float32(1.0) * sqrt(Float32(Float32(Float32(Float32(2.0) - maxCos) * ux) - Float32(Float32(t_0 * maxCos) * ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.999549985

   1. Initial program 88.2%

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

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

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

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. lift-+.f32 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. lower-neg.f32 88.6

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

      13. lift-+.f32 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f32 88.6

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

   4. Applied rewrites 88.6%

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

   5. Taylor expanded in uy around 0

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

   7. Applied rewrites 74.0%

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

   if 0.999549985 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 38.9%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 33.2%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate--r+ N/A

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

      6. lower--.f32 N/A

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

      7. lower--.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lift-+.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

   7. Applied rewrites 31.1%

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

   8. Taylor expanded in ux around 0

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

   10. Applied rewrites 74.0%

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

Alternative 10: 80.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(1.0) * sqrt(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(Float32(-1.0) + maxCos)), Float32(2.0)) - Float32(Float32(2.0) * maxCos)) * ux)))
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 47.5%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. distribute-rgt-in N/A

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

   5. associate--r+ N/A

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

   6. lower--.f32 N/A

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

   7. lower--.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-+.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. +-commutative N/A

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

   14. lift-fma.f32 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

7. Applied rewrites 46.2%

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

8. Taylor expanded in ux around 0

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

10. Applied rewrites 79.4%

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

Alternative 11: 49.4% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(1.0) * sqrt(Float32(Float32(1.0) - Float32(Float32(fma(maxCos, ux, Float32(1.0)) - ux) * Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))))))
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 47.5%

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

   1. lift-+.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. lift--.f32 N/A

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

   6. associate-+r- N/A

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

   7. lower--.f32 N/A

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

   8. lower-fma.f32 47.5

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

7. Applied rewrites 47.5%

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

Alternative 12: 49.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(maxCos, ux, 1 - ux\right)\\ 1 \cdot \sqrt{1 - t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (fma maxCos ux (- 1.0 ux)))) (* 1.0 (sqrt (- 1.0 (* t_0 t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = fmaf(maxCos, ux, (1.0f - ux));
	return 1.0f * sqrtf((1.0f - (t_0 * t_0)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = fma(maxCos, ux, Float32(Float32(1.0) - ux))
	return Float32(Float32(1.0) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 56.3

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

   6. lift-+.f32 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f32 56.3

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

4. Applied rewrites 56.3%

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

5. Taylor expanded in uy around 0

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

7. Applied rewrites 47.5%

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

Alternative 13: 49.4% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(ux, uy, maxCos)
	return sqrt(fma(Float32(ux - fma(maxCos, ux, Float32(1.0))), fma(maxCos, ux, Float32(Float32(1.0) - 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. 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. lift-*.f32 N/A

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

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

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-neg.f32 56.4

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

   13. lift-+.f32 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-fma.f32 56.4

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

4. Applied rewrites 56.4%

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

5. Taylor expanded in uy around 0

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

7. Applied rewrites 47.5%

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

Alternative 14: 40.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(1.0) * sqrt(Float32(Float32(1.0) - fma(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 47.5%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 39.6%

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

Alternative 15: 39.7% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(1.0) * sqrt(Float32(Float32(1.0) - fma(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 47.5%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 39.6%

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

9. Taylor expanded in maxCos around 0

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

11. Applied rewrites 39.0%

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

Alternative 16: 6.6% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Julia

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

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(1.0) * sqrt((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 47.5%

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

6. Taylor expanded in ux around 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. Add Preprocessing

Reproduce

herbie shell --seed 2025018 
(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)))))))