UniformSampleCone 2

Percentage Accurate: 98.9% → 98.9%

Time: 14.9s

Alternatives: 15

Speedup: 1.0×

Specification

\[\left(\left(\left(\left(\left(-10000 \leq xi \land xi \leq 10000\right) \land \left(-10000 \leq yi \land yi \leq 10000\right)\right) \land \left(-10000 \leq zi \land zi \leq 10000\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(\sin t\_2 \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (sqrt (- 1.0 (* t_0 t_0))))
        (t_2 (* (* uy 2.0) (PI))))
   (+ (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi)) (* t_0 zi))))

Initial Program: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\\ \left(\left(\cos t\_2 \cdot t\_1\right) \cdot xi + \left(\sin t\_2 \cdot t\_1\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux))
        (t_1 (sqrt (- 1.0 (* t_0 t_0))))
        (t_2 (* (* uy 2.0) (PI))))
   (+ (+ (* (* (cos t_2) t_1) xi) (* (* (sin t_2) t_1) yi)) (* t_0 zi))))

Alternative 1: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\\ t_1 := \sqrt{1 - t\_0 \cdot t\_0}\\ t_2 := \mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\\ \left(yi \cdot \left(\sin t\_2 \cdot t\_1\right) + xi \cdot \left(t\_1 \cdot \cos t\_2\right)\right) - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* maxCos (- 1.0 ux)) ux))
        (t_1 (sqrt (- 1.0 (* t_0 t_0))))
        (t_2 (* (PI) (* 2.0 uy))))
   (-
    (+ (* yi (* (sin t_2) t_1)) (* xi (* t_1 (cos t_2))))
    (* (* (* (- ux 1.0) maxCos) ux) zi))))

Derivation

1. Initial program 99.0%

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

3. Final simplification 99.0%

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

Alternative 2: 98.8% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-cos.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 98.9

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

5. Applied rewrites 98.9%

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

6. Final simplification 98.9%

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

Alternative 3: 98.7% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (PI) (* 2.0 uy))))
   (-
    (+ (* yi (sin t_0)) (* xi (cos t_0)))
    (* (* (* (- ux 1.0) maxCos) ux) zi))))

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-cos.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-sin.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. lower-PI.f32 32.1

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

5. Applied rewrites 31.9%

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

7. Applied rewrites 98.7%

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

8. Final simplification 98.7%

   \[\leadsto \left(yi \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) + xi \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right)\right) - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
9. Add Preprocessing

Alternative 4: 84.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{if}\;2 \cdot uy \leq 0.11500000208616257:\\ \;\;\;\;\left(\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2 + xi\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot xi - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (* (- ux 1.0) maxCos) ux) zi)))
   (if (<= (* 2.0 uy) 0.11500000208616257)
     (- (+ (* (* (* yi (PI)) uy) 2.0) xi) t_0)
     (- (* (cos (* (* (PI) uy) 2.0)) xi) t_0))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.115000002

   1. Initial program 99.2%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 35.1

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

   5. Applied rewrites 35.1%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 55.7%

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

   10. Applied rewrites 89.6%

       \[\leadsto \left(\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2 + xi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

   if 0.115000002 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 97.5%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 7.7

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

   5. Applied rewrites 7.7%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 17.7%

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

   9. Taylor expanded in xi around inf

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

   11. Applied rewrites 58.3%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.11500000208616257:\\ \;\;\;\;\left(\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2 + xi\right) - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot xi - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ t_1 := \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2 - t\_0\\ \mathbf{if}\;yi \leq -1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;yi \leq 8.00000002901995 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot yi, 2, xi\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (* (- ux 1.0) maxCos) ux) zi))
        (t_1 (- (* (* (* yi (PI)) uy) 2.0) t_0)))
   (if (<= yi -1.0000000036274937e-15)
     t_1
     (if (<= yi 8.00000002901995e-15)
       (- (fma (* (* (PI) uy) yi) 2.0 xi) t_0)
       t_1))))

Derivation

1. Split input into 2 regimes

2. if yi < -1e-15 or 8.00000003e-15 < yi

   1. Initial program 98.5%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 46.3

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

   5. Applied rewrites 46.3%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 25.7%

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

   9. Taylor expanded in xi around 0

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

   11. Applied rewrites 54.7%

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

   if -1e-15 < yi < 8.00000003e-15

   1. Initial program 99.4%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 20.0

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

   5. Applied rewrites 20.0%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 72.0%

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

   10. Applied rewrites 72.0%

       \[\leadsto \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot yi, 2, xi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2 - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{elif}\;yi \leq 8.00000002901995 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot yi, 2, xi\right) - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2 - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.6% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ t_1 := \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2 - t\_0\\ \mathbf{if}\;yi \leq -1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;yi \leq 8.00000002901995 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(yi \cdot 2, \mathsf{PI}\left(\right) \cdot uy, xi\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (* (- ux 1.0) maxCos) ux) zi))
        (t_1 (- (* (* (* yi (PI)) uy) 2.0) t_0)))
   (if (<= yi -1.0000000036274937e-15)
     t_1
     (if (<= yi 8.00000002901995e-15)
       (- (fma (* yi 2.0) (* (PI) uy) xi) t_0)
       t_1))))

Derivation

1. Split input into 2 regimes

2. if yi < -1e-15 or 8.00000003e-15 < yi

   1. Initial program 98.5%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 46.3

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

   5. Applied rewrites 46.9%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 25.7%

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

   9. Taylor expanded in xi around 0

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

   11. Applied rewrites 54.7%

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

   if -1e-15 < yi < 8.00000003e-15

   1. Initial program 99.4%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 20.0

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

   5. Applied rewrites 20.0%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 72.0%

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

   10. Applied rewrites 72.0%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2 - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{elif}\;yi \leq 8.00000002901995 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(yi \cdot 2, \mathsf{PI}\left(\right) \cdot uy, xi\right) - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2 - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\\ t_1 := \mathsf{fma}\left(t\_0, 2, xi\right)\\ t_2 := maxCos \cdot \left(1 - ux\right)\\ \mathbf{if}\;xi \leq -5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(zi \cdot t\_2, ux, t\_1\right)\\ \mathbf{elif}\;xi \leq 6.000000233523199 \cdot 10^{-16}:\\ \;\;\;\;t\_0 \cdot 2 - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(zi \cdot ux, t\_2, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* yi (PI)) uy))
        (t_1 (fma t_0 2.0 xi))
        (t_2 (* maxCos (- 1.0 ux))))
   (if (<= xi -5.000000229068525e-19)
     (fma (* zi t_2) ux t_1)
     (if (<= xi 6.000000233523199e-16)
       (- (* t_0 2.0) (* (* (* (- ux 1.0) maxCos) ux) zi))
       (fma (* zi ux) t_2 t_1)))))

Derivation

1. Split input into 3 regimes

2. if xi < -5.00000023e-19

   1. Initial program 99.1%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 12.9

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

   5. Applied rewrites 12.9%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 69.5%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

   10. Applied rewrites 67.1%

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

   if -5.00000023e-19 < xi < 6.0000002e-16

   1. Initial program 98.9%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 51.2

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 36.6%

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

   9. Taylor expanded in xi around 0

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

   11. Applied rewrites 61.2%

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

   if 6.0000002e-16 < xi

   1. Initial program 99.1%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 17.6

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

   5. Applied rewrites 17.6%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 54.9%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

   10. Applied rewrites 54.6%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;xi \leq -5.000000229068525 \cdot 10^{-19}:\\ \;\;\;\;\mathsf{fma}\left(zi \cdot \left(maxCos \cdot \left(1 - ux\right)\right), ux, \mathsf{fma}\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy, 2, xi\right)\right)\\ \mathbf{elif}\;xi \leq 6.000000233523199 \cdot 10^{-16}:\\ \;\;\;\;\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2 - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(zi \cdot ux, maxCos \cdot \left(1 - ux\right), \mathsf{fma}\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy, 2, xi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.5% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\\ t_1 := \mathsf{fma}\left(\left(\left(1 - ux\right) \cdot ux\right) \cdot maxCos, zi, t\_0 \cdot 2\right)\\ \mathbf{if}\;yi \leq -1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;yi \leq 8.00000002901995 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(zi \cdot ux, maxCos \cdot \left(1 - ux\right), \mathsf{fma}\left(t\_0, 2, xi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* yi (PI)) uy))
        (t_1 (fma (* (* (- 1.0 ux) ux) maxCos) zi (* t_0 2.0))))
   (if (<= yi -1.0000000036274937e-15)
     t_1
     (if (<= yi 8.00000002901995e-15)
       (fma (* zi ux) (* maxCos (- 1.0 ux)) (fma t_0 2.0 xi))
       t_1))))

Derivation

1. Split input into 2 regimes

2. if yi < -1e-15 or 8.00000003e-15 < yi

   1. Initial program 98.5%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 46.3

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

   5. Applied rewrites 46.3%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 25.7%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

   10. Applied rewrites 25.6%

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

   11. Taylor expanded in xi around 0

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

   13. Applied rewrites 53.9%

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

   if -1e-15 < yi < 8.00000003e-15

   1. Initial program 99.4%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 20.0

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

   5. Applied rewrites 20.0%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 72.0%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

   10. Applied rewrites 61.3%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(1 - ux\right) \cdot ux\right) \cdot maxCos, zi, \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right)\\ \mathbf{elif}\;yi \leq 8.00000002901995 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(zi \cdot ux, maxCos \cdot \left(1 - ux\right), \mathsf{fma}\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy, 2, xi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(1 - ux\right) \cdot ux\right) \cdot maxCos, zi, \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 38.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\\ t_1 := \mathsf{fma}\left(\left(\left(1 - ux\right) \cdot ux\right) \cdot maxCos, zi, t\_0 \cdot 2\right)\\ \mathbf{if}\;yi \leq -1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;yi \leq 8.00000002901995 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi, \mathsf{fma}\left(t\_0, 2, xi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* yi (PI)) uy))
        (t_1 (fma (* (* (- 1.0 ux) ux) maxCos) zi (* t_0 2.0))))
   (if (<= yi -1.0000000036274937e-15)
     t_1
     (if (<= yi 8.00000002901995e-15)
       (fma (* maxCos ux) zi (fma t_0 2.0 xi))
       t_1))))

Derivation

1. Split input into 2 regimes

2. if yi < -1e-15 or 8.00000003e-15 < yi

   1. Initial program 98.5%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 46.3

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

   5. Applied rewrites 46.3%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 25.6%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

   10. Applied rewrites 25.6%

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

   11. Taylor expanded in xi around 0

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

   13. Applied rewrites 54.7%

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

   if -1e-15 < yi < 8.00000003e-15

   1. Initial program 99.4%

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

   3. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-cos.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 20.0

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

   5. Applied rewrites 20.0%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 72.6%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

   10. Applied rewrites 61.3%

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

   11. Taylor expanded in ux around 0

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

       1. lower-*.f32 61.8

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

   13. Applied rewrites 61.8%

       \[\leadsto \mathsf{fma}\left(\color{blue}{maxCos \cdot ux}, zi, \mathsf{fma}\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy, 2, xi\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -1.0000000036274937 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(1 - ux\right) \cdot ux\right) \cdot maxCos, zi, \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right)\\ \mathbf{elif}\;yi \leq 8.00000002901995 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi, \mathsf{fma}\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy, 2, xi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(1 - ux\right) \cdot ux\right) \cdot maxCos, zi, \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.5% accurate, 8.8× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (- (+ (* (* (* yi (PI)) uy) 2.0) xi) (* (* (* (- ux 1.0) maxCos) ux) zi)))

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-cos.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-sin.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. lower-PI.f32 31.9

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

5. Applied rewrites 31.9%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 51.3%

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

10. Applied rewrites 82.2%

    \[\leadsto \left(\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2 + xi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]

11. Final simplification 82.2%

    \[\leadsto \left(\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2 + xi\right) - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
12. Add Preprocessing

Alternative 11: 50.7% accurate, 9.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(\left(1 - ux\right) \cdot ux\right) \cdot maxCos, zi, \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2 + xi\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma (* (* (- 1.0 ux) ux) maxCos) zi (+ (* (* (* yi (PI)) uy) 2.0) xi)))

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-cos.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-sin.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. lower-PI.f32 31.9

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

5. Applied rewrites 31.9%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 51.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

10. Applied rewrites 45.4%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\left(ux \cdot \left(1 - ux\right)\right) \cdot maxCos, zi, \mathsf{fma}\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy, 2, xi\right)\right)} \]
11. Step-by-step derivation

12. Applied rewrites 75.2%

    \[\leadsto \mathsf{fma}\left(\left(ux \cdot \left(1 - ux\right)\right) \cdot maxCos, zi, \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2 + xi\right) \]

13. Final simplification 75.6%

    \[\leadsto \mathsf{fma}\left(\left(\left(1 - ux\right) \cdot ux\right) \cdot maxCos, zi, \left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2 + xi\right) \]
14. Add Preprocessing

Alternative 12: 32.2% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(maxCos \cdot ux, zi, \mathsf{fma}\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy, 2, xi\right)\right) \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (fma (* maxCos ux) zi (fma (* (* yi (PI)) uy) 2.0 xi)))

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-cos.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-sin.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. lower-PI.f32 31.9

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

5. Applied rewrites 31.9%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 51.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

10. Applied rewrites 45.4%

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

11. Taylor expanded in ux around 0

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

    1. lower-*.f32 45.4

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

13. Applied rewrites 45.4%

    \[\leadsto \mathsf{fma}\left(\color{blue}{maxCos \cdot ux}, zi, \mathsf{fma}\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy, 2, xi\right)\right) \]
14. Add Preprocessing

Alternative 13: 13.6% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(zi - zi \cdot ux\right) \cdot ux\right) \cdot maxCos \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (* (* (- zi (* zi ux)) ux) maxCos))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return ((zi - (zi * ux)) * ux) * maxCos;
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = ((zi - (zi * ux)) * ux) * maxcos
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(Float32(zi - Float32(zi * ux)) * ux) * maxCos)
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = ((zi - (zi * ux)) * ux) * maxCos;
end

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in zi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 13.0

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

5. Applied rewrites 13.0%

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

6. Taylor expanded in ux around 0

   \[\leadsto \left(\left(zi + -1 \cdot \left(ux \cdot zi\right)\right) \cdot ux\right) \cdot maxCos \]
7. Step-by-step derivation

8. Applied rewrites 13.0%

   \[\leadsto \left(\left(zi - zi \cdot ux\right) \cdot ux\right) \cdot maxCos \]
9. Add Preprocessing

Alternative 14: 13.6% accurate, 18.6× speedup

Math

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

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (* (* (* zi (- 1.0 ux)) maxCos) ux))

C

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

Fortran

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

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(Float32(zi * Float32(Float32(1.0) - ux)) * maxCos) * ux)
end

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in zi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 13.0

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

5. Applied rewrites 13.0%

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

7. Applied rewrites 13.0%

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

8. Final simplification 13.0%

   \[\leadsto \left(\left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos\right) \cdot ux \]
9. Add Preprocessing

Alternative 15: 12.1% accurate, 32.1× speedup

Math

\[\begin{array}{l} \\ \left(zi \cdot ux\right) \cdot maxCos \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos) :precision binary32 (* (* zi ux) maxCos))

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return (zi * ux) * maxCos;
}

Fortran

real(4) function code(xi, yi, zi, ux, uy, maxcos)
    real(4), intent (in) :: xi
    real(4), intent (in) :: yi
    real(4), intent (in) :: zi
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = (zi * ux) * maxcos
end function

Julia

function code(xi, yi, zi, ux, uy, maxCos)
	return Float32(Float32(zi * ux) * maxCos)
end

MATLAB

function tmp = code(xi, yi, zi, ux, uy, maxCos)
	tmp = (zi * ux) * maxCos;
end

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in zi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 13.0

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

5. Applied rewrites 13.0%

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

6. Taylor expanded in ux around 0

   \[\leadsto \left(ux \cdot zi\right) \cdot maxCos \]
7. Step-by-step derivation

8. Applied rewrites 12.1%

   \[\leadsto \left(zi \cdot ux\right) \cdot maxCos \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024285 
(FPCore (xi yi zi ux uy maxCos)
  :name "UniformSampleCone 2"
  :precision binary32
  :pre (and (and (and (and (and (and (<= -10000.0 xi) (<= xi 10000.0)) (and (<= -10000.0 yi) (<= yi 10000.0))) (and (<= -10000.0 zi) (<= zi 10000.0))) (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) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) xi) (* (* (sin (* (* uy 2.0) (PI))) (sqrt (- 1.0 (* (* (* (- 1.0 ux) maxCos) ux) (* (* (- 1.0 ux) maxCos) ux))))) yi)) (* (* (* (- 1.0 ux) maxCos) ux) zi)))