UniformSampleCone 2

Percentage Accurate: 99.0% → 98.9%

Time: 13.9s

Alternatives: 11

Speedup: N/A×

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: 99.0% 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, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sin.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 98.9

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

5. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 98.9

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

7. Applied rewrites 98.9%

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

Alternative 2: 98.7% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sin.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 98.9

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

5. Applied rewrites 98.9%

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

6. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-cos.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 98.8

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

8. Applied rewrites 98.8%

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

Alternative 3: 89.9% accurate, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-sin.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 98.9

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

5. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 98.9

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

7. Applied rewrites 98.9%

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

8. Taylor expanded in uy around 0

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

10. Applied rewrites 89.4%

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

Alternative 4: 65.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\\ \mathbf{if}\;yi \leq -8.499999921696144 \cdot 10^{-7} \lor \neg \left(yi \leq 9.9999998245167 \cdot 10^{-14}\right):\\ \;\;\;\;\sin t\_0 \cdot yi\\ \mathbf{else}:\\ \;\;\;\;\cos t\_0 \cdot xi\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (PI) uy) 2.0)))
   (if (or (<= yi -8.499999921696144e-7) (not (<= yi 9.9999998245167e-14)))
     (* (sin t_0) yi)
     (* (cos t_0) xi))))

Derivation

1. Split input into 2 regimes

2. if yi < -8.49999992e-7 or 9.99999982e-14 < yi

   1. Initial program 98.6%

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

   3. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-cos.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f32 N/A

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

   5. Applied rewrites 8.5%

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

   6. Taylor expanded in maxCos around inf

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

   8. Applied rewrites 12.6%

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

   9. Taylor expanded in yi around inf

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

   11. Applied rewrites 78.7%

       \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot yi \]

   if -8.49999992e-7 < yi < 9.99999982e-14

   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 maxCos around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-cos.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f32 N/A

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

   5. Applied rewrites 12.7%

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

   6. Taylor expanded in xi around inf

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

   8. Applied rewrites 68.1%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -8.499999921696144 \cdot 10^{-7} \lor \neg \left(yi \leq 9.9999998245167 \cdot 10^{-14}\right):\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot yi\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot xi\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;yi \leq -8.499999921696144 \cdot 10^{-7} \lor \neg \left(yi \leq 9.9999998245167 \cdot 10^{-14}\right):\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot yi\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy, 2, \left(\left(zi \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot maxCos\right) + xi\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (or (<= yi -8.499999921696144e-7) (not (<= yi 9.9999998245167e-14)))
   (* (sin (* (* (PI) uy) 2.0)) yi)
   (+ (fma (* (* yi (PI)) uy) 2.0 (* (* (* zi (- 1.0 ux)) ux) maxCos)) xi)))

Derivation

1. Split input into 2 regimes

2. if yi < -8.49999992e-7 or 9.99999982e-14 < yi

   1. Initial program 98.6%

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

   3. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-cos.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f32 N/A

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

   5. Applied rewrites 8.4%

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

   6. Taylor expanded in maxCos around inf

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

   8. Applied rewrites 12.5%

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

   9. Taylor expanded in yi around inf

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

   11. Applied rewrites 78.7%

       \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot yi \]

   if -8.49999992e-7 < yi < 9.99999982e-14

   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 maxCos around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-cos.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f32 N/A

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

   5. Applied rewrites 12.7%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 65.3%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -8.499999921696144 \cdot 10^{-7} \lor \neg \left(yi \leq 9.9999998245167 \cdot 10^{-14}\right):\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot yi\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy, 2, \left(\left(zi \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot maxCos\right) + xi\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.4% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;yi \leq -8.499999921696144 \cdot 10^{-7} \lor \neg \left(yi \leq 9.9999998245167 \cdot 10^{-14}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(1 - ux\right) \cdot zi, ux, \frac{xi}{maxCos}\right), maxCos, \left(\left(\mathsf{PI}\left(\right) \cdot yi\right) \cdot uy\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy, 2, \left(\left(zi \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot maxCos\right) + xi\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (or (<= yi -8.499999921696144e-7) (not (<= yi 9.9999998245167e-14)))
   (fma
    (fma (* (- 1.0 ux) zi) ux (/ xi maxCos))
    maxCos
    (* (* (* (PI) yi) uy) 2.0))
   (+ (fma (* (* yi (PI)) uy) 2.0 (* (* (* zi (- 1.0 ux)) ux) maxCos)) xi)))

Derivation

1. Split input into 2 regimes

2. if yi < -8.49999992e-7 or 9.99999982e-14 < yi

   1. Initial program 98.6%

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

   3. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-cos.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f32 N/A

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

   5. Applied rewrites 8.4%

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

   6. Taylor expanded in maxCos around inf

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

   8. Applied rewrites 12.5%

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

   9. Taylor expanded in uy around 0

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

   11. Applied rewrites 58.7%

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

   if -8.49999992e-7 < yi < 9.99999982e-14

   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 maxCos around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-cos.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f32 N/A

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

   5. Applied rewrites 12.7%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 65.3%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -8.499999921696144 \cdot 10^{-7} \lor \neg \left(yi \leq 9.9999998245167 \cdot 10^{-14}\right):\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(1 - ux\right) \cdot zi, ux, \frac{xi}{maxCos}\right), maxCos, \left(\left(\mathsf{PI}\left(\right) \cdot yi\right) \cdot uy\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy, 2, \left(\left(zi \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot maxCos\right) + xi\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.6% accurate, 9.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-cos.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 N/A

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

5. Applied rewrites 11.4%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 51.1%

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

9. Final simplification 51.1%

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

Alternative 8: 33.3% accurate, 17.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.8%

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

3. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-cos.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 N/A

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

5. Applied rewrites 11.4%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 44.1%

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

9. Final simplification 44.2%

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

Alternative 9: 13.3% accurate, 18.6× speedup

Math

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

FPCore

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

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return (((1.0f - ux) * 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 = (((1.0e0 - ux) * zi) * ux) * maxcos
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
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 14.0

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

5. Applied rewrites 14.0%

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

Alternative 10: 13.4% accurate, 18.6× speedup

Math

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

FPCore

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

C

float code(float xi, float yi, float zi, float ux, float uy, float maxCos) {
	return (maxCos * (zi * (1.0f - ux))) * 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 = (maxcos * (zi * (1.0e0 - ux))) * ux
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.8%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
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 14.0

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

5. Applied rewrites 14.0%

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

7. Applied rewrites 14.0%

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

Alternative 11: 11.9% 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 98.8%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
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 14.0

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

5. Applied rewrites 14.0%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 12.2%

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

Reproduce

herbie shell --seed 2024324 
(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)))