UniformSampleCone 2

Percentage Accurate: 99.0% → 99.0%

Time: 16.1s

Alternatives: 14

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 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: 99.0% accurate, 0.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

4. Applied rewrites 98.7%

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

5. Final simplification 98.7%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in ux around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 98.7

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

5. Applied rewrites 98.7%

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

6. Final simplification 98.7%

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

Alternative 3: 98.9% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
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 98.5

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

5. Applied rewrites 98.5%

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

6. Final simplification 98.5%

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

Alternative 4: 98.8% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in ux around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 98.7

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

5. Applied rewrites 98.7%

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

6. Taylor expanded in ux around 0

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

   1. lower-cos.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 98.4

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

8. Applied rewrites 98.4%

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

9. Final simplification 98.4%

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

Alternative 5: 98.8% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-cos.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 98.4

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

5. Applied rewrites 98.4%

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

6. Final simplification 98.4%

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

Alternative 6: 33.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(ux \cdot ux\right) \cdot {\left(1 - ux\right)}^{2}\\ t_1 := \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{if}\;xi \leq -5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{1 - {\left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)}^{2}} \cdot xi - t\_1\\ \mathbf{elif}\;xi \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(-maxCos\right) \cdot maxCos, t\_0, 1\right)} \cdot \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot yi\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot t\_0} \cdot xi + \left(\left(maxCos - maxCos \cdot ux\right) \cdot ux\right) \cdot zi\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* ux ux) (pow (- 1.0 ux) 2.0)))
        (t_1 (* (* (* (- ux 1.0) maxCos) ux) zi)))
   (if (<= xi -5.0000000843119176e-17)
     (- (* (sqrt (- 1.0 (pow (* (* maxCos (- 1.0 ux)) ux) 2.0))) xi) t_1)
     (if (<= xi 1.9999999920083944e-11)
       (-
        (*
         (sqrt (fma (* (- maxCos) maxCos) t_0 1.0))
         (* (sin (* (* (PI) uy) 2.0)) yi))
        t_1)
       (+
        (* (sqrt (- 1.0 (* (* maxCos maxCos) t_0))) xi)
        (* (* (- maxCos (* maxCos ux)) ux) zi))))))

Derivation

1. Split input into 3 regimes

2. if xi < -5.00000008e-17

   1. Initial program 99.2%

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

   3. Taylor expanded in ux around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   8. Applied rewrites 7.3%

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

   10. Applied rewrites 63.5%

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

   if -5.00000008e-17 < xi < 1.99999999e-11

   1. Initial program 98.5%

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

   3. Taylor expanded in xi around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   5. Applied rewrites 9.5%

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

   if 1.99999999e-11 < xi

   1. Initial program 98.7%

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

   3. Taylor expanded in ux around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   8. Applied rewrites 66.6%

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

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;xi \leq -5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{1 - {\left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)}^{2}} \cdot xi - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{elif}\;xi \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\left(-maxCos\right) \cdot maxCos, \left(ux \cdot ux\right) \cdot {\left(1 - ux\right)}^{2}, 1\right)} \cdot \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot yi\right) - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot {\left(1 - ux\right)}^{2}\right)} \cdot xi + \left(\left(maxCos - maxCos \cdot ux\right) \cdot ux\right) \cdot zi\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ t_1 := \sqrt{1 - {\left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)}^{2}}\\ \mathbf{if}\;xi \leq -5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;t\_1 \cdot xi - t\_0\\ \mathbf{elif}\;xi \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;\sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(t\_1 \cdot yi\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot {\left(1 - ux\right)}^{2}\right)} \cdot xi + \left(\left(maxCos - maxCos \cdot ux\right) \cdot ux\right) \cdot zi\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (* (- ux 1.0) maxCos) ux) zi))
        (t_1 (sqrt (- 1.0 (pow (* (* maxCos (- 1.0 ux)) ux) 2.0)))))
   (if (<= xi -5.0000000843119176e-17)
     (- (* t_1 xi) t_0)
     (if (<= xi 1.9999999920083944e-11)
       (- (* (sin (* (* 2.0 uy) (PI))) (* t_1 yi)) t_0)
       (+
        (*
         (sqrt
          (- 1.0 (* (* maxCos maxCos) (* (* ux ux) (pow (- 1.0 ux) 2.0)))))
         xi)
        (* (* (- maxCos (* maxCos ux)) ux) zi))))))

Derivation

1. Split input into 3 regimes

2. if xi < -5.00000008e-17

   1. Initial program 99.2%

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

   3. Taylor expanded in ux around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   8. Applied rewrites 7.3%

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

   10. Applied rewrites 63.5%

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

   if -5.00000008e-17 < xi < 1.99999999e-11

   1. Initial program 98.5%

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

   3. Taylor expanded in xi around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   5. Applied rewrites 9.6%

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

   7. Applied rewrites 69.5%

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

   if 1.99999999e-11 < xi

   1. Initial program 98.7%

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

   3. Taylor expanded in ux around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   8. Applied rewrites 66.6%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;xi \leq -5.0000000843119176 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{1 - {\left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)}^{2}} \cdot xi - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{elif}\;xi \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;\sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\sqrt{1 - {\left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)}^{2}} \cdot yi\right) - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot {\left(1 - ux\right)}^{2}\right)} \cdot xi + \left(\left(maxCos - maxCos \cdot ux\right) \cdot ux\right) \cdot zi\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot yi\\ \mathbf{if}\;yi \leq -1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;yi \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot {\left(1 - ux\right)}^{2}\right)} \cdot xi + \left(\left(maxCos - maxCos \cdot ux\right) \cdot ux\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\mathsf{fma}\left(-2, ux, 1\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (sin (* (* (PI) uy) 2.0)) yi)))
   (if (<= yi -1.999999936531045e-19)
     t_0
     (if (<= yi 4.99999991225835e-15)
       (+
        (*
         (sqrt
          (- 1.0 (* (* maxCos maxCos) (* (* ux ux) (pow (- 1.0 ux) 2.0)))))
         xi)
        (* (* (- maxCos (* maxCos ux)) ux) zi))
       (*
        (sqrt (- 1.0 (* (* (fma -2.0 ux 1.0) (* ux ux)) (* maxCos maxCos))))
        t_0)))))

Derivation

1. Split input into 3 regimes

2. if yi < -1.99999994e-19

   1. Initial program 98.7%

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

   3. Taylor expanded in xi around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   5. Applied rewrites 5.8%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 66.1%

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

   if -1.99999994e-19 < yi < 4.99999991e-15

   1. Initial program 98.9%

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

   3. Taylor expanded in ux around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   8. Applied rewrites 66.2%

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

   if 4.99999991e-15 < yi

   1. Initial program 98.4%

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

   3. Taylor expanded in xi around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   5. Applied rewrites 6.7%

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

   6. Taylor expanded in yi around inf

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

   8. Applied rewrites 66.3%

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

   9. Taylor expanded in ux around 0

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

   11. Applied rewrites 62.6%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot yi\\ \mathbf{elif}\;yi \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{1 - \left(maxCos \cdot maxCos\right) \cdot \left(\left(ux \cdot ux\right) \cdot {\left(1 - ux\right)}^{2}\right)} \cdot xi + \left(\left(maxCos - maxCos \cdot ux\right) \cdot ux\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\mathsf{fma}\left(-2, ux, 1\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot yi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot yi\\ \mathbf{if}\;yi \leq -1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;yi \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{1 - {\left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)}^{2}} \cdot xi - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\mathsf{fma}\left(-2, ux, 1\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (sin (* (* (PI) uy) 2.0)) yi)))
   (if (<= yi -1.999999936531045e-19)
     t_0
     (if (<= yi 4.99999991225835e-15)
       (-
        (* (sqrt (- 1.0 (pow (* (* maxCos (- 1.0 ux)) ux) 2.0))) xi)
        (* (* (* (- ux 1.0) maxCos) ux) zi))
       (*
        (sqrt (- 1.0 (* (* (fma -2.0 ux 1.0) (* ux ux)) (* maxCos maxCos))))
        t_0)))))

Derivation

1. Split input into 3 regimes

2. if yi < -1.99999994e-19

   1. Initial program 98.7%

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

   3. Taylor expanded in xi around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   5. Applied rewrites 5.9%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 66.1%

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

   if -1.99999994e-19 < yi < 4.99999991e-15

   1. Initial program 98.9%

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

   3. Taylor expanded in ux around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   8. Applied rewrites 17.3%

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

   10. Applied rewrites 66.2%

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

   if 4.99999991e-15 < yi

   1. Initial program 98.4%

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

   3. Taylor expanded in xi around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   5. Applied rewrites 6.8%

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

   6. Taylor expanded in yi around inf

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

   8. Applied rewrites 66.3%

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

   9. Taylor expanded in ux around 0

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

   11. Applied rewrites 62.6%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot yi\\ \mathbf{elif}\;yi \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{1 - {\left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)}^{2}} \cdot xi - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\mathsf{fma}\left(-2, ux, 1\right) \cdot \left(ux \cdot ux\right)\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot yi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot yi\\ \mathbf{if}\;yi \leq -1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;yi \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{1 - {\left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)}^{2}} \cdot xi - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (sin (* (* (PI) uy) 2.0)) yi)))
   (if (<= yi -1.999999936531045e-19)
     t_0
     (if (<= yi 4.99999991225835e-15)
       (-
        (* (sqrt (- 1.0 (pow (* (* maxCos (- 1.0 ux)) ux) 2.0))) xi)
        (* (* (* (- ux 1.0) maxCos) ux) zi))
       t_0))))

Derivation

1. Split input into 2 regimes

2. if yi < -1.99999994e-19 or 4.99999991e-15 < 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 xi around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   5. Applied rewrites 6.2%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 66.2%

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

   if -1.99999994e-19 < yi < 4.99999991e-15

   1. Initial program 98.9%

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

   3. Taylor expanded in ux around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   8. Applied rewrites 17.3%

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

   10. Applied rewrites 66.2%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -1.999999936531045 \cdot 10^{-19}:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot yi\\ \mathbf{elif}\;yi \leq 4.99999991225835 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{1 - {\left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)}^{2}} \cdot xi - \left(\left(\left(ux - 1\right) \cdot maxCos\right) \cdot ux\right) \cdot zi\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot yi\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(ux, \left(zi \cdot \left(1 - ux\right)\right) \cdot maxCos, \sqrt{1 - {\left(\left(maxCos \cdot \left(1 - ux\right)\right) \cdot ux\right)}^{2}} \cdot xi\right)\\ \mathbf{if}\;xi \leq -4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;xi \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot yi\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0
         (fma
          ux
          (* (* zi (- 1.0 ux)) maxCos)
          (* (sqrt (- 1.0 (pow (* (* maxCos (- 1.0 ux)) ux) 2.0))) xi))))
   (if (<= xi -4.99999991225835e-14)
     t_0
     (if (<= xi 1.9999999920083944e-11)
       (* (sin (* (* (PI) uy) 2.0)) yi)
       t_0))))

Derivation

1. Split input into 2 regimes

2. if xi < -4.99999991e-14 or 1.99999999e-11 < xi

   1. Initial program 99.0%

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

   3. Taylor expanded in ux around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   8. Applied rewrites 7.4%

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

   10. Applied rewrites 65.6%

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

   if -4.99999991e-14 < xi < 1.99999999e-11

   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 xi around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   5. Applied rewrites 9.4%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 59.8%

      \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot \color{blue}{yi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 49.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(zi \cdot ux, maxCos, xi\right)\\ \mathbf{if}\;xi \leq -4.99999991225835 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;xi \leq 1.9999999920083944 \cdot 10^{-11}:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot yi\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (fma (* zi ux) maxCos xi)))
   (if (<= xi -4.99999991225835e-14)
     t_0
     (if (<= xi 1.9999999920083944e-11)
       (* (sin (* (* (PI) uy) 2.0)) yi)
       t_0))))

Derivation

1. Split input into 2 regimes

2. if xi < -4.99999991e-14 or 1.99999999e-11 < xi

   1. Initial program 99.0%

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

   3. Taylor expanded in ux around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   8. Applied rewrites 7.4%

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

   9. Taylor expanded in ux around 0

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

   11. Applied rewrites 65.6%

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

   if -4.99999991e-14 < xi < 1.99999999e-11

   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 xi around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   5. Applied rewrites 9.3%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 59.8%

      \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot \color{blue}{yi} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 30.8% accurate, 29.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in ux around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 98.7

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

5. Applied rewrites 98.7%

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

6. Taylor expanded in uy around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

8. Applied rewrites 12.7%

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

9. Taylor expanded in ux around 0

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

11. Applied rewrites 39.5%

    \[\leadsto \mathsf{fma}\left(zi \cdot ux, \color{blue}{maxCos}, xi\right) \]
12. Add Preprocessing

Alternative 14: 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 98.7%

   \[\left(\left(\cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot xi + \left(\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right)}\right) \cdot yi\right) + \left(\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\right) \cdot zi \]
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 12.7

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

5. Applied rewrites 12.7%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 11.6%

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

Reproduce

herbie shell --seed 2024295 
(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)))