UniformSampleCone 2

Percentage Accurate: 98.9% → 98.8%

Time: 17.3s

Alternatives: 22

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux\\ t_1 := \left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\\ \left(\left(\cos t\_1 \cdot \sqrt{1 - t\_0 \cdot t\_0}\right) \cdot xi + \left(\sin t\_1 \cdot \sqrt{\left(\frac{\frac{\frac{1}{ux \cdot ux} - maxCos \cdot maxCos}{ux} + \left(maxCos \cdot maxCos\right) \cdot 2}{ux} - maxCos \cdot maxCos\right) \cdot {ux}^{4}}\right) \cdot yi\right) + t\_0 \cdot zi \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (- 1.0 ux) maxCos) ux)) (t_1 (* (* uy 2.0) (PI))))
   (+
    (+
     (* (* (cos t_1) (sqrt (- 1.0 (* t_0 t_0)))) xi)
     (*
      (*
       (sin t_1)
       (sqrt
        (*
         (-
          (/
           (+
            (/ (- (/ 1.0 (* ux ux)) (* maxCos maxCos)) ux)
            (* (* maxCos maxCos) 2.0))
           ux)
          (* maxCos maxCos))
         (pow ux 4.0))))
      yi))
    (* t_0 zi))))

Derivation

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 -inf

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

5. Applied rewrites 98.9%

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

6. Final simplification 98.9%

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

Alternative 2: 98.9% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

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. Add Preprocessing

Alternative 3: 98.8% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

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

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

5. Applied rewrites 98.8%

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

Alternative 4: 98.7% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in ux around 0

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

5. Applied rewrites 98.2%

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

Alternative 5: 98.7% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in maxCos around 0

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

5. Applied rewrites 98.2%

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

Alternative 6: 98.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot uy\\ \mathbf{if}\;uy \leq 0.0022799998987466097:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), xi\right) + \mathsf{fma}\left(\mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{3} \cdot yi\right) \cdot uy, -1.3333333333333333, \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot xi\right) \cdot -2\right), uy, \left(\mathsf{PI}\left(\right) \cdot yi\right) \cdot 2\right) \cdot uy\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(zi \cdot ux, maxCos, \mathsf{fma}\left(\cos \left(t\_0 \cdot -2\right), xi, \sin \left(t\_0 \cdot 2\right) \cdot yi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (PI) uy)))
   (if (<= uy 0.0022799998987466097)
     (+
      (fma (* maxCos ux) (* zi (- 1.0 ux)) xi)
      (*
       (fma
        (fma
         (* (* (pow (PI) 3.0) yi) uy)
         -1.3333333333333333
         (* (* (* (PI) (PI)) xi) -2.0))
        uy
        (* (* (PI) yi) 2.0))
       uy))
     (fma
      (* zi ux)
      maxCos
      (fma (cos (* t_0 -2.0)) xi (* (sin (* t_0 2.0)) yi))))))

Derivation

1. Split input into 2 regimes

2. if uy < 0.0022799999

   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 maxCos around 0

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in uy around 0

      \[\leadsto xi + \color{blue}{\left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.4%

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), xi\right) + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{3} \cdot yi\right) \cdot uy, -1.3333333333333333, \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot xi\right) \cdot -2\right), uy, \left(\mathsf{PI}\left(\right) \cdot yi\right) \cdot 2\right) \cdot uy} \]

   if 0.0022799999 < uy

   1. Initial program 98.1%

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

   3. Taylor expanded in ux around -inf

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

   5. Applied rewrites 98.2%

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

   7. Applied rewrites 97.9%

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

   8. Taylor expanded in ux around 0

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

   10. Applied rewrites 93.2%

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

Alternative 7: 98.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot uy\\ \mathbf{if}\;uy \leq 0.0022799998987466097:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), xi\right) + \mathsf{fma}\left(\mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{3} \cdot yi\right) \cdot uy, -1.3333333333333333, \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot xi\right) \cdot -2\right), uy, \left(\mathsf{PI}\left(\right) \cdot yi\right) \cdot 2\right) \cdot uy\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi, \mathsf{fma}\left(\cos \left(-2 \cdot t\_0\right), xi, \sin \left(t\_0 \cdot 2\right) \cdot yi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (PI) uy)))
   (if (<= uy 0.0022799998987466097)
     (+
      (fma (* maxCos ux) (* zi (- 1.0 ux)) xi)
      (*
       (fma
        (fma
         (* (* (pow (PI) 3.0) yi) uy)
         -1.3333333333333333
         (* (* (* (PI) (PI)) xi) -2.0))
        uy
        (* (* (PI) yi) 2.0))
       uy))
     (fma
      (* maxCos ux)
      zi
      (fma (cos (* -2.0 t_0)) xi (* (sin (* t_0 2.0)) yi))))))

Derivation

1. Split input into 2 regimes

2. if uy < 0.0022799999

   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 maxCos around 0

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in uy around 0

      \[\leadsto xi + \color{blue}{\left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.4%

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), xi\right) + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{3} \cdot yi\right) \cdot uy, -1.3333333333333333, \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot xi\right) \cdot -2\right), uy, \left(\mathsf{PI}\left(\right) \cdot yi\right) \cdot 2\right) \cdot uy} \]

   if 0.0022799999 < uy

   1. Initial program 98.1%

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

   3. Taylor expanded in ux around -inf

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in ux around 0

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

   8. Applied rewrites 93.2%

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

Alternative 8: 98.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0022799998987466097:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), xi\right) + \mathsf{fma}\left(\mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{3} \cdot yi\right) \cdot uy, -1.3333333333333333, \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot xi\right) \cdot -2\right), uy, \left(\mathsf{PI}\left(\right) \cdot yi\right) \cdot 2\right) \cdot uy\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot uy\right)\right), xi, \mathsf{fma}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right), yi, \left(zi \cdot ux\right) \cdot maxCos\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0022799998987466097)
   (+
    (fma (* maxCos ux) (* zi (- 1.0 ux)) xi)
    (*
     (fma
      (fma
       (* (* (pow (PI) 3.0) yi) uy)
       -1.3333333333333333
       (* (* (* (PI) (PI)) xi) -2.0))
      uy
      (* (* (PI) yi) 2.0))
     uy))
   (fma
    (cos (* -2.0 (* (PI) uy)))
    xi
    (fma (sin (* (PI) (* 2.0 uy))) yi (* (* zi ux) maxCos)))))

Derivation

1. Split input into 2 regimes

2. if uy < 0.0022799999

   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 maxCos around 0

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in uy around 0

      \[\leadsto xi + \color{blue}{\left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.4%

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), xi\right) + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{3} \cdot yi\right) \cdot uy, -1.3333333333333333, \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot xi\right) \cdot -2\right), uy, \left(\mathsf{PI}\left(\right) \cdot yi\right) \cdot 2\right) \cdot uy} \]

   if 0.0022799999 < uy

   1. Initial program 98.1%

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

   3. Taylor expanded in ux around 0

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

   5. Applied rewrites 93.2%

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

Alternative 9: 97.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.010599999688565731:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), xi\right) + \mathsf{fma}\left(\mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{3} \cdot yi\right) \cdot uy, -1.3333333333333333, \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot xi\right) \cdot -2\right), uy, \left(\mathsf{PI}\left(\right) \cdot yi\right) \cdot 2\right) \cdot uy\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot uy\right)\right), xi, \sin \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.010599999688565731)
   (+
    (fma (* maxCos ux) (* zi (- 1.0 ux)) xi)
    (*
     (fma
      (fma
       (* (* (pow (PI) 3.0) yi) uy)
       -1.3333333333333333
       (* (* (* (PI) (PI)) xi) -2.0))
      uy
      (* (* (PI) yi) 2.0))
     uy))
   (fma (cos (* -2.0 (* (PI) uy))) xi (* (sin (* (PI) (* 2.0 uy))) yi))))

Derivation

1. Split input into 2 regimes

2. if uy < 0.0105999997

   1. Initial program 99.1%

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

   3. Taylor expanded in maxCos around 0

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in uy around 0

      \[\leadsto xi + \color{blue}{\left(maxCos \cdot \left(ux \cdot \left(zi \cdot \left(1 - ux\right)\right)\right) + uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 98.3%

      \[\leadsto \mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), xi\right) + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left({\mathsf{PI}\left(\right)}^{3} \cdot yi\right) \cdot uy, -1.3333333333333333, \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot xi\right) \cdot -2\right), uy, \left(\mathsf{PI}\left(\right) \cdot yi\right) \cdot 2\right) \cdot uy} \]

   if 0.0105999997 < uy

   1. Initial program 98.0%

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

   3. Taylor expanded in ux around 0

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

   5. Applied rewrites 88.1%

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

Alternative 10: 92.8% accurate, 2.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in maxCos around 0

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

5. Applied rewrites 98.2%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 92.2%

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

Alternative 11: 91.3% accurate, 2.2× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

2. if yi < -8.00000003e-15

   1. Initial program 98.6%

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

   3. Taylor expanded in maxCos around 0

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 93.9%

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

   if -8.00000003e-15 < yi

   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 maxCos around 0

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 93.5%

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

Alternative 12: 87.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := zi \cdot \left(1 - ux\right)\\ \mathbf{if}\;uy \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, t\_0, xi\right) + \mathsf{fma}\left(\mathsf{PI}\left(\right), \left(xi \cdot \mathsf{PI}\left(\right)\right) \cdot \left(uy \cdot -2\right), \left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot uy\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot uy\right)\right), xi, \left(maxCos \cdot ux\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* zi (- 1.0 ux))))
   (if (<= uy 0.05000000074505806)
     (+
      (fma (* maxCos ux) t_0 xi)
      (* (fma (PI) (* (* xi (PI)) (* uy -2.0)) (* (* yi (PI)) 2.0)) uy))
     (fma (cos (* -2.0 (* (PI) uy))) xi (* (* maxCos ux) t_0)))))

Derivation

1. Split input into 2 regimes

2. if uy < 0.0500000007

   1. Initial program 99.1%

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

   3. Taylor expanded in maxCos around 0

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 93.4%

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

   10. Applied rewrites 93.4%

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

   if 0.0500000007 < uy

   1. Initial program 97.9%

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

   3. Taylor expanded in maxCos around 0

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

   5. Applied rewrites 97.7%

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

   6. Taylor expanded in yi around 0

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

   8. Applied rewrites 63.4%

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

Alternative 13: 87.9% accurate, 2.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in maxCos around 0

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

5. Applied rewrites 98.2%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 88.0%

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

Alternative 14: 83.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(1 - ux\right) \cdot maxCos\right) \cdot ux \leq 7.000000051862236 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot xi\right) \cdot uy - \mathsf{PI}\left(\right) \cdot yi\right), uy, \left(zi \cdot ux\right) \cdot maxCos\right) + xi\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - ux\right) \cdot ux, maxCos, \frac{\mathsf{fma}\left(uy \cdot yi, \mathsf{PI}\left(\right) \cdot 2, xi\right)}{zi}\right) \cdot zi\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= (* (* (- 1.0 ux) maxCos) ux) 7.000000051862236e-17)
   (+
    (fma
     (* -2.0 (- (* (* (* (PI) (PI)) xi) uy) (* (PI) yi)))
     uy
     (* (* zi ux) maxCos))
    xi)
   (*
    (fma (* (- 1.0 ux) ux) maxCos (/ (fma (* uy yi) (* (PI) 2.0) xi) zi))
    zi)))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 (-.f32 #s(literal 1 binary32) ux) maxCos) ux) < 7.00000005e-17

   1. Initial program 98.8%

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

   3. Taylor expanded in maxCos around 0

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 85.3%

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

   9. Taylor expanded in ux around 0

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

   11. Applied rewrites 85.0%

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

   if 7.00000005e-17 < (*.f32 (*.f32 (-.f32 #s(literal 1 binary32) ux) maxCos) ux)

   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 maxCos around 0

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in xi around inf

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

   8. Applied rewrites 40.2%

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

   9. Taylor expanded in zi around inf

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

   11. Applied rewrites 97.2%

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

   12. Taylor expanded in uy around 0

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

   14. Applied rewrites 86.1%

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

Alternative 15: 85.5% accurate, 6.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in maxCos around 0

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

5. Applied rewrites 98.2%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 85.8%

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

10. Applied rewrites 85.8%

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

Alternative 16: 85.5% accurate, 6.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in maxCos around 0

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

5. Applied rewrites 98.2%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 85.8%

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

9. Taylor expanded in ux around 0

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

11. Applied rewrites 85.8%

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

Alternative 17: 85.5% accurate, 6.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in maxCos around 0

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

5. Applied rewrites 98.2%

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

6. Taylor expanded in xi around inf

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

8. Applied rewrites 50.8%

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

9. Taylor expanded in uy around 0

   \[\leadsto xi \]
10. Step-by-step derivation

11. Applied rewrites 43.6%

    \[\leadsto xi \]

12. Taylor expanded in uy around 0

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

14. Applied rewrites 85.8%

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

Alternative 18: 81.0% accurate, 9.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in maxCos around 0

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

5. Applied rewrites 98.2%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 85.8%

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

9. Taylor expanded in xi around 0

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

11. Applied rewrites 82.1%

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

Alternative 19: 78.5% accurate, 11.8× speedup

Math

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

FPCore

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

Derivation

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 uy around 0

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

5. Applied rewrites 82.7%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 78.0%

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

Alternative 20: 56.7% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;xi \leq -5.000000156871975 \cdot 10^{-23} \lor \neg \left(xi \leq 2.00000009162741 \cdot 10^{-18}\right):\\ \;\;\;\;xi\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot yi\right) \cdot uy\right) \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (or (<= xi -5.000000156871975e-23) (not (<= xi 2.00000009162741e-18)))
   xi
   (* (* (* (PI) yi) uy) 2.0)))

Derivation

1. Split input into 2 regimes

2. if xi < -5.00000016e-23 or 2.00000009e-18 < xi

   1. Initial program 99.1%

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

   3. Taylor expanded in maxCos around 0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in xi around inf

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

   8. Applied rewrites 71.8%

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

   9. Taylor expanded in uy around 0

      \[\leadsto xi \]
   10. Step-by-step derivation

   11. Applied rewrites 60.6%

       \[\leadsto xi \]

   if -5.00000016e-23 < xi < 2.00000009e-18

   1. Initial program 98.6%

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

   3. Taylor expanded in maxCos around 0

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 86.7%

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

   9. Taylor expanded in yi around inf

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

   11. Applied rewrites 52.7%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;xi \leq -5.000000156871975 \cdot 10^{-23} \lor \neg \left(xi \leq 2.00000009162741 \cdot 10^{-18}\right):\\ \;\;\;\;xi\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot yi\right) \cdot uy\right) \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 73.9% accurate, 20.8× speedup

Math

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

FPCore

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

Derivation

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 uy around 0

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

5. Applied rewrites 82.7%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 72.2%

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

Alternative 22: 45.9% accurate, 353.0× speedup

Math

\[\begin{array}{l} \\ xi \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(4) function code(xi, yi, zi, ux, uy, maxcos)
use fmin_fmax_functions
    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 = xi
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 98.9%

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

3. Taylor expanded in maxCos around 0

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

5. Applied rewrites 98.2%

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

6. Taylor expanded in xi around inf

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

8. Applied rewrites 50.8%

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

9. Taylor expanded in uy around 0

   \[\leadsto xi \]
10. Step-by-step derivation

11. Applied rewrites 43.6%

    \[\leadsto xi \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025019 
(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)))