UniformSampleCone 2

Percentage Accurate: 98.9% → 99.0%

Time: 10.6s

Alternatives: 17

Speedup: 0.5×

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: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

4. Applied rewrites 99.1%

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

Alternative 2: 98.8% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

5. Applied rewrites 99.1%

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

Alternative 3: 95.8% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in ux around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

5. Applied rewrites 96.1%

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

Alternative 4: 97.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\\ t_1 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;uy \leq 0.012000000104308128:\\ \;\;\;\;xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot t\_1, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(t\_1 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos t\_0, xi, \sin t\_0 \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (PI) (* 2.0 uy))) (t_1 (* (PI) (PI))))
   (if (<= uy 0.012000000104308128)
     (+
      xi
      (fma
       maxCos
       (* ux (* zi (- 1.0 ux)))
       (*
        uy
        (fma
         2.0
         (* yi (PI))
         (*
          uy
          (fma
           -2.0
           (* xi t_1)
           (* -1.3333333333333333 (* uy (* yi (* t_1 (PI)))))))))))
     (fma (cos t_0) xi (* (sin t_0) yi)))))

Derivation

1. Split input into 2 regimes

2. if uy < 0.0120000001

   1. Initial program 99.3%

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

   3. Taylor expanded in maxCos around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. *-commutative N/A

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

   5. Applied rewrites 99.3%

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

   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

      1. lower-+.f32 N/A

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

      2. lower-fma.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      5. lift--.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

   8. Applied rewrites 99.2%

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

      1. lift-PI.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      2. lift-pow.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      3. unpow3 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      4. pow2 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      6. pow2 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      7. lift-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      8. lift-PI.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      9. lift-PI.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      10. lift-PI.f32 99.2

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

   10. Applied rewrites 99.2%

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

   if 0.0120000001 < 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

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. Applied rewrites 93.7%

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

Alternative 5: 93.1% accurate, 2.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

5. Applied rewrites 99.1%

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

   1. lift-fma.f32 N/A

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

   2. lift-cos.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-sin.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lower-+.f32 N/A

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

7. Applied rewrites 99.0%

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

8. Taylor expanded in uy around 0

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

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

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

   2. lower--.f32 N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. pow2 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-*.f32 92.5

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

10. Applied rewrites 92.5%

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

Alternative 6: 92.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\\ t_1 := \left(1 - ux\right) \cdot zi\\ \mathbf{if}\;yi \leq 1.000000031374395 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, t\_1, \mathsf{fma}\left(\cos t\_0, xi, \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot yi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, t\_1, xi + \sin t\_0 \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (PI) (* 2.0 uy))) (t_1 (* (- 1.0 ux) zi)))
   (if (<= yi 1.000000031374395e-22)
     (fma (* maxCos ux) t_1 (fma (cos t_0) xi (* (* 2.0 (* uy (PI))) yi)))
     (fma (* maxCos ux) t_1 (+ xi (* (sin t_0) yi))))))

Derivation

1. Split input into 2 regimes

2. if yi < 1.00000003e-22

   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

      1. associate-*r* N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. *-commutative N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lift-PI.f32 91.3

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

   8. Applied rewrites 91.3%

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

   if 1.00000003e-22 < 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

      1. associate-*r* N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. *-commutative N/A

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

   5. Applied rewrites 98.9%

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

      1. lift-fma.f32 N/A

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

      2. lift-cos.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-sin.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. lower-+.f32 N/A

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in uy around 0

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

   10. Applied rewrites 93.9%

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

Alternative 7: 92.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;uy \leq 0.07000000029802322:\\ \;\;\;\;xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot t\_0, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(t\_0 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, \left(1 - ux\right) \cdot zi, xi + \sin \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (PI) (PI))))
   (if (<= uy 0.07000000029802322)
     (+
      xi
      (fma
       maxCos
       (* ux (* zi (- 1.0 ux)))
       (*
        uy
        (fma
         2.0
         (* yi (PI))
         (*
          uy
          (fma
           -2.0
           (* xi t_0)
           (* -1.3333333333333333 (* uy (* yi (* t_0 (PI)))))))))))
     (fma
      (* maxCos ux)
      (* (- 1.0 ux) zi)
      (+ xi (* (sin (* (PI) (* 2.0 uy))) yi))))))

Derivation

1. Split input into 2 regimes

2. if uy < 0.0700000003

   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

      1. associate-*r* N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. *-commutative N/A

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

   5. Applied rewrites 99.3%

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

   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

      1. lower-+.f32 N/A

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

      2. lower-fma.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      5. lift--.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

   8. Applied rewrites 97.0%

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

      1. lift-PI.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      2. lift-pow.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      3. unpow3 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      4. pow2 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      6. pow2 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      7. lift-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      8. lift-PI.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      9. lift-PI.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      10. lift-PI.f32 97.0

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

   10. Applied rewrites 97.0%

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

   if 0.0700000003 < uy

   1. Initial program 97.6%

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

   3. Taylor expanded in maxCos around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. *-commutative N/A

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

   5. Applied rewrites 97.8%

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

      1. lift-fma.f32 N/A

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

      2. lift-cos.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-sin.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. lower-+.f32 N/A

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in uy around 0

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

   10. Applied rewrites 61.3%

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

Alternative 8: 91.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;uy \leq 0.07000000029802322:\\ \;\;\;\;xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot t\_0, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(t\_0 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, xi, \sin \left(\mathsf{PI}\left(\right) \cdot \left(2 \cdot uy\right)\right) \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (PI) (PI))))
   (if (<= uy 0.07000000029802322)
     (+
      xi
      (fma
       maxCos
       (* ux (* zi (- 1.0 ux)))
       (*
        uy
        (fma
         2.0
         (* yi (PI))
         (*
          uy
          (fma
           -2.0
           (* xi t_0)
           (* -1.3333333333333333 (* uy (* yi (* t_0 (PI)))))))))))
     (fma 1.0 xi (* (sin (* (PI) (* 2.0 uy))) yi)))))

Derivation

1. Split input into 2 regimes

2. if uy < 0.0700000003

   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

      1. associate-*r* N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lift--.f32 N/A

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

      7. *-commutative N/A

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

   5. Applied rewrites 99.3%

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

   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

      1. lower-+.f32 N/A

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

      2. lower-fma.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      3. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      4. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      5. lift--.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      6. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

   8. Applied rewrites 97.0%

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

      1. lift-PI.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      2. lift-pow.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

      3. unpow3 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      4. pow2 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      6. pow2 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      7. lift-*.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      8. lift-PI.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      9. lift-PI.f32 N/A

         \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

      10. lift-PI.f32 97.0

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

   10. Applied rewrites 97.0%

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

   if 0.0700000003 < uy

   1. Initial program 97.6%

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

   3. Taylor expanded in 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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

   5. Applied rewrites 93.4%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 59.5%

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

Alternative 9: 89.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot t\_0, -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(t\_0 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (PI) (PI))))
   (+
    xi
    (fma
     maxCos
     (* ux (* zi (- 1.0 ux)))
     (*
      uy
      (fma
       2.0
       (* yi (PI))
       (*
        uy
        (fma
         -2.0
         (* xi t_0)
         (* -1.3333333333333333 (* uy (* yi (* t_0 (PI)))))))))))))

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

5. Applied rewrites 99.1%

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

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \color{blue}{\left(zi \cdot \left(1 - ux\right)\right)}, uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \color{blue}{\left(1 - ux\right)}\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - \color{blue}{ux}\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

   5. lift--.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \left(2 \cdot \left(yi \cdot \mathsf{PI}\left(\right)\right) + uy \cdot \left(-2 \cdot \left(xi \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

8. Applied rewrites 87.5%

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

   1. lift-PI.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

   2. lift-pow.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)\right)\right)\right) \]

   3. unpow3 N/A

      \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

   4. pow2 N/A

      \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

   5. lower-*.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

   6. pow2 N/A

      \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

   7. lift-*.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

   8. lift-PI.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

   9. lift-PI.f32 N/A

      \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \frac{-4}{3} \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]

   10. lift-PI.f32 87.5

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

10. Applied rewrites 87.5%

    \[\leadsto xi + \mathsf{fma}\left(maxCos, ux \cdot \left(zi \cdot \left(1 - ux\right)\right), uy \cdot \mathsf{fma}\left(2, yi \cdot \mathsf{PI}\left(\right), uy \cdot \mathsf{fma}\left(-2, xi \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), -1.3333333333333333 \cdot \left(uy \cdot \left(yi \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 10: 86.0% accurate, 5.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

5. Applied rewrites 99.1%

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

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

8. Applied rewrites 83.5%

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

9. Taylor expanded in maxCos around inf

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

    1. lower-*.f32 N/A

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

    2. lower-fma.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. lift--.f32 N/A

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

    5. lower-/.f32 N/A

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

11. Applied rewrites 83.5%

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

Alternative 11: 86.1% accurate, 6.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

5. Applied rewrites 99.1%

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

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

8. Applied rewrites 83.5%

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

Alternative 12: 83.4% accurate, 6.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

5. Applied rewrites 99.1%

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

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

8. Applied rewrites 83.5%

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

9. Taylor expanded in ux around 0

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

11. Applied rewrites 80.9%

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

12. Final simplification 80.9%

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

Alternative 13: 82.0% accurate, 9.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in uy around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 79.3

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

8. Applied rewrites 79.3%

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

Alternative 14: 82.0% accurate, 9.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in uy around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lift--.f32 79.2

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

8. Applied rewrites 79.2%

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

Alternative 15: 74.7% accurate, 18.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

5. Applied rewrites 99.1%

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

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

8. Applied rewrites 83.5%

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

9. Taylor expanded in yi around inf

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

    1. lower-*.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. lift-PI.f32 71.6

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

11. Applied rewrites 71.6%

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

Alternative 16: 32.2% accurate, 22.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. *-commutative N/A

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

5. Applied rewrites 99.1%

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

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-fma.f32 N/A

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

8. Applied rewrites 83.5%

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

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

    1. lower-*.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. lift-PI.f32 28.4

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

11. Applied rewrites 28.4%

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

Alternative 17: 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

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 = (zi * ux) * maxcos
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in zi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

      \[\leadsto \left(\left(zi \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot maxCos \]

   4. lower-*.f32 N/A

      \[\leadsto \left(\left(zi \cdot \left(1 - ux\right)\right) \cdot ux\right) \cdot maxCos \]

   5. *-commutative N/A

      \[\leadsto \left(\left(\left(1 - ux\right) \cdot zi\right) \cdot ux\right) \cdot maxCos \]

   6. lower-*.f32 N/A

      \[\leadsto \left(\left(\left(1 - ux\right) \cdot zi\right) \cdot ux\right) \cdot maxCos \]

   7. lift--.f32 13.5

      \[\leadsto \left(\left(\left(1 - ux\right) \cdot zi\right) \cdot ux\right) \cdot maxCos \]

5. Applied rewrites 13.5%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 12.1%

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

Reproduce

herbie shell --seed 2025051 
(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)))