UniformSampleCone 2

Percentage Accurate: 99.0% → 98.8%

Time: 17.0s

Alternatives: 24

Speedup: N/A×

Specification

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

Math

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

FPCore

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

Initial Program: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.8% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f32 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. *-commutative N/A

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

5. Applied rewrites 98.8%

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

7. Applied rewrites 99.0%

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

Alternative 2: 98.8% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f32 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. *-commutative N/A

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

5. Applied rewrites 98.8%

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

7. Applied rewrites 98.9%

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

Alternative 3: 98.8% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f32 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. *-commutative N/A

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

5. Applied rewrites 98.8%

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

7. Applied rewrites 98.8%

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

Alternative 4: 98.8% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f32 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. *-commutative N/A

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

5. Applied rewrites 98.8%

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

Alternative 5: 95.7% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f32 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

5. Applied rewrites 96.1%

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

7. Applied rewrites 96.2%

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

Alternative 6: 95.7% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f32 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

5. Applied rewrites 96.1%

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

Alternative 7: 97.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\\ \mathbf{if}\;uy \leq 0.010999999940395355:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right), xi, \mathsf{fma}\left(t\_0, yi, \left(\left(\left(1 - ux\right) \cdot zi\right) \cdot ux\right) \cdot maxCos\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, uy, 0.5\right)\right), xi, t\_0 \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sin (* (* (PI) uy) 2.0))))
   (if (<= uy 0.010999999940395355)
     (fma
      (fma (* -2.0 (* uy uy)) (* (PI) (PI)) 1.0)
      xi
      (fma t_0 yi (* (* (* (- 1.0 ux) zi) ux) maxCos)))
     (fma (sin (* (PI) (fma -2.0 uy 0.5))) xi (* t_0 yi)))))

Derivation

1. Split input into 2 regimes

2. if uy < 0.0109999999

   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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f32 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. *-commutative N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 99.3%

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

   if 0.0109999999 < uy

   1. Initial program 96.7%

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

   3. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f32 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. *-commutative N/A

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

   5. Applied rewrites 96.9%

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in zi around 0

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

   10. Applied rewrites 90.6%

       \[\leadsto \mathsf{fma}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, uy, 0.5\right)\right), \color{blue}{xi}, \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot yi\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.010999999940395355:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right), xi, \mathsf{fma}\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right), yi, \left(\left(\left(1 - ux\right) \cdot zi\right) \cdot ux\right) \cdot maxCos\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(-2, uy, 0.5\right)\right), xi, \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot yi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot uy\\ t_1 := \sin \left(t\_0 \cdot 2\right)\\ \mathbf{if}\;uy \leq 0.010999999940395355:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-2 \cdot \left(uy \cdot uy\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right), xi, \mathsf{fma}\left(t\_1, yi, \left(\left(\left(1 - ux\right) \cdot zi\right) \cdot ux\right) \cdot maxCos\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(-2 \cdot t\_0\right), xi, t\_1 \cdot yi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (PI) uy)) (t_1 (sin (* t_0 2.0))))
   (if (<= uy 0.010999999940395355)
     (fma
      (fma (* -2.0 (* uy uy)) (* (PI) (PI)) 1.0)
      xi
      (fma t_1 yi (* (* (* (- 1.0 ux) zi) ux) maxCos)))
     (fma (cos (* -2.0 t_0)) xi (* t_1 yi)))))

Derivation

1. Split input into 2 regimes

2. if uy < 0.0109999999

   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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f32 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. *-commutative N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 99.3%

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

   if 0.0109999999 < uy

   1. Initial program 96.7%

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

   3. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f32 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. *-commutative N/A

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

   5. Applied rewrites 96.9%

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

   6. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f32 N/A

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

      3. cos-neg-rev N/A

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

      4. lower-cos.f32 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

      13. lower-sin.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower-PI.f32 90.0

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

   8. Applied rewrites 90.0%

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

Alternative 9: 93.0% accurate, 2.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f32 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. *-commutative N/A

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 93.9%

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

Alternative 10: 90.0% accurate, 2.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f32 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

5. Applied rewrites 96.1%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 91.3%

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

Alternative 11: 88.3% accurate, 2.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f32 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. *-commutative N/A

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in zi around inf

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

8. Applied rewrites 98.7%

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

9. Taylor expanded in uy around 0

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

11. Applied rewrites 89.4%

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

Alternative 12: 90.1% accurate, 2.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f32 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. *-commutative N/A

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 88.7%

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

Alternative 13: 90.1% accurate, 2.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f32 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. *-commutative N/A

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 88.7%

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

Alternative 14: 88.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0042500002309679985:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - ux\right) \cdot ux, zi \cdot maxCos, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot xi\right) \cdot uy, -2, \left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right), uy, xi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, xi, \mathsf{fma}\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right), yi, \left(zi \cdot ux\right) \cdot maxCos\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0042500002309679985)
   (fma
    (* (- 1.0 ux) ux)
    (* zi maxCos)
    (fma (fma (* (* (* (PI) (PI)) xi) uy) -2.0 (* (* yi (PI)) 2.0)) uy xi))
   (fma 1.0 xi (fma (sin (* (* (PI) uy) 2.0)) yi (* (* zi ux) maxCos)))))

Derivation

1. Split input into 2 regimes

2. if uy < 0.00425000023

   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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f32 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. *-commutative N/A

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 97.8%

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

   10. Applied rewrites 97.9%

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

   if 0.00425000023 < uy

   1. Initial program 97.0%

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

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f32 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f32 N/A

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

   5. Applied rewrites 95.8%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 68.3%

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

Alternative 15: 88.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.05999999865889549:\\ \;\;\;\;\mathsf{fma}\left(\left(1 - ux\right) \cdot ux, zi \cdot maxCos, \mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot xi\right) \cdot uy, -2, \left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right), uy, xi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right), yi, \left(zi \cdot ux\right) \cdot maxCos\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (<= uy 0.05999999865889549)
   (fma
    (* (- 1.0 ux) ux)
    (* zi maxCos)
    (fma (fma (* (* (* (PI) (PI)) xi) uy) -2.0 (* (* yi (PI)) 2.0)) uy xi))
   (fma (sin (* (* (PI) uy) 2.0)) yi (* (* zi ux) maxCos))))

Derivation

1. Split input into 2 regimes

2. if uy < 0.0599999987

   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. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f32 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. *-commutative N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 92.5%

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

   10. Applied rewrites 92.6%

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

   if 0.0599999987 < uy

   1. Initial program 95.7%

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

   3. Taylor expanded in ux around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f32 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f32 N/A

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in xi around 0

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

   8. Applied rewrites 61.0%

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

Alternative 16: 85.8% accurate, 6.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f32 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. *-commutative N/A

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 85.3%

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

10. Applied rewrites 85.4%

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

Alternative 17: 85.8% accurate, 6.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f32 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. *-commutative N/A

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in xi around 0

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

8. Applied rewrites 47.4%

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

9. Taylor expanded in uy around 0

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

11. Applied rewrites 85.4%

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

Alternative 18: 82.9% accurate, 7.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f32 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

5. Applied rewrites 96.1%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 82.9%

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

Alternative 19: 81.7% accurate, 9.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f32 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. *-commutative N/A

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in xi around 0

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

8. Applied rewrites 47.4%

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

9. Taylor expanded in uy around 0

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

11. Applied rewrites 81.8%

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

Alternative 20: 60.8% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;yi \leq -2.7819999637301862 \cdot 10^{-9} \lor \neg \left(yi \leq 1.1999999620013451 \cdot 10^{-14}\right):\\ \;\;\;\;\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), xi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (or (<= yi -2.7819999637301862e-9) (not (<= yi 1.1999999620013451e-14)))
   (* (* (* yi (PI)) uy) 2.0)
   (fma (* maxCos ux) (* zi (- 1.0 ux)) xi)))

Derivation

1. Split input into 2 regimes

2. if yi < -2.78199996e-9 or 1.19999996e-14 < yi

   1. Initial program 98.5%

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

   3. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f32 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. *-commutative N/A

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 80.2%

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

   9. Taylor expanded in yi around inf

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

   11. Applied rewrites 59.2%

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

   if -2.78199996e-9 < yi < 1.19999996e-14

   1. Initial program 98.8%

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

   3. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f32 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. *-commutative N/A

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 67.0%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -2.7819999637301862 \cdot 10^{-9} \lor \neg \left(yi \leq 1.1999999620013451 \cdot 10^{-14}\right):\\ \;\;\;\;\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi \cdot \left(1 - ux\right), xi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 58.9% accurate, 12.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;yi \leq -2.7819999637301862 \cdot 10^{-9} \lor \neg \left(yi \leq 1.1999999620013451 \cdot 10^{-14}\right):\\ \;\;\;\;\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi, xi\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (xi yi zi ux uy maxCos)
 :precision binary32
 (if (or (<= yi -2.7819999637301862e-9) (not (<= yi 1.1999999620013451e-14)))
   (* (* (* yi (PI)) uy) 2.0)
   (fma (* maxCos ux) zi xi)))

Derivation

1. Split input into 2 regimes

2. if yi < -2.78199996e-9 or 1.19999996e-14 < yi

   1. Initial program 98.5%

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

   3. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f32 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. *-commutative N/A

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 80.2%

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

   9. Taylor expanded in yi around inf

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

   11. Applied rewrites 59.2%

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

   if -2.78199996e-9 < yi < 1.19999996e-14

   1. Initial program 98.8%

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

   3. Taylor expanded in ux around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f32 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f32 N/A

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

      7. distribute-lft-neg-in N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-PI.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f32 N/A

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in uy around 0

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

   8. Applied rewrites 64.3%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;yi \leq -2.7819999637301862 \cdot 10^{-9} \lor \neg \left(yi \leq 1.1999999620013451 \cdot 10^{-14}\right):\\ \;\;\;\;\left(\left(yi \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(maxCos \cdot ux, zi, xi\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 78.9% accurate, 12.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f32 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

5. Applied rewrites 96.1%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 79.3%

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

Alternative 23: 50.1% accurate, 29.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in ux around 0

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

   1. +-commutative N/A

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

   2. associate-+l+ N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f32 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f32 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

5. Applied rewrites 96.1%

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

6. Taylor expanded in uy around 0

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

8. Applied rewrites 47.9%

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

Alternative 24: 12.4% 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 98.7%

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

3. Taylor expanded in zi around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 13.6

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

5. Applied rewrites 13.6%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 12.5%

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

Reproduce

herbie shell --seed 2024356 
(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)))