Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.3% → 99.3%

Time: 15.5s

Alternatives: 5

Speedup: 1.5×

Specification

\[\left(\left(\left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 0.5\right)\right) \land \left(0.0001 \leq alphax \land alphax \leq 1\right)\right) \land \left(0.0001 \leq alphay \land alphay \leq 1\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{t\_2 \cdot t\_2}{alphax \cdot alphax} + \frac{t\_1 \cdot t\_1}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \end{array} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0
         (atan
          (* (/ alphay alphax) (tan (+ (* (* 2.0 (PI)) u1) (* 0.5 (PI)))))))
        (t_1 (sin t_0))
        (t_2 (cos t_0)))
   (/
    1.0
    (sqrt
     (+
      1.0
      (/
       (*
        (/
         1.0
         (+
          (/ (* t_2 t_2) (* alphax alphax))
          (/ (* t_1 t_1) (* alphay alphay))))
        u0)
       (- 1.0 u0)))))))

Initial Program: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{t\_2 \cdot t\_2}{alphax \cdot alphax} + \frac{t\_1 \cdot t\_1}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \end{array} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0
         (atan
          (* (/ alphay alphax) (tan (+ (* (* 2.0 (PI)) u1) (* 0.5 (PI)))))))
        (t_1 (sin t_0))
        (t_2 (cos t_0)))
   (/
    1.0
    (sqrt
     (+
      1.0
      (/
       (*
        (/
         1.0
         (+
          (/ (* t_2 t_2) (* alphax alphax))
          (/ (* t_1 t_1) (* alphay alphay))))
        u0)
       (- 1.0 u0)))))))

Alternative 1: 99.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot 0.5\\ \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + t\_0\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\mathsf{PI}\left(\right) \cdot \left(u1 \cdot 2\right) + t\_0\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \end{array} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0 (* (PI) 0.5)))
   (/
    1.0
    (sqrt
     (+
      (/
       (/ u0 (- 1.0 u0))
       (+
        (pow
         (/
          (sin (atan (* (tan (+ (* (* (PI) 2.0) u1) t_0)) (/ alphay alphax))))
          alphay)
         2.0)
        (pow
         (/
          (cos (atan (* (tan (+ (* (PI) (* u1 2.0)) t_0)) (/ alphay alphax))))
          alphax)
         2.0)))
      1.0)))))

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
2. Add Preprocessing

4. Applied rewrites 87.3%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}}} \]
5. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)} \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   2. +-commutative N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \color{blue}{\left(u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   3. lower-+.f32 N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \color{blue}{\left(u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right) + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\color{blue}{u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   8. lower-*.f32 93.7

      \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \color{blue}{\mathsf{PI}\left(\right) \cdot 0.5}\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

6. Applied rewrites 93.7%

   \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \mathsf{PI}\left(\right) \cdot 0.5\right)} \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]
7. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)} \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \color{blue}{u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)}\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   5. *-commutative N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1}\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2} + \color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1}\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   7. +-commutative N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right)} \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   8. lift-+.f32 99.4

      \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \mathsf{PI}\left(\right) \cdot 0.5\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \mathsf{PI}\left(\right) \cdot 0.5\right)} \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1} + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   10. lift-*.f32 N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot u1 + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   11. associate-*l* N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \left(2 \cdot u1\right)} + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   12. lift-*.f32 N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(2 \cdot u1\right)} + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   13. lower-*.f32 99.4

       \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \mathsf{PI}\left(\right) \cdot 0.5\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \left(2 \cdot u1\right)} + \mathsf{PI}\left(\right) \cdot 0.5\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   14. lift-*.f32 N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(2 \cdot u1\right)} + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   15. *-commutative N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(u1 \cdot 2\right)} + \mathsf{PI}\left(\right) \cdot \frac{1}{2}\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

   16. lower-*.f32 99.4

       \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \mathsf{PI}\left(\right) \cdot 0.5\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(u1 \cdot 2\right)} + \mathsf{PI}\left(\right) \cdot 0.5\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]

8. Applied rewrites 99.4%

   \[\leadsto \frac{1}{\sqrt{\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u1 + \mathsf{PI}\left(\right) \cdot 0.5\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u1 \cdot 2\right) + \mathsf{PI}\left(\right) \cdot 0.5\right)} \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}} + 1}} \]
9. Add Preprocessing

Alternative 2: 98.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(\tan \left(\mathsf{fma}\left(u1, 2, 0.5\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)\\ \sqrt{\frac{1}{\frac{\frac{u0}{1 - u0}}{{\left(\frac{alphax}{\cos t\_0}\right)}^{-2} + {\left(\frac{alphay}{\sin t\_0}\right)}^{-2}} + 1}} \end{array} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0 (atan (* (tan (* (fma u1 2.0 0.5) (PI))) (/ alphay alphax)))))
   (sqrt
    (/
     1.0
     (+
      (/
       (/ u0 (- 1.0 u0))
       (+ (pow (/ alphax (cos t_0)) -2.0) (pow (/ alphay (sin t_0)) -2.0)))
      1.0)))))

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{1 + \frac{u0}{\left(\frac{{\cos \tan^{-1} \left(\frac{alphay \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u1 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphax \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u1 \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2}}{{alphax}^{2}} + \frac{{\sin \tan^{-1} \left(\frac{alphay \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u1 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphax \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u1 \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2}}{{alphay}^{2}}\right) \cdot \left(1 - u0\right)}}}} \]

5. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{\frac{u0}{1 - u0}}{\frac{{\sin \tan^{-1} \left(\frac{\frac{alphay}{alphax} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)}{\cos \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)}\right)}^{2}}{alphay \cdot alphay} + \frac{{\cos \tan^{-1} \left(\frac{\frac{alphay}{alphax} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)}{\cos \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)}\right)}^{2}}{alphax \cdot alphax}} + 1}}} \]

7. Applied rewrites 97.7%

   \[\leadsto \sqrt{\frac{1}{\frac{\frac{u0}{1 - u0}}{{\left(\frac{alphax}{\cos \tan^{-1} \left(\tan \left(\mathsf{fma}\left(u1, 2, 0.5\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)}\right)}^{-2} + {\left(\frac{alphay}{\sin \tan^{-1} \left(\tan \left(\mathsf{fma}\left(u1, 2, 0.5\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)}\right)}^{-2}} + 1}} \]
8. Add Preprocessing

Alternative 3: 97.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, 0.5\right)\\ \frac{1}{\sqrt{1 + \frac{\frac{\left(alphay \cdot alphay\right) \cdot u0}{{\sin \tan^{-1} \left(\frac{\frac{alphay}{alphax} \cdot \sin t\_0}{\cos t\_0}\right)}^{2}}}{1 - u0}}} \end{array} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0 (* (PI) (fma 2.0 u1 0.5))))
   (/
    1.0
    (sqrt
     (+
      1.0
      (/
       (/
        (* (* alphay alphay) u0)
        (pow (sin (atan (/ (* (/ alphay alphax) (sin t_0)) (cos t_0)))) 2.0))
       (- 1.0 u0)))))))

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
2. Add Preprocessing

3. Taylor expanded in alphax around inf

   \[\leadsto \frac{1}{\sqrt{1 + \frac{\color{blue}{\frac{{alphay}^{2} \cdot u0}{{\sin \tan^{-1} \left(\frac{alphay \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u1 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphax \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u1 \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2}}}}{1 - u0}}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{1}{\sqrt{1 + \frac{\color{blue}{\frac{{alphay}^{2} \cdot u0}{{\sin \tan^{-1} \left(\frac{alphay \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u1 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphax \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u1 \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2}}}}{1 - u0}}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{1}{\sqrt{1 + \frac{\frac{\color{blue}{{alphay}^{2} \cdot u0}}{{\sin \tan^{-1} \left(\frac{alphay \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u1 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphax \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u1 \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2}}}{1 - u0}}} \]

   3. unpow2 N/A

      \[\leadsto \frac{1}{\sqrt{1 + \frac{\frac{\color{blue}{\left(alphay \cdot alphay\right)} \cdot u0}{{\sin \tan^{-1} \left(\frac{alphay \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u1 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphax \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u1 \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2}}}{1 - u0}}} \]

   4. lower-*.f32 N/A

      \[\leadsto \frac{1}{\sqrt{1 + \frac{\frac{\color{blue}{\left(alphay \cdot alphay\right)} \cdot u0}{{\sin \tan^{-1} \left(\frac{alphay \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u1 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphax \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u1 \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2}}}{1 - u0}}} \]

   5. lower-pow.f32 N/A

      \[\leadsto \frac{1}{\sqrt{1 + \frac{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\color{blue}{{\sin \tan^{-1} \left(\frac{alphay \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u1 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphax \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u1 \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2}}}}{1 - u0}}} \]

5. Applied rewrites 97.2%

   \[\leadsto \frac{1}{\sqrt{1 + \frac{\color{blue}{\frac{\left(alphay \cdot alphay\right) \cdot u0}{{\sin \tan^{-1} \left(\frac{\frac{alphay}{alphax} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)}{\cos \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)}\right)}^{2}}}}{1 - u0}}} \]
6. Add Preprocessing

Alternative 4: 96.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{1 - {\left(\frac{alphay}{\sin \tan^{-1} \left(\tan \left(\mathsf{fma}\left(u1, 2, 0.5\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)}\right)}^{2} \cdot \frac{u0}{1 - u0}} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (sqrt
  (-
   1.0
   (*
    (pow
     (/
      alphay
      (sin (atan (* (tan (* (fma u1 2.0 0.5) (PI))) (/ alphay alphax)))))
     2.0)
    (/ u0 (- 1.0 u0))))))

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{1 + \frac{u0}{\left(\frac{{\cos \tan^{-1} \left(\frac{alphay \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u1 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphax \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u1 \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2}}{{alphax}^{2}} + \frac{{\sin \tan^{-1} \left(\frac{alphay \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u1 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphax \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + 2 \cdot \left(u1 \cdot \mathsf{PI}\left(\right)\right)\right)}\right)}^{2}}{{alphay}^{2}}\right) \cdot \left(1 - u0\right)}}}} \]

5. Applied rewrites 97.7%

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\frac{\frac{u0}{1 - u0}}{\frac{{\sin \tan^{-1} \left(\frac{\frac{alphay}{alphax} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)}{\cos \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)}\right)}^{2}}{alphay \cdot alphay} + \frac{{\cos \tan^{-1} \left(\frac{\frac{alphay}{alphax} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)}{\cos \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, u1, 0.5\right)\right)}\right)}^{2}}{alphax \cdot alphax}} + 1}}} \]

6. Taylor expanded in alphay around 0

   \[\leadsto \sqrt{1 + -1 \cdot \frac{{alphay}^{2} \cdot u0}{{\sin \tan^{-1} \left(\frac{alphay \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}\right)}^{2} \cdot \left(1 - u0\right)}} \]

8. Applied rewrites 96.2%

   \[\leadsto \sqrt{1 - \frac{alphay \cdot alphay}{{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \frac{\sin \left(\mathsf{fma}\left(u1, 2, 0.5\right) \cdot \mathsf{PI}\left(\right)\right)}{\cos \left(\mathsf{fma}\left(u1, 2, 0.5\right) \cdot \mathsf{PI}\left(\right)\right)}\right)}^{2}} \cdot \frac{u0}{1 - u0}} \]

10. Applied rewrites 96.2%

    \[\leadsto \sqrt{1 - \frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2}}} \]

12. Applied rewrites 96.2%

    \[\leadsto \color{blue}{\sqrt{1 - {\left(\frac{alphay}{\sin \tan^{-1} \left(\tan \left(\mathsf{fma}\left(u1, 2, 0.5\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)}\right)}^{2} \cdot \frac{u0}{1 - u0}}} \]
13. Add Preprocessing

Alternative 5: 91.6% accurate, 1436.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay) :precision binary32 1.0)

C

float code(float u0, float u1, float alphax, float alphay) {
	return 1.0f;
}

Fortran

real(4) function code(u0, u1, alphax, alphay)
    real(4), intent (in) :: u0
    real(4), intent (in) :: u1
    real(4), intent (in) :: alphax
    real(4), intent (in) :: alphay
    code = 1.0e0
end function

Julia

function code(u0, u1, alphax, alphay)
	return Float32(1.0)
end

MATLAB

function tmp = code(u0, u1, alphax, alphay)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
2. Add Preprocessing

3. Taylor expanded in u0 around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 91.3%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024322 
(FPCore (u0 u1 alphax alphay)
  :name "Trowbridge-Reitz Sample, sample surface normal, cosTheta"
  :precision binary32
  :pre (and (and (and (and (<= 2.328306437e-10 u0) (<= u0 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 0.5))) (and (<= 0.0001 alphax) (<= alphax 1.0))) (and (<= 0.0001 alphay) (<= alphay 1.0)))
  (/ 1.0 (sqrt (+ 1.0 (/ (* (/ 1.0 (+ (/ (* (cos (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 (PI)) u1) (* 0.5 (PI))))))) (cos (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 (PI)) u1) (* 0.5 (PI)))))))) (* alphax alphax)) (/ (* (sin (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 (PI)) u1) (* 0.5 (PI))))))) (sin (atan (* (/ alphay alphax) (tan (+ (* (* 2.0 (PI)) u1) (* 0.5 (PI)))))))) (* alphay alphay)))) u0) (- 1.0 u0))))))