Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.4%

Time: 21.9s

Alternatives: 7

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: 99.4% 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: 99.4% accurate, 1.5× speedup

Math

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

FPCore

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

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

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

5. Applied rewrites 94.4%

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

7. Applied rewrites 85.7%

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

9. Applied rewrites 99.4%

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

10. Final simplification 99.4%

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

Alternative 2: 98.3% accurate, 1.5× speedup

Math

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

FPCore

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

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

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

5. Applied rewrites 94.4%

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

7. Applied rewrites 85.7%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 98.4%

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

11. Final simplification 98.4%

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

Alternative 3: 98.2% accurate, 3.0× speedup

Math

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

FPCore

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

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 alphay around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

5. Applied rewrites 95.2%

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

7. Applied rewrites 80.3%

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

9. Applied rewrites 98.2%

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

10. Final simplification 98.2%

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

Alternative 4: 98.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ {\left(1 + \frac{alphay \cdot \left(alphay \cdot u0\right)}{\left(1 - u0\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot 0.5\right)\right)\right)\right)}\right)}^{-0.5} \end{array} \]

FPCore

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

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 alphay around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

5. Applied rewrites 95.2%

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

7. Applied rewrites 80.3%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 98.0%

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

Alternative 5: 97.8% accurate, 3.6× speedup

Math

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

FPCore

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

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 alphay around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

5. Applied rewrites 95.2%

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

7. Applied rewrites 97.7%

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

9. Applied rewrites 97.9%

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

10. Final simplification 97.9%

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

Alternative 6: 97.8% accurate, 3.7× speedup

Math

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

FPCore

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

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 alphay around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

5. Applied rewrites 95.2%

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

7. Applied rewrites 97.7%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 97.7%

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

11. Final simplification 97.7%

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

Alternative 7: 91.8% 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 alphay around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

5. Applied rewrites 95.2%

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

6. Taylor expanded in alphay around 0

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

8. Applied rewrites 91.6%

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

Reproduce

herbie shell --seed 2024221 
(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))))))