Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.4%

Time: 25.7s

Alternatives: 5

Speedup: 1.0×

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, 1.0× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0
         (atan
          (* (/ alphay alphax) (tan (+ (* (* 2.0 (PI)) u1) (* (PI) 0.5))))))
        (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)))))))

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. Final simplification 99.4%

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

Alternative 2: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 97.6

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

5. Applied rewrites 97.6%

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

6. Taylor expanded in u1 around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 97.8

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

8. Applied rewrites 97.8%

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

9. Final simplification 97.8%

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

Alternative 3: 98.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \left(u1 \cdot u1\right)\\ \frac{1}{\sqrt{1 + \frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(2 \cdot u1, t\_0, 0.125\right)}{0.25 + \left(t\_0 - u1\right)}\right)\right)\right)\right)}}} \end{array} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0 (* 4.0 (* u1 u1))))
   (/
    1.0
    (sqrt
     (+
      1.0
      (/
       (* (* alphay alphay) u0)
       (*
        (- 1.0 u0)
        (+
         0.5
         (*
          -0.5
          (cos
           (*
            2.0
            (atan
             (*
              (/ alphay alphax)
              (tan
               (*
                (PI)
                (/
                 (fma (* 2.0 u1) t_0 0.125)
                 (+ 0.25 (- t_0 u1))))))))))))))))))

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 92.8%

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

7. Applied rewrites 96.8%

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

   1. +-commutative N/A

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

   2. flip3-+ N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{{\frac{1}{2}}^{3} + {\left(2 \cdot u1\right)}^{3}}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - \frac{1}{2} \cdot \left(2 \cdot u1\right)\right)}}\right)\right)\right)\right)} + 1}} \]

   3. +-commutative N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\color{blue}{{\left(2 \cdot u1\right)}^{3} + {\frac{1}{2}}^{3}}}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - \frac{1}{2} \cdot \left(2 \cdot u1\right)\right)}\right)\right)\right)\right)} + 1}} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{{\left(2 \cdot u1\right)}^{3} + {\frac{1}{2}}^{3}}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - \frac{1}{2} \cdot \left(2 \cdot u1\right)\right)}}\right)\right)\right)\right)} + 1}} \]

   5. cube-mult N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\color{blue}{\left(2 \cdot u1\right) \cdot \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right)\right)} + {\frac{1}{2}}^{3}}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - \frac{1}{2} \cdot \left(2 \cdot u1\right)\right)}\right)\right)\right)\right)} + 1}} \]

   6. lower-fma.f32 N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\color{blue}{\mathsf{fma}\left(2 \cdot u1, \left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right), {\frac{1}{2}}^{3}\right)}}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - \frac{1}{2} \cdot \left(2 \cdot u1\right)\right)}\right)\right)\right)\right)} + 1}} \]

   7. lower-*.f32 N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(\color{blue}{2 \cdot u1}, \left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right), {\frac{1}{2}}^{3}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - \frac{1}{2} \cdot \left(2 \cdot u1\right)\right)}\right)\right)\right)\right)} + 1}} \]

   8. swap-sqr N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(2 \cdot u1, \color{blue}{\left(2 \cdot 2\right) \cdot \left(u1 \cdot u1\right)}, {\frac{1}{2}}^{3}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - \frac{1}{2} \cdot \left(2 \cdot u1\right)\right)}\right)\right)\right)\right)} + 1}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(2 \cdot u1, \color{blue}{4} \cdot \left(u1 \cdot u1\right), {\frac{1}{2}}^{3}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - \frac{1}{2} \cdot \left(2 \cdot u1\right)\right)}\right)\right)\right)\right)} + 1}} \]

   10. metadata-eval N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(2 \cdot u1, \color{blue}{\left(2 + 2\right)} \cdot \left(u1 \cdot u1\right), {\frac{1}{2}}^{3}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - \frac{1}{2} \cdot \left(2 \cdot u1\right)\right)}\right)\right)\right)\right)} + 1}} \]

   11. lower-*.f32 N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(2 \cdot u1, \color{blue}{\left(2 + 2\right) \cdot \left(u1 \cdot u1\right)}, {\frac{1}{2}}^{3}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - \frac{1}{2} \cdot \left(2 \cdot u1\right)\right)}\right)\right)\right)\right)} + 1}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(2 \cdot u1, \color{blue}{4} \cdot \left(u1 \cdot u1\right), {\frac{1}{2}}^{3}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - \frac{1}{2} \cdot \left(2 \cdot u1\right)\right)}\right)\right)\right)\right)} + 1}} \]

   13. lower-*.f32 N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(2 \cdot u1, 4 \cdot \color{blue}{\left(u1 \cdot u1\right)}, {\frac{1}{2}}^{3}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - \frac{1}{2} \cdot \left(2 \cdot u1\right)\right)}\right)\right)\right)\right)} + 1}} \]

   14. metadata-eval N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(2 \cdot u1, 4 \cdot \left(u1 \cdot u1\right), \color{blue}{\frac{1}{8}}\right)}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - \frac{1}{2} \cdot \left(2 \cdot u1\right)\right)}\right)\right)\right)\right)} + 1}} \]

   15. lower-+.f32 N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(2 \cdot u1, 4 \cdot \left(u1 \cdot u1\right), \frac{1}{8}\right)}{\color{blue}{\frac{1}{2} \cdot \frac{1}{2} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - \frac{1}{2} \cdot \left(2 \cdot u1\right)\right)}}\right)\right)\right)\right)} + 1}} \]

   16. metadata-eval N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(2 \cdot u1, 4 \cdot \left(u1 \cdot u1\right), \frac{1}{8}\right)}{\color{blue}{\frac{1}{4}} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - \frac{1}{2} \cdot \left(2 \cdot u1\right)\right)}\right)\right)\right)\right)} + 1}} \]

   17. *-commutative N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(2 \cdot u1, 4 \cdot \left(u1 \cdot u1\right), \frac{1}{8}\right)}{\frac{1}{4} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - \color{blue}{\left(2 \cdot u1\right) \cdot \frac{1}{2}}\right)}\right)\right)\right)\right)} + 1}} \]

   18. *-commutative N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(2 \cdot u1, 4 \cdot \left(u1 \cdot u1\right), \frac{1}{8}\right)}{\frac{1}{4} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - \color{blue}{\left(u1 \cdot 2\right)} \cdot \frac{1}{2}\right)}\right)\right)\right)\right)} + 1}} \]

   19. associate-*l* N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(2 \cdot u1, 4 \cdot \left(u1 \cdot u1\right), \frac{1}{8}\right)}{\frac{1}{4} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - \color{blue}{u1 \cdot \left(2 \cdot \frac{1}{2}\right)}\right)}\right)\right)\right)\right)} + 1}} \]

   20. metadata-eval N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(2 \cdot u1, 4 \cdot \left(u1 \cdot u1\right), \frac{1}{8}\right)}{\frac{1}{4} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - u1 \cdot \color{blue}{1}\right)}\right)\right)\right)\right)} + 1}} \]

   21. *-commutative N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(2 \cdot u1, 4 \cdot \left(u1 \cdot u1\right), \frac{1}{8}\right)}{\frac{1}{4} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - \color{blue}{1 \cdot u1}\right)}\right)\right)\right)\right)} + 1}} \]

   22. *-lft-identity N/A

       \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(2 \cdot u1, 4 \cdot \left(u1 \cdot u1\right), \frac{1}{8}\right)}{\frac{1}{4} + \left(\left(2 \cdot u1\right) \cdot \left(2 \cdot u1\right) - \color{blue}{u1}\right)}\right)\right)\right)\right)} + 1}} \]

9. Applied rewrites 97.2%

   \[\leadsto \frac{1}{\sqrt{\frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{\mathsf{fma}\left(2 \cdot u1, 4 \cdot \left(u1 \cdot u1\right), 0.125\right)}{0.25 + \left(4 \cdot \left(u1 \cdot u1\right) - u1\right)}}\right)\right)\right)\right)} + 1}} \]

10. Final simplification 97.2%

    \[\leadsto \frac{1}{\sqrt{1 + \frac{\left(alphay \cdot alphay\right) \cdot u0}{\left(1 - u0\right) \cdot \left(0.5 + -0.5 \cdot \cos \left(2 \cdot \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{PI}\left(\right) \cdot \frac{\mathsf{fma}\left(2 \cdot u1, 4 \cdot \left(u1 \cdot u1\right), 0.125\right)}{0.25 + \left(4 \cdot \left(u1 \cdot u1\right) - u1\right)}\right)\right)\right)\right)}}} \]
11. Add Preprocessing

Alternative 4: 97.8% accurate, 3.7× speedup

Math

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

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 92.8%

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

7. Applied rewrites 96.8%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 97.1%

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

11. Final simplification 97.1%

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

Alternative 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 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 92.8%

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

6. Taylor expanded in alphay around 0

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

8. Applied rewrites 91.7%

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

Reproduce

herbie shell --seed 2024216 
(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))))))