Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.4%

Time: 17.0s

Alternatives: 5

Speedup: 1.4×

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, 1.4× speedup

Math

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

FPCore

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

   1. lift-*.f32 N/A

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

   2. lift-cos.f32 N/A

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

   3. lift-atan.f32 N/A

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

   4. cos-atan N/A

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

   5. lift-cos.f32 N/A

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

   6. lift-atan.f32 N/A

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

   7. cos-atan N/A

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

4. Applied rewrites 93.3%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. distribute-rgt-in N/A

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

   8. +-commutative N/A

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

   9. lift-fma.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 98.3

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

   12. lift-fma.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 98.3

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

6. Applied rewrites 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 98.3

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

8. Applied rewrites 98.3%

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

   1. lift-fma.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 99.4

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

10. Applied rewrites 99.4%

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

11. Final simplification 99.4%

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

Alternative 2: 97.7% accurate, 1.4× speedup

Math

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

FPCore

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

   1. lift-*.f32 N/A

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

   2. lift-cos.f32 N/A

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

   3. lift-atan.f32 N/A

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

   4. cos-atan N/A

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

   5. lift-cos.f32 N/A

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

   6. lift-atan.f32 N/A

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

   7. cos-atan N/A

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

4. Applied rewrites 93.3%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. distribute-rgt-in N/A

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

   8. +-commutative N/A

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

   9. lift-fma.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 98.3

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

   12. lift-fma.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 98.3

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

6. Applied rewrites 98.3%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 98.3

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

8. Applied rewrites 98.3%

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

9. Taylor expanded in u1 around 0

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

11. Applied rewrites 98.3%

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

12. Final simplification 98.3%

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

Alternative 3: 97.7% accurate, 2.8× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0 (* (fma 2.0 u1 0.5) (PI))))
   (/
    1.0
    (sqrt
     (-
      1.0
      (/
       (/
        (* (* u0 (* alphay alphay)) 2.0)
        (-
         1.0
         (cos (* (atan (* (/ (sin t_0) (cos t_0)) (/ alphay alphax))) 2.0))))
       (- u0 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 52.3%

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

5. Taylor expanded in alphax around inf

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-cos.f32 N/A

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

7. Applied rewrites 98.0%

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

8. Final simplification 98.0%

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

Alternative 4: 96.5% accurate, 4.0× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (-
  1.0
  (/
   (* u0 (* alphay alphay))
   (*
    (- (cos (* (atan (* (tan (* 0.5 (PI))) (/ alphay alphax))) 2.0)) 1.0)
    (- u0 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 52.7%

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

5. Taylor expanded in alphay around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. associate-/r* N/A

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

7. Applied rewrites 75.6%

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

9. Applied rewrites 76.4%

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

10. Taylor expanded in u1 around 0

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

12. Applied rewrites 96.6%

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

13. Final simplification 96.6%

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

Alternative 5: 91.3% 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 91.9%

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

Reproduce

herbie shell --seed 2024308 
(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))))))