Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.9%

Time: 23.1s

Alternatives: 8

Speedup: 1.3×

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: 99.9% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

4. Applied rewrites 78.7%

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

6. Applied rewrites 79.5%

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

7. Taylor expanded in alphax around 0

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

9. Applied rewrites 98.5%

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

10. Taylor expanded in u1 around inf

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

12. Applied rewrites 99.9%

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

13. Final simplification 99.9%

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

Alternative 2: 99.2% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

4. Applied rewrites 78.7%

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

6. Applied rewrites 78.8%

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

7. Taylor expanded in alphax around 0

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

9. Applied rewrites 98.5%

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

10. Taylor expanded in alphax around 0

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

12. Applied rewrites 99.0%

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

13. Final simplification 99.0%

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

Alternative 3: 98.7% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

4. Applied rewrites 78.6%

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

6. Applied rewrites 79.1%

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

7. Taylor expanded in alphax around 0

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

9. Applied rewrites 98.5%

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

10. Taylor expanded in u1 around 0

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

12. Applied rewrites 98.5%

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

13. Final simplification 98.5%

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

Alternative 4: 98.4% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

4. Applied rewrites 79.5%

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

6. Applied rewrites 79.1%

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

7. Taylor expanded in alphax around 0

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

9. Applied rewrites 98.5%

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

10. Taylor expanded in u1 around 0

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

12. Applied rewrites 97.9%

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

13. Final simplification 97.9%

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

Alternative 5: 98.4% accurate, 2.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

4. Applied rewrites 87.5%

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

5. Taylor expanded in u1 around 0

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

7. Applied rewrites 97.6%

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

8. Taylor expanded in alphax around inf

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

10. Applied rewrites 97.6%

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

11. Final simplification 97.6%

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

Alternative 6: 91.8% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

4. Applied rewrites 79.2%

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

6. Applied rewrites 78.7%

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

7. Taylor expanded in alphax around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

9. Applied rewrites 91.6%

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

11. Applied rewrites 90.6%

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

12. Final simplification 90.3%

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

Alternative 7: 97.9% accurate, 2.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

4. Applied rewrites 79.0%

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

6. Applied rewrites 79.2%

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

7. Taylor expanded in alphax around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

9. Applied rewrites 91.6%

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

11. Applied rewrites 90.3%

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

12. Final simplification 90.3%

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

Alternative 8: 91.7% 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 99.3%

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

3. Taylor expanded in alphax around 0

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

5. Applied rewrites 90.3%

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

Reproduce

herbie shell --seed 2024278 
(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))))))