Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.9%

Time: 20.3s

Alternatives: 6

Speedup: 1.7×

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, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 99.8%

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

11. Applied rewrites 99.8%

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

Alternative 2: 99.3% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 99.8%

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

10. Taylor expanded in u1 around 0

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

12. Applied rewrites 99.6%

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

Alternative 3: 98.6% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0 (* 2.0 (* u1 (PI)))))
   (sqrt
    (/
     1.0
     (+
      1.0
      (/
       u0
       (*
        (+
         (/
          (*
           4.0
           (/
            (* (* alphax alphax) (* (* u1 u1) (* (PI) (PI))))
            (* alphay alphay)))
          (* alphax alphax))
         (/
          (pow
           (sin (atan (* (/ (- alphay) alphax) (/ (cos t_0) (sin t_0)))))
           2.0)
          (* alphay alphay)))
        (- 1.0 u0))))))))

Derivation

1. Initial program 99.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-/.f32 N/A

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 99.8%

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

10. Taylor expanded in u1 around 0

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

12. Applied rewrites 99.0%

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

13. Final simplification 99.0%

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

Alternative 4: 98.2% accurate, 2.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in alphax around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-pow.f32 N/A

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

5. Applied rewrites 97.8%

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

7. Applied rewrites 98.4%

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

Alternative 5: 97.8% accurate, 2.9× speedup

Math

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

FPCore

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

3. Taylor expanded in alphax around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-pow.f32 N/A

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

5. Applied rewrites 97.8%

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

7. Applied rewrites 98.1%

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

Alternative 6: 91.6% accurate, 1436.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in alphax around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-pow.f32 N/A

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

5. Applied rewrites 41.0%

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

6. Taylor expanded in u0 around 0

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

8. Applied rewrites 90.7%

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

Reproduce

herbie shell --seed 2024305 
(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))))))