Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.9%

Time: 17.1s

Alternatives: 9

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, 1.5× speedup

Math

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

FPCore

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

Alternative 2: 99.9% accurate, 1.7× speedup

Math

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

FPCore

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

8. Applied rewrites 99.8%

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

10. Applied rewrites 99.8%

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

Alternative 3: 99.0% accurate, 1.7× speedup

Math

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

FPCore

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in alphax around 0

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

    1. lower-/.f32 N/A

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

    2. distribute-rgt-in N/A

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

    3. associate-*r* N/A

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

    4. lower-cos.f32 N/A

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

    5. associate-*r* N/A

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

    6. distribute-rgt-in N/A

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

    7. lower-*.f32 N/A

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

    8. lower-PI.f32 N/A

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

    9. +-commutative N/A

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

    10. lower-fma.f32 N/A

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

    11. lower-*.f32 N/A

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

    12. distribute-rgt-in N/A

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

    13. associate-*r* N/A

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

    14. lower-sin.f32 N/A

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

11. Applied rewrites 98.7%

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

Alternative 4: 98.0% accurate, 2.4× speedup

Math

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

FPCore

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in alphay around 0

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

11. Applied rewrites 98.4%

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

Alternative 5: 98.0% accurate, 2.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

4. Applied rewrites 100.0%

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

6. Applied rewrites 100.0%

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in alphax around inf

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

11. Applied rewrites 98.4%

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

Alternative 6: 97.0% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (fma
  (*
   (/
    (* alphay alphay)
    (-
     0.5
     (*
      0.5
      (cos
       (*
        2.0
        (atan
         (*
          (/ (sin (* (PI) 0.5)) (cos (* (PI) (fma -2.0 u1 -0.5))))
          (/ alphay alphax))))))))
   (/ u0 (- 1.0 u0)))
  -0.5
  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 alphay around 0

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

5. Applied rewrites 96.4%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 96.8%

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

10. Applied rewrites 96.8%

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

Alternative 7: 96.4% accurate, 3.1× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (fma
  (/
   (* alphay alphay)
   (pow (sin (atan (* (tan (* (PI) 0.5)) (/ alphay alphax)))) 2.0))
  (* (/ u0 (- 1.0 u0)) -0.5)
  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 alphay around 0

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

5. Applied rewrites 96.4%

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

7. Applied rewrites 96.4%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 95.9%

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

12. Applied rewrites 95.9%

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

Alternative 8: 96.4% accurate, 3.8× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (fma
  (*
   (/
    (* alphay alphay)
    (fma
     (cos (* (atan (* (tan (* (PI) 0.5)) (/ alphay alphax))) -2.0))
     -0.5
     0.5))
   (/ u0 (- 1.0 u0)))
  -0.5
  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 alphay around 0

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

5. Applied rewrites 96.4%

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

7. Applied rewrites 96.4%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 95.9%

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

12. Applied rewrites 95.9%

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

Alternative 9: 91.3% accurate, 1436.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in u0 around 0

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

5. Applied rewrites 91.4%

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

Reproduce

herbie shell --seed 2025017 
(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))))))