Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.9%

Time: 17.8s

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, 1.7× speedup

Math

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

FPCore

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

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 100.0%

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

   1. lift-cos.f32 N/A

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

   2. lift-atan.f32 N/A

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

   3. cos-atan N/A

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

   4. lower-/.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f32 N/A

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

6. Applied rewrites 100.0%

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

8. Applied rewrites 100.0%

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

10. Applied rewrites 100.0%

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

Alternative 2: 98.1% accurate, 2.1× speedup

Math

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

FPCore

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

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 100.0%

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

5. Taylor expanded in alphax around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-fma.f32 N/A

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

7. Applied rewrites 98.6%

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

8. Taylor expanded in u1 around 0

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

10. Applied rewrites 99.0%

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

Alternative 3: 98.2% accurate, 2.5× speedup

Math

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

FPCore

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

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 100.0%

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

5. Taylor expanded in alphax around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-fma.f32 N/A

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

7. Applied rewrites 98.6%

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

9. Applied rewrites 98.6%

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

Alternative 4: 98.2% accurate, 2.6× speedup

Math

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

FPCore

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

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 100.0%

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

5. Taylor expanded in alphax around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-fma.f32 N/A

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

7. Applied rewrites 98.6%

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

9. Applied rewrites 98.6%

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

Alternative 5: 98.1% 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 99.3%

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

4. Applied rewrites 100.0%

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

   1. lift-cos.f32 N/A

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

   2. lift-atan.f32 N/A

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

   3. cos-atan N/A

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

   4. lower-/.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f32 N/A

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

6. Applied rewrites 100.0%

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

8. Applied rewrites 100.0%

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

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 98.5%

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

Alternative 6: 97.8% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (/
  1.0
  (sqrt
   (fma
    (* (/ alphay (- 1.0 u0)) alphay)
    (/
     u0
     (pow
      (sin (atan (* (tan (* (PI) (fma u1 2.0 0.5))) (/ alphay alphax))))
      2.0))
    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 100.0%

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

5. Taylor expanded in alphax around inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-fma.f32 N/A

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

7. Applied rewrites 98.6%

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

9. Applied rewrites 98.1%

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

    1. lift-pow.f32 N/A

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

    2. unpow1/2 N/A

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

    3. lower-sqrt.f32 98.1

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

11. Applied rewrites 98.1%

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

Alternative 7: 96.4% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (/
  1.0
  (fma
   (* u0 (* alphay alphay))
   (/
    0.5
    (*
     (pow
      (sin (atan (* (tan (* (PI) (fma u1 2.0 0.5))) (/ alphay alphax))))
      2.0)
     (- 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. 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 45.3%

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

6. Taylor expanded in alphay around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. times-frac N/A

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

   4. lower-fma.f32 N/A

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

8. Applied rewrites 96.7%

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

10. Applied rewrites 96.7%

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

12. Applied rewrites 96.7%

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

Alternative 8: 95.5% accurate, 3.0× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (/
  1.0
  (fma
   (* u0 0.5)
   (/
    (* alphay (fma u0 alphay alphay))
    (pow
     (sin (atan (* (tan (* (PI) (fma u1 2.0 0.5))) (/ alphay alphax))))
     2.0))
   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. 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 45.3%

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

6. Taylor expanded in alphay around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. times-frac N/A

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

   4. lower-fma.f32 N/A

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

8. Applied rewrites 96.7%

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

9. Taylor expanded in u0 around 0

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

11. Applied rewrites 95.9%

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

13. Applied rewrites 95.9%

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

Alternative 9: 91.4% 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 99.3%

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

3. Taylor expanded in u0 around 0

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

5. Applied rewrites 91.9%

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

Reproduce

herbie shell --seed 2025010 
(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))))))