Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.3% → 100.0%

Time: 16.9s

Alternatives: 8

Speedup: 1.6×

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: 99.3% 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: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(\tan \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(u1, 2, 0.5\right)\right) \cdot \frac{alphay}{alphax}\right)\\ e^{\mathsf{log1p}\left(\frac{u0}{\left({\left(\frac{\sin t\_0}{alphay}\right)}^{2} + {\left(\frac{\cos t\_0}{alphax}\right)}^{2}\right) \cdot \left(1 - u0\right)}\right) \cdot -0.5} \end{array} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0 (atan (* (tan (* (PI) (fma u1 2.0 0.5))) (/ alphay alphax)))))
   (exp
    (*
     (log1p
      (/
       u0
       (*
        (+ (pow (/ (sin t_0) alphay) 2.0) (pow (/ (cos t_0) alphax) 2.0))
        (- 1.0 u0))))
     -0.5))))

Derivation

1. Initial program 99.2%

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

4. Applied rewrites 99.9%

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

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{u0}{{\left(\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \mathsf{PI}\left(\right)\right)\right)}{alphax}\right)}^{2} + {\left(\frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \mathsf{PI}\left(\right)\right)\right)}{alphay}\right)}^{2}}}{1 - u0}\right) \cdot -0.5}} \]

8. Applied rewrites 100.0%

   \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{u0}{\left({\left(\frac{\sin \tan^{-1} \left(\tan \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(u1, 2, 0.5\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(u1, 2, 0.5\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}\right) \cdot \left(1 - u0\right)}}\right) \cdot -0.5} \]
9. Add Preprocessing

Alternative 2: 99.9% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (pow
  (-
   (/
    (/
     u0
     (+
      (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 (* (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 99.2%

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{{\left(\frac{\frac{u0}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{\cos \tan^{-1} \left(\tan \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), u1 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphax}\right)}^{2}}}{1 - u0} - -1\right)}^{-0.5}} \]
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 99.9%

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

Alternative 3: 99.4% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.2%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

4. Applied rewrites 99.3%

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

   1. lift-cos.f32 N/A

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

   2. lift-atan.f32 N/A

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

   3. cos-atan N/A

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

   4. lower-/.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f32 N/A

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

6. Applied rewrites 99.3%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. distribute-rgt-in N/A

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

   8. +-commutative N/A

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

   9. lift-fma.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 99.3

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

   12. lift-fma.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 99.3

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

8. Applied rewrites 99.3%

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

Alternative 4: 98.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \left(0.5 + 2 \cdot u1\right)\\ e^{\mathsf{log1p}\left(\frac{alphay \cdot alphay}{{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \frac{\sin t\_0}{\cos t\_0}\right)}^{2}} \cdot \frac{u0}{1 - u0}\right) \cdot -0.5} \end{array} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0 (* (PI) (+ 0.5 (* 2.0 u1)))))
   (exp
    (*
     (log1p
      (*
       (/
        (* alphay alphay)
        (pow (sin (atan (* (/ alphay alphax) (/ (sin t_0) (cos t_0))))) 2.0))
       (/ u0 (- 1.0 u0))))
     -0.5))))

Derivation

1. Initial program 99.2%

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

4. Applied rewrites 99.9%

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

6. Applied rewrites 100.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{u0}{{\left(\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \mathsf{PI}\left(\right)\right)\right)}{alphax}\right)}^{2} + {\left(\frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\mathsf{fma}\left(2, u1, 0.5\right) \cdot \mathsf{PI}\left(\right)\right)\right)}{alphay}\right)}^{2}}}{1 - u0}\right) \cdot -0.5}} \]

7. Taylor expanded in alphax around inf

   \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{{alphay}^{2} \cdot u0}{{\sin \tan^{-1} \left(\frac{alphay \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}{alphax \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot u1\right)\right)}\right)}^{2} \cdot \left(1 - u0\right)}}\right) \cdot \frac{-1}{2}} \]

9. Applied rewrites 98.7%

   \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{alphay \cdot alphay}{{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \frac{\sin \left(\mathsf{PI}\left(\right) \cdot \left(0.5 + 2 \cdot u1\right)\right)}{\cos \left(\mathsf{PI}\left(\right) \cdot \left(0.5 + 2 \cdot u1\right)\right)}\right)}^{2}} \cdot \frac{u0}{1 - u0}}\right) \cdot -0.5} \]
10. Add Preprocessing

Alternative 5: 98.0% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (pow
  (-
   (/
    (/
     (* (* 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 99.2%

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

3. Taylor expanded in alphax around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-pow.f32 N/A

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

5. Applied rewrites 98.1%

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

7. Applied rewrites 98.6%

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

Alternative 6: 97.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{\frac{\left(alphay \cdot alphay\right) \cdot u0}{0.5 - 0.5 \cdot \cos \left(2 \cdot \tan^{-1} \left(\tan \left(1.5 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)\right)}}{1 - u0} - -1\right)}^{-0.5} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (pow
  (-
   (/
    (/
     (* (* alphay alphay) u0)
     (-
      0.5
      (* 0.5 (cos (* 2.0 (atan (* (tan (* 1.5 (PI))) (/ alphay alphax))))))))
    (- 1.0 u0))
   -1.0)
  -0.5))

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in alphax around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-pow.f32 N/A

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

5. Applied rewrites 98.1%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 97.8%

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

10. Applied rewrites 98.3%

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

12. Applied rewrites 98.3%

    \[\leadsto {\left(\frac{\frac{\left(alphay \cdot alphay\right) \cdot u0}{0.5 - \color{blue}{0.5 \cdot \cos \left(2 \cdot \tan^{-1} \left(\tan \left(1.5 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)\right)}}}{1 - u0} - -1\right)}^{-0.5} \]
13. Add Preprocessing

Alternative 7: 97.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{\frac{\frac{\left(alphay \cdot alphay\right) \cdot u0}{0.5 - 0.5 \cdot \cos \left(2 \cdot \tan^{-1} \left(\tan \left(1.5 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)\right)}}{1 - u0} - -1}} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (/
  1.0
  (sqrt
   (-
    (/
     (/
      (* (* alphay alphay) u0)
      (-
       0.5
       (* 0.5 (cos (* 2.0 (atan (* (tan (* 1.5 (PI))) (/ alphay alphax))))))))
     (- 1.0 u0))
    -1.0))))

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in alphax around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-pow.f32 N/A

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

5. Applied rewrites 98.1%

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

6. Taylor expanded in u1 around 0

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

8. Applied rewrites 97.8%

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

10. Applied rewrites 97.8%

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

12. Applied rewrites 97.8%

    \[\leadsto \frac{1}{\sqrt{\frac{\frac{\left(alphay \cdot alphay\right) \cdot u0}{0.5 - \color{blue}{0.5 \cdot \cos \left(2 \cdot \tan^{-1} \left(\tan \left(1.5 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)\right)}}}{1 - u0} - -1}} \]
13. Add Preprocessing

Alternative 8: 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 99.2%

   \[\frac{1}{\sqrt{1 + \frac{\frac{1}{\frac{\cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \cos \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphax \cdot alphax} + \frac{\sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{alphay}{alphax} \cdot \tan \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u1 + 0.5 \cdot \mathsf{PI}\left(\right)\right)\right)}{alphay \cdot alphay}} \cdot u0}{1 - u0}}} \]
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 2025015 
(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))))))