Trowbridge-Reitz Sample, sample surface normal, cosTheta

Percentage Accurate: 99.4% → 99.4%

Time: 19.4s

Alternatives: 8

Speedup: 1.1×

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: 99.4% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0 (* (+ (* u1 2.0) 0.5) (PI)))
        (t_1
         (sin
          (atan
           (* (tan (+ (* (PI) 0.5) (* (* (PI) 2.0) u1))) (/ alphay alphax))))))
   (/
    1.0
    (sqrt
     (+
      (/
       (*
        u0
        (/
         1.0
         (+
          (/ (* t_1 t_1) (* alphay alphay))
          (/
           (pow (cos (atan (/ (* (sin t_0) (/ alphay alphax)) (cos t_0)))) 2.0)
           (* alphax 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 0

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

   1. lower-pow.f32 N/A

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

5. Applied rewrites 97.1%

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

7. Applied rewrites 98.4%

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

9. Applied rewrites 99.2%

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

10. Final simplification 99.2%

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

Alternative 2: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1} \left(\tan \left(\left(u1 \cdot 2 + 0.5\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)\\ {\left({\left(\frac{\frac{u0}{1 - u0}}{{\left(\frac{alphax}{\cos t\_0}\right)}^{-2} + {\left(\frac{\sin t\_0}{alphay}\right)}^{2}} + 1\right)}^{-0.25}\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0 (atan (* (tan (* (+ (* u1 2.0) 0.5) (PI))) (/ alphay alphax)))))
   (pow
    (pow
     (+
      (/
       (/ u0 (- 1.0 u0))
       (+ (pow (/ alphax (cos t_0)) -2.0) (pow (/ (sin t_0) alphay) 2.0)))
      1.0)
     -0.25)
    2.0)))

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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

4. Applied rewrites 78.5%

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

6. Applied rewrites 89.7%

   \[\leadsto \color{blue}{{\left({\left(\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\mathsf{fma}\left(u1, 2, 0.5\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{alphax}{\cos \tan^{-1} \left(\tan \left(\mathsf{fma}\left(u1, 2, 0.5\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)}\right)}^{-2}} + 1\right)}^{-0.25}\right)}^{2}} \]
7. Step-by-step derivation

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 86.0

      \[\leadsto {\left({\left(\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\mathsf{fma}\left(u1, 2, 0.5\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{alphax}{\cos \tan^{-1} \left(\tan \left(\color{blue}{\left(u1 \cdot 2 + 0.5\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)}\right)}^{-2}} + 1\right)}^{-0.25}\right)}^{2} \]

8. Applied rewrites 86.4%

   \[\leadsto {\left({\left(\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\mathsf{fma}\left(u1, 2, 0.5\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{alphax}{\cos \tan^{-1} \left(\tan \left(\color{blue}{\left(u1 \cdot 2 + 0.5\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)}\right)}^{-2}} + 1\right)}^{-0.25}\right)}^{2} \]
9. Step-by-step derivation

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 99.2

      \[\leadsto {\left({\left(\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\color{blue}{\left(u1 \cdot 2 + 0.5\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{alphax}{\cos \tan^{-1} \left(\tan \left(\left(u1 \cdot 2 + 0.5\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)}\right)}^{-2}} + 1\right)}^{-0.25}\right)}^{2} \]

10. Applied rewrites 99.2%

    \[\leadsto {\left({\left(\frac{\frac{u0}{1 - u0}}{{\left(\frac{\sin \tan^{-1} \left(\tan \left(\color{blue}{\left(u1 \cdot 2 + 0.5\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2} + {\left(\frac{alphax}{\cos \tan^{-1} \left(\tan \left(\left(u1 \cdot 2 + 0.5\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)}\right)}^{-2}} + 1\right)}^{-0.25}\right)}^{2} \]

11. Final simplification 99.2%

    \[\leadsto {\left({\left(\frac{\frac{u0}{1 - u0}}{{\left(\frac{alphax}{\cos \tan^{-1} \left(\tan \left(\left(u1 \cdot 2 + 0.5\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)}\right)}^{-2} + {\left(\frac{\sin \tan^{-1} \left(\tan \left(\left(u1 \cdot 2 + 0.5\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{alphay}{alphax}\right)}{alphay}\right)}^{2}} + 1\right)}^{-0.25}\right)}^{2} \]
12. Add Preprocessing

Alternative 3: 97.1% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0 (* (+ (* u1 2.0) 0.5) (PI))))
   (/
    1.0
    (sqrt
     (+
      (/
       (*
        (/
         1.0
         (+
          (/
           (- 1.0 (cos (* (atan (* (tan t_0) (/ alphay alphax))) 2.0)))
           (* (* alphay alphay) 2.0))
          (/
           (pow
            (cos
             (atan
              (/
               (* (sin (* (fma u1 2.0 0.5) (PI))) (/ alphay alphax))
               (cos t_0))))
            2.0)
           (* alphax alphax))))
        u0)
       (- 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 0

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

   1. lower-pow.f32 N/A

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

5. Applied rewrites 97.1%

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

7. Applied rewrites 98.4%

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

   1. lift-/.f32 N/A

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

9. Applied rewrites 97.5%

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

    1. lift-fma.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. lift-+.f32 98.4

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

11. Applied rewrites 98.4%

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

12. Final simplification 96.9%

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

Alternative 4: 89.1% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (let* ((t_0 (* (+ (* u1 2.0) 0.5) (PI))))
   (/
    1.0
    (sqrt
     (+
      (/
       (*
        (/
         1.0
         (+
          (/
           (-
            1.0
            (cos
             (*
              (atan (* (tan (* (fma u1 2.0 0.5) (PI))) (/ alphay alphax)))
              2.0)))
           (* (* alphay alphay) 2.0))
          (/
           (pow (cos (atan (/ (* (sin t_0) (/ alphay alphax)) (cos t_0)))) 2.0)
           (* alphax alphax))))
        u0)
       (- 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 0

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

   1. lower-pow.f32 N/A

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

5. Applied rewrites 97.1%

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

7. Applied rewrites 98.3%

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

   1. lift-/.f32 N/A

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

9. Applied rewrites 97.4%

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

11. Applied rewrites 62.1%

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

12. Final simplification 98.0%

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

Alternative 5: 85.8% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (/
  1.0
  (sqrt
   (+
    (/
     (*
      (/
       (* alphay alphay)
       (pow
        (sin
         (atan
          (*
           (/ alphay (cos (* (+ (* u1 2.0) 0.5) (PI))))
           (/ (sin (* (fma u1 2.0 0.5) (PI))) alphax))))
        2.0))
      u0)
     (- 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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

4. Applied rewrites 78.4%

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

5. Taylor expanded in alphax around inf

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

   1. lower-/.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

7. Applied rewrites 96.9%

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

9. Applied rewrites 97.7%

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

10. Final simplification 62.9%

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

Alternative 6: 87.3% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (u0 u1 alphax alphay)
 :precision binary32
 (/
  1.0
  (sqrt
   (+
    (/
     (*
      (/
       (* alphay alphay)
       (pow
        (sin
         (atan
          (*
           (/ alphay (cos (* (fma 2.0 u1 0.5) (PI))))
           (/ (sin (* (fma u1 2.0 0.5) (PI))) alphax))))
        2.0))
      u0)
     (- 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. Step-by-step derivation

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

4. Applied rewrites 78.6%

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

5. Taylor expanded in alphax around inf

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

   1. lower-/.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

7. Applied rewrites 97.0%

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

9. Applied rewrites 87.0%

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

10. Final simplification 97.0%

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

Alternative 7: 79.2% accurate, 3.0× speedup

Math

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

FPCore

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

4. Applied rewrites 78.5%

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

5. Taylor expanded in alphax around inf

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

   1. lower-/.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-pow.f32 N/A

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

7. Applied rewrites 96.9%

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

9. Applied rewrites 6.7%

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

    1. lift-fma.f32 N/A

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

    2. lift-*.f32 N/A

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

11. Applied rewrites 97.0%

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

Alternative 8: 91.2% accurate, 1436.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in alphax around 0

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

5. Applied rewrites 90.4%

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

Reproduce

herbie shell --seed 2024264 
(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))))))