GTR1 distribution

Percentage Accurate: 98.5% → 98.5%

Time: 9.7s

Alternatives: 9

Speedup: 1.0×

Specification

\[\left(0 \leq cosTheta \land cosTheta \leq 1\right) \land \left(0.0001 \leq \alpha \land \alpha \leq 1\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha \cdot \alpha - 1\\ \frac{t\_0}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(t\_0 \cdot cosTheta\right) \cdot cosTheta\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/
    t_0
    (* (* (PI) (log (* alpha alpha))) (+ 1.0 (* (* t_0 cosTheta) cosTheta))))))

Initial Program: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha \cdot \alpha - 1\\ \frac{t\_0}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(t\_0 \cdot cosTheta\right) \cdot cosTheta\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/
    t_0
    (* (* (PI) (log (* alpha alpha))) (+ 1.0 (* (* t_0 cosTheta) cosTheta))))))

Alternative 1: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := cosTheta \cdot \left(\alpha \cdot \alpha\right)\\ \frac{\alpha \cdot \alpha - 1}{\left(\frac{t\_0 \cdot t\_0 - cosTheta \cdot cosTheta}{t\_0 + cosTheta} \cdot cosTheta - -1\right) \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot \mathsf{PI}\left(\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (* cosTheta (* alpha alpha))))
   (/
    (- (* alpha alpha) 1.0)
    (*
     (-
      (* (/ (- (* t_0 t_0) (* cosTheta cosTheta)) (+ t_0 cosTheta)) cosTheta)
      -1.0)
     (* (log (* alpha alpha)) (PI))))))

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right)} \cdot cosTheta\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(cosTheta \cdot \left(\alpha \cdot \alpha - 1\right)\right)} \cdot cosTheta\right)} \]

   3. lift--.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(cosTheta \cdot \color{blue}{\left(\alpha \cdot \alpha - 1\right)}\right) \cdot cosTheta\right)} \]

   4. sub-neg N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(cosTheta \cdot \color{blue}{\left(\alpha \cdot \alpha + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot cosTheta\right)} \]

   5. distribute-rgt-in N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta + \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)} \cdot cosTheta\right)} \]

   6. flip-+ N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta}} \cdot cosTheta\right)} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta}} \cdot cosTheta\right)} \]

   8. lower--.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right)} - \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right)} \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   11. lower-*.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right)} - \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   12. lower-*.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(\color{blue}{-1} \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   14. neg-mul-1 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \color{blue}{\left(\mathsf{neg}\left(cosTheta\right)\right)} \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   15. lower-neg.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \color{blue}{\left(-cosTheta\right)} \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   16. metadata-eval N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(-cosTheta\right) \cdot \left(\color{blue}{-1} \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   17. neg-mul-1 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(-cosTheta\right) \cdot \color{blue}{\left(\mathsf{neg}\left(cosTheta\right)\right)}}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   18. lower-neg.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(-cosTheta\right) \cdot \color{blue}{\left(-cosTheta\right)}}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   19. lower--.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(-cosTheta\right) \cdot \left(-cosTheta\right)}{\color{blue}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta}} \cdot cosTheta\right)} \]

4. Applied rewrites 98.5%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(-cosTheta\right) \cdot \left(-cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(-cosTheta\right)}} \cdot cosTheta\right)} \]

5. Final simplification 98.5%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\frac{\left(cosTheta \cdot \left(\alpha \cdot \alpha\right)\right) \cdot \left(cosTheta \cdot \left(\alpha \cdot \alpha\right)\right) - cosTheta \cdot cosTheta}{cosTheta \cdot \left(\alpha \cdot \alpha\right) + cosTheta} \cdot cosTheta - -1\right) \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot \mathsf{PI}\left(\right)\right)} \]
6. Add Preprocessing

Alternative 2: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha \cdot \alpha - 1\\ \frac{t\_0}{\left(\left(cosTheta \cdot t\_0\right) \cdot cosTheta - -1\right) \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot \mathsf{PI}\left(\right)\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/
    t_0
    (*
     (- (* (* cosTheta t_0) cosTheta) -1.0)
     (* (log (* alpha alpha)) (PI))))))

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing

3. Final simplification 98.5%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\left(cosTheta \cdot \left(\alpha \cdot \alpha - 1\right)\right) \cdot cosTheta - -1\right) \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot \mathsf{PI}\left(\right)\right)} \]
4. Add Preprocessing

Alternative 3: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha \cdot \alpha - 1\\ \frac{t\_0}{\left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot \log \alpha\right) \cdot \left(\left(cosTheta \cdot t\_0\right) \cdot cosTheta - -1\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/
    t_0
    (* (* (+ (PI) (PI)) (log alpha)) (- (* (* cosTheta t_0) cosTheta) -1.0)))))

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   2. lift-log.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\log \left(\alpha \cdot \alpha\right)}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \color{blue}{\left(\alpha \cdot \alpha\right)}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   4. log-prod N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\log \alpha + \log \alpha\right)}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   5. distribute-rgt-in N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\log \alpha \cdot \mathsf{PI}\left(\right) + \log \alpha \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   6. distribute-lft-out N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\log \alpha \cdot \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   7. lower-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\log \alpha \cdot \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   8. lower-log.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\color{blue}{\log \alpha} \cdot \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   9. lower-+.f32 98.4

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\log \alpha \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

4. Applied rewrites 98.4%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(\log \alpha \cdot \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

5. Final simplification 98.4%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot \log \alpha\right) \cdot \left(\left(cosTheta \cdot \left(\alpha \cdot \alpha - 1\right)\right) \cdot cosTheta - -1\right)} \]
6. Add Preprocessing

Alternative 4: 97.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\alpha \cdot \alpha - 1}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot \mathsf{PI}\left(\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (- (* alpha alpha) 1.0)
  (* (- 1.0 (* cosTheta cosTheta)) (* (log (* alpha alpha)) (PI)))))

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing

3. Taylor expanded in alpha around 0

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \color{blue}{\left(1 + -1 \cdot {cosTheta}^{2}\right)}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left({cosTheta}^{2}\right)\right)}\right)} \]

   2. unsub-neg N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \color{blue}{\left(1 - {cosTheta}^{2}\right)}} \]

   3. lower--.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \color{blue}{\left(1 - {cosTheta}^{2}\right)}} \]

   4. unpow2 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 - \color{blue}{cosTheta \cdot cosTheta}\right)} \]

   5. lower-*.f32 97.7

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 - \color{blue}{cosTheta \cdot cosTheta}\right)} \]

5. Applied rewrites 97.7%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \color{blue}{\left(1 - cosTheta \cdot cosTheta\right)}} \]

6. Final simplification 97.7%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot \mathsf{PI}\left(\right)\right)} \]
7. Add Preprocessing

Alternative 5: 97.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{\alpha \cdot \alpha - 1}{\left(\left(1 - cosTheta \cdot cosTheta\right) \cdot \log \alpha\right) \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (- (* alpha alpha) 1.0)
  (* (* (- 1.0 (* cosTheta cosTheta)) (log alpha)) (* 2.0 (PI)))))

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right)} \cdot cosTheta\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(cosTheta \cdot \left(\alpha \cdot \alpha - 1\right)\right)} \cdot cosTheta\right)} \]

   3. lift--.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(cosTheta \cdot \color{blue}{\left(\alpha \cdot \alpha - 1\right)}\right) \cdot cosTheta\right)} \]

   4. sub-neg N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(cosTheta \cdot \color{blue}{\left(\alpha \cdot \alpha + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot cosTheta\right)} \]

   5. distribute-rgt-in N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta + \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)} \cdot cosTheta\right)} \]

   6. flip-+ N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta}} \cdot cosTheta\right)} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta}} \cdot cosTheta\right)} \]

   8. lower--.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right)} - \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right)} \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   11. lower-*.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right)} - \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   12. lower-*.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(\color{blue}{-1} \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   14. neg-mul-1 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \color{blue}{\left(\mathsf{neg}\left(cosTheta\right)\right)} \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   15. lower-neg.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \color{blue}{\left(-cosTheta\right)} \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   16. metadata-eval N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(-cosTheta\right) \cdot \left(\color{blue}{-1} \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   17. neg-mul-1 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(-cosTheta\right) \cdot \color{blue}{\left(\mathsf{neg}\left(cosTheta\right)\right)}}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   18. lower-neg.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(-cosTheta\right) \cdot \color{blue}{\left(-cosTheta\right)}}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   19. lower--.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(-cosTheta\right) \cdot \left(-cosTheta\right)}{\color{blue}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta}} \cdot cosTheta\right)} \]

4. Applied rewrites 98.5%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(-cosTheta\right) \cdot \left(-cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(-cosTheta\right)}} \cdot cosTheta\right)} \]

5. Taylor expanded in alpha around 0

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)}} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)} \]

   4. lower-PI.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\left(1 + -1 \cdot {cosTheta}^{2}\right) \cdot \log \alpha\right)}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\left(\left(1 + -1 \cdot {cosTheta}^{2}\right) \cdot \log \alpha\right)}} \]

   7. mul-1-neg N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(1 + \color{blue}{\left(\mathsf{neg}\left({cosTheta}^{2}\right)\right)}\right) \cdot \log \alpha\right)} \]

   8. sub-neg N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{\left(1 - {cosTheta}^{2}\right)} \cdot \log \alpha\right)} \]

   9. lower--.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\color{blue}{\left(1 - {cosTheta}^{2}\right)} \cdot \log \alpha\right)} \]

   10. unpow2 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(1 - \color{blue}{cosTheta \cdot cosTheta}\right) \cdot \log \alpha\right)} \]

   11. lower-*.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(1 - \color{blue}{cosTheta \cdot cosTheta}\right) \cdot \log \alpha\right)} \]

   12. lower-log.f32 97.7

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(1 - cosTheta \cdot cosTheta\right) \cdot \color{blue}{\log \alpha}\right)} \]

7. Applied rewrites 97.7%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(1 - cosTheta \cdot cosTheta\right) \cdot \log \alpha\right)}} \]

8. Final simplification 97.7%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\left(1 - cosTheta \cdot cosTheta\right) \cdot \log \alpha\right) \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)} \]
9. Add Preprocessing

Alternative 6: 95.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\alpha \cdot \alpha - 1}{\log \left(\alpha \cdot \alpha\right) \cdot \mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ (- (* alpha alpha) 1.0) (* (log (* alpha alpha)) (PI))))

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right)} \cdot cosTheta\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(cosTheta \cdot \left(\alpha \cdot \alpha - 1\right)\right)} \cdot cosTheta\right)} \]

   3. lift--.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(cosTheta \cdot \color{blue}{\left(\alpha \cdot \alpha - 1\right)}\right) \cdot cosTheta\right)} \]

   4. sub-neg N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(cosTheta \cdot \color{blue}{\left(\alpha \cdot \alpha + \left(\mathsf{neg}\left(1\right)\right)\right)}\right) \cdot cosTheta\right)} \]

   5. distribute-rgt-in N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta + \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)} \cdot cosTheta\right)} \]

   6. flip-+ N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta}} \cdot cosTheta\right)} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta}} \cdot cosTheta\right)} \]

   8. lower--.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right)} - \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right)} \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   11. lower-*.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right)} - \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   12. lower-*.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \color{blue}{\left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   13. metadata-eval N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(\color{blue}{-1} \cdot cosTheta\right) \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   14. neg-mul-1 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \color{blue}{\left(\mathsf{neg}\left(cosTheta\right)\right)} \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   15. lower-neg.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \color{blue}{\left(-cosTheta\right)} \cdot \left(\left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   16. metadata-eval N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(-cosTheta\right) \cdot \left(\color{blue}{-1} \cdot cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   17. neg-mul-1 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(-cosTheta\right) \cdot \color{blue}{\left(\mathsf{neg}\left(cosTheta\right)\right)}}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   18. lower-neg.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(-cosTheta\right) \cdot \color{blue}{\left(-cosTheta\right)}}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta} \cdot cosTheta\right)} \]

   19. lower--.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(-cosTheta\right) \cdot \left(-cosTheta\right)}{\color{blue}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(\mathsf{neg}\left(1\right)\right) \cdot cosTheta}} \cdot cosTheta\right)} \]

4. Applied rewrites 98.5%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \color{blue}{\frac{\left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot cosTheta\right) - \left(-cosTheta\right) \cdot \left(-cosTheta\right)}{\left(\alpha \cdot \alpha\right) \cdot cosTheta - \left(-cosTheta\right)}} \cdot cosTheta\right)} \]

5. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\mathsf{PI}\left(\right) \cdot \log \left({\alpha}^{2}\right)}} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left({\alpha}^{2}\right) \cdot \mathsf{PI}\left(\right)}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left({\alpha}^{2}\right) \cdot \mathsf{PI}\left(\right)}} \]

   3. lower-log.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left({\alpha}^{2}\right)} \cdot \mathsf{PI}\left(\right)} \]

   4. unpow2 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \mathsf{PI}\left(\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \mathsf{PI}\left(\right)} \]

   6. lower-PI.f32 95.4

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\log \left(\alpha \cdot \alpha\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}} \]

7. Applied rewrites 95.4%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\log \left(\alpha \cdot \alpha\right) \cdot \mathsf{PI}\left(\right)}} \]
8. Add Preprocessing

Alternative 7: 65.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\log \left(\alpha \cdot \alpha\right) \cdot \mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ -1.0 (* (log (* alpha alpha)) (PI))))

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing

3. Taylor expanded in alpha around 0

   \[\leadsto \frac{\color{blue}{-1}}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
4. Step-by-step derivation

5. Applied rewrites 66.3%

   \[\leadsto \frac{\color{blue}{-1}}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

6. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{-1}{\color{blue}{\mathsf{PI}\left(\right) \cdot \log \left({\alpha}^{2}\right)}} \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{-1}{\color{blue}{\log \left({\alpha}^{2}\right) \cdot \mathsf{PI}\left(\right)}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{-1}{\color{blue}{\log \left({\alpha}^{2}\right) \cdot \mathsf{PI}\left(\right)}} \]

   3. lower-log.f32 N/A

      \[\leadsto \frac{-1}{\color{blue}{\log \left({\alpha}^{2}\right)} \cdot \mathsf{PI}\left(\right)} \]

   4. unpow2 N/A

      \[\leadsto \frac{-1}{\log \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \mathsf{PI}\left(\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{-1}{\log \color{blue}{\left(\alpha \cdot \alpha\right)} \cdot \mathsf{PI}\left(\right)} \]

   6. lower-PI.f32 64.5

      \[\leadsto \frac{-1}{\log \left(\alpha \cdot \alpha\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}} \]

8. Applied rewrites 64.5%

   \[\leadsto \frac{-1}{\color{blue}{\log \left(\alpha \cdot \alpha\right) \cdot \mathsf{PI}\left(\right)}} \]
9. Add Preprocessing

Alternative 8: 7.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \alpha \cdot \alpha - 1\\ \frac{t\_0}{\left(\mathsf{fma}\left(0, \frac{1}{0}, 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(cosTheta \cdot t\_0\right) \cdot cosTheta - -1\right)} \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/
    t_0
    (*
     (* (fma 0.0 (/ 1.0 0.0) 0.0) (PI))
     (- (* (* cosTheta t_0) cosTheta) -1.0)))))

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\log \left(\alpha \cdot \alpha\right)}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   2. *-rgt-identity N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \color{blue}{\left(\left(\alpha \cdot \alpha\right) \cdot 1\right)}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   3. log-prod N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\log \left(\alpha \cdot \alpha\right) + \log 1\right)}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \left(\log \color{blue}{\left(\alpha \cdot \alpha\right)} + \log 1\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   5. log-prod N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\left(\log \alpha + \log \alpha\right)} + \log 1\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   6. flip-+ N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\frac{\log \alpha \cdot \log \alpha - \log \alpha \cdot \log \alpha}{\log \alpha - \log \alpha}} + \log 1\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   7. div-inv N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\left(\log \alpha \cdot \log \alpha - \log \alpha \cdot \log \alpha\right) \cdot \frac{1}{\log \alpha - \log \alpha}} + \log 1\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   8. +-inverses N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{0} \cdot \frac{1}{\log \alpha - \log \alpha} + \log 1\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   9. metadata-eval N/A

      \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \left(0 \cdot \frac{1}{\log \alpha - \log \alpha} + \color{blue}{0}\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   10. lower-fma.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{fma}\left(0, \frac{1}{\log \alpha - \log \alpha}, 0\right)}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   11. +-inverses N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(0, \frac{1}{\color{blue}{0}}, 0\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   12. lower-/.f32 1.4

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(0, \color{blue}{\frac{1}{0}}, 0\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

4. Applied rewrites 1.4%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{fma}\left(0, \frac{1}{0}, 0\right)}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

5. Final simplification 1.4%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\left(\mathsf{fma}\left(0, \frac{1}{0}, 0\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(cosTheta \cdot \left(\alpha \cdot \alpha - 1\right)\right) \cdot cosTheta - -1\right)} \]
6. Add Preprocessing

Alternative 9: -0.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (cosTheta alpha) :precision binary32 (/ -1.0 (* (/ 0.0 0.0) (PI))))

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \frac{\color{blue}{\alpha \cdot \alpha - 1}}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   2. sub-neg N/A

      \[\leadsto \frac{\color{blue}{\alpha \cdot \alpha + \left(\mathsf{neg}\left(1\right)\right)}}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\alpha \cdot \alpha} + \left(\mathsf{neg}\left(1\right)\right)}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   4. lower-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\alpha, \alpha, \mathsf{neg}\left(1\right)\right)}}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   5. metadata-eval 7.0

      \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, \color{blue}{-1}\right)}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)}} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   8. associate-*l* N/A

      \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\log \left(\alpha \cdot \alpha\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right)}} \]

   9. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right) \cdot \log \left(\alpha \cdot \alpha\right)\right)}} \]

   10. associate-*r* N/A

       \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right) \cdot \log \left(\alpha \cdot \alpha\right)}} \]

   11. lower-*.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)\right) \cdot \log \left(\alpha \cdot \alpha\right)}} \]

4. Applied rewrites -0.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right)\right) \cdot \frac{0}{0}}} \]

5. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{0}{0}} \]
6. Step-by-step derivation

   1. lower-PI.f32 -0.0

      \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{0}{0}} \]

7. Applied rewrites -0.0%

   \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{0}{0}} \]

8. Taylor expanded in alpha around 0

   \[\leadsto \frac{\color{blue}{-1}}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}} \]
9. Step-by-step derivation

10. Applied rewrites -0.0%

    \[\leadsto \frac{\color{blue}{-1}}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}} \]

11. Final simplification -0.0%

    \[\leadsto \frac{-1}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024254 
(FPCore (cosTheta alpha)
  :name "GTR1 distribution"
  :precision binary32
  :pre (and (and (<= 0.0 cosTheta) (<= cosTheta 1.0)) (and (<= 0.0001 alpha) (<= alpha 1.0)))
  (/ (- (* alpha alpha) 1.0) (* (* (PI) (log (* alpha alpha))) (+ 1.0 (* (* (- (* alpha alpha) 1.0) cosTheta) cosTheta)))))