GTR1 distribution

Percentage Accurate: 98.6% → 98.6%

Time: 9.5s

Alternatives: 12

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: 98.6% 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: 98.6% accurate, 1.0× speedup

Math

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

FPCore

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

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%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

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

   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 98.5

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

4. Applied rewrites 98.5%

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

5. Final simplification 98.5%

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

Alternative 3: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

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 97.5

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

5. Applied rewrites 97.5%

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

6. Final simplification 97.5%

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

Alternative 4: 95.3% accurate, 1.2× speedup

Math

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

FPCore

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

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 cosTheta around 0

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

5. Applied rewrites 95.4%

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

6. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 95.4

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

8. Applied rewrites 95.4%

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

9. Final simplification 95.4%

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

Alternative 5: 66.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{\log \alpha \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - cosTheta \cdot cosTheta\right)\right)} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ -0.5 (* (log alpha) (* (PI) (- 1.0 (* cosTheta cosTheta))))))

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. +-commutative N/A

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

   4. flip3-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{neg}\left(1\right)\right)}^{3} + {\left(\alpha \cdot \alpha\right)}^{3}}{\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) + \left(\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) - \left(\mathsf{neg}\left(1\right)\right) \cdot \left(\alpha \cdot \alpha\right)\right)}}}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{{\left(\mathsf{neg}\left(1\right)\right)}^{3} + {\left(\alpha \cdot \alpha\right)}^{3}}{\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) + \left(\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) - \left(\mathsf{neg}\left(1\right)\right) \cdot \left(\alpha \cdot \alpha\right)\right)}}}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{{\color{blue}{-1}}^{3} + {\left(\alpha \cdot \alpha\right)}^{3}}{\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) + \left(\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) - \left(\mathsf{neg}\left(1\right)\right) \cdot \left(\alpha \cdot \alpha\right)\right)}}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{-1} + {\left(\alpha \cdot \alpha\right)}^{3}}{\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) + \left(\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) - \left(\mathsf{neg}\left(1\right)\right) \cdot \left(\alpha \cdot \alpha\right)\right)}}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{neg}\left(1\right)\right)} + {\left(\alpha \cdot \alpha\right)}^{3}}{\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) + \left(\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) - \left(\mathsf{neg}\left(1\right)\right) \cdot \left(\alpha \cdot \alpha\right)\right)}}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   9. lower-+.f32 N/A

      \[\leadsto \frac{\frac{\color{blue}{\left(\mathsf{neg}\left(1\right)\right) + {\left(\alpha \cdot \alpha\right)}^{3}}}{\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) + \left(\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) - \left(\mathsf{neg}\left(1\right)\right) \cdot \left(\alpha \cdot \alpha\right)\right)}}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{\color{blue}{-1} + {\left(\alpha \cdot \alpha\right)}^{3}}{\left(\mathsf{neg}\left(1\right)\right) \cdot \left(\mathsf{neg}\left(1\right)\right) + \left(\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right) - \left(\mathsf{neg}\left(1\right)\right) \cdot \left(\alpha \cdot \alpha\right)\right)}}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   11. cube-mult N/A

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

   12. lower-*.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. lower-+.f32 N/A

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

   18. metadata-eval N/A

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

   19. neg-mul-1 N/A

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in alpha around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-log.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. mul-1-neg N/A

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

   8. unsub-neg N/A

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

   9. lower--.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 70.3

       \[\leadsto \frac{-0.5}{\log \alpha \cdot \left(\left(1 - cosTheta \cdot cosTheta\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)} \]

7. Applied rewrites 70.3%

   \[\leadsto \color{blue}{\frac{-0.5}{\log \alpha \cdot \left(\left(1 - cosTheta \cdot cosTheta\right) \cdot \mathsf{PI}\left(\right)\right)}} \]

8. Final simplification 70.3%

   \[\leadsto \frac{-0.5}{\log \alpha \cdot \left(\mathsf{PI}\left(\right) \cdot \left(1 - cosTheta \cdot cosTheta\right)\right)} \]
9. Add Preprocessing

Alternative 6: 66.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{-0.5}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ -0.5 (* (- 1.0 (* cosTheta cosTheta)) (* (PI) (log alpha)))))

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

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

   1. lower-/.f32 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

   7. lower--.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-log.f32 70.3

       \[\leadsto \frac{-0.5}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\log \alpha}\right)} \]

5. Applied rewrites 70.3%

   \[\leadsto \color{blue}{\frac{-0.5}{\left(1 - cosTheta \cdot cosTheta\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \log \alpha\right)}} \]
6. Add Preprocessing

Alternative 7: 65.6% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ -1.0 (* (* (PI) (log (* alpha alpha))) 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. Taylor expanded in cosTheta around 0

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

5. Applied rewrites 95.4%

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

6. Taylor expanded in alpha around 0

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

8. Applied rewrites 69.0%

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

Alternative 8: 7.2% accurate, 1.4× speedup

Math

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

FPCore

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

4. Applied rewrites 23.7%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lower-+.f32 23.6

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

6. Applied rewrites 23.7%

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

7. Taylor expanded in alpha around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{\mathsf{PI}\left(\right)}, \frac{\alpha \cdot \alpha}{\mathsf{fma}\left(\alpha \cdot \alpha + -1, cosTheta \cdot cosTheta, 1\right) \cdot \frac{0}{0}}, \frac{-1}{\frac{0}{0} \cdot \mathsf{fma}\left(\color{blue}{-1 \cdot {cosTheta}^{2}}, \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)}\right) \]
8. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{\mathsf{PI}\left(\right)}, \frac{\alpha \cdot \alpha}{\mathsf{fma}\left(\alpha \cdot \alpha + -1, cosTheta \cdot cosTheta, 1\right) \cdot \frac{0}{0}}, \frac{-1}{\frac{0}{0} \cdot \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left({cosTheta}^{2}\right)}, \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)}\right) \]

   2. lower-neg.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{\mathsf{PI}\left(\right)}, \frac{\alpha \cdot \alpha}{\mathsf{fma}\left(\alpha \cdot \alpha + -1, cosTheta \cdot cosTheta, 1\right) \cdot \frac{0}{0}}, \frac{-1}{\frac{0}{0} \cdot \mathsf{fma}\left(\color{blue}{-{cosTheta}^{2}}, \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)}\right) \]

   3. unpow2 N/A

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

   4. lower-*.f32 23.7

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

9. Applied rewrites 23.6%

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

10. Final simplification -0.0%

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

Alternative 9: 7.2% accurate, 1.5× speedup

Math

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

FPCore

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

4. Applied rewrites 21.2%

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

5. Taylor expanded in alpha around inf

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\alpha \cdot {cosTheta}^{2}}}, \frac{\alpha}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}, \frac{-1}{\frac{0}{0} \cdot \mathsf{fma}\left(cosTheta \cdot \left(\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta\right), \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)}\right) \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\alpha \cdot {cosTheta}^{2}}}, \frac{\alpha}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}, \frac{-1}{\frac{0}{0} \cdot \mathsf{fma}\left(cosTheta \cdot \left(\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta\right), \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{\alpha \cdot {cosTheta}^{2}}}, \frac{\alpha}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}, \frac{-1}{\frac{0}{0} \cdot \mathsf{fma}\left(cosTheta \cdot \left(\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta\right), \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)}\right) \]

   3. unpow2 N/A

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

   4. lower-*.f32 -0.0

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

7. Applied rewrites 7.7%

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

8. Final simplification -0.0%

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

Alternative 10: 7.1% accurate, 1.5× speedup

Math

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

FPCore

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

4. Applied rewrites 21.2%

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

5. Taylor expanded in alpha around 0

   \[\leadsto \mathsf{fma}\left(\frac{\alpha}{\color{blue}{1 + -1 \cdot {cosTheta}^{2}}}, \frac{\alpha}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}, \frac{-1}{\frac{0}{0} \cdot \mathsf{fma}\left(cosTheta \cdot \left(\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta\right), \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)}\right) \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{\alpha}{1 + \color{blue}{\left(\mathsf{neg}\left({cosTheta}^{2}\right)\right)}}, \frac{\alpha}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}, \frac{-1}{\frac{0}{0} \cdot \mathsf{fma}\left(cosTheta \cdot \left(\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta\right), \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)}\right) \]

   2. unsub-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{\alpha}{\color{blue}{1 - {cosTheta}^{2}}}, \frac{\alpha}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}, \frac{-1}{\frac{0}{0} \cdot \mathsf{fma}\left(cosTheta \cdot \left(\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta\right), \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)}\right) \]

   3. lower--.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\alpha}{\color{blue}{1 - {cosTheta}^{2}}}, \frac{\alpha}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}, \frac{-1}{\frac{0}{0} \cdot \mathsf{fma}\left(cosTheta \cdot \left(\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta\right), \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)}\right) \]

   4. unpow2 N/A

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

   5. lower-*.f32 25.0

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

7. Applied rewrites 25.0%

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

8. Final simplification -0.0%

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

Alternative 11: 7.1% accurate, 1.7× speedup

Math

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

FPCore

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

4. Applied rewrites 21.2%

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

5. Taylor expanded in cosTheta around 0

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

7. Applied rewrites 25.0%

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

8. Final simplification -0.0%

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

Alternative 12: -0.0% accurate, 3.8× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ -1.0 (* (/ 0.0 0.0) (fma (* cosTheta (- cosTheta)) (PI) (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 7.1

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

   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. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lift-log.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. log-prod N/A

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

   14. flip-+ N/A

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

   15. +-inverses N/A

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

   16. +-inverses N/A

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

   17. lower-/.f32 N/A

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

   18. lift-+.f32 N/A

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

   19. +-commutative N/A

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

4. Applied rewrites -0.0%

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

5. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 -0.0

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

7. Applied rewrites -0.0%

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

8. Taylor expanded in alpha around 0

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

10. Applied rewrites -0.0%

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

Reproduce

herbie shell --seed 2024221 
(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)))))