GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 4.8s

Alternatives: 8

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: 98.7% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.6%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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. *-commutative N/A

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

   3. lower-*.f32 98.6

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

   4. lift-+.f32 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-fma.f32 98.6

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 98.6

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

   11. lift--.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. difference-of-sqr-1 N/A

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

   14. difference-of-sqr--1-rev N/A

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

   15. lower-fma.f32 98.6

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

   16. lift-*.f32 N/A

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

   17. lift-log.f32 N/A

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

   18. log-pow-rev N/A

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

   19. lower-log.f32 N/A

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

   20. lower-pow.f32 98.7

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

4. Applied rewrites 98.7%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.6%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. difference-of-sqr-1 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower--.f32 N/A

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

   7. metadata-eval N/A

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

   8. fp-cancel-sign-sub-inv N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower--.f32 98.6

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

4. Applied rewrites 98.6%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lift--.f32 N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. fp-cancel-sign-sub-inv N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. lift--.f32 N/A

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

   13. difference-of-sqr--1 N/A

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

   14. lift-*.f32 N/A

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

   15. distribute-lft-in N/A

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

   16. lift-*.f32 N/A

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

   17. associate-*r* N/A

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

   18. *-commutative N/A

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

   19. mul-1-neg N/A

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

   20. lower-fma.f32 N/A

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

   21. lower-*.f32 N/A

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

   22. lower-neg.f32 98.6

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

6. Applied rewrites 98.6%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lift-fma.f32 N/A

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

   6. distribute-rgt-in N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. associate-*r* N/A

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

   12. lift-*.f32 N/A

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

   13. lift-neg.f32 N/A

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

   14. distribute-lft-neg-out N/A

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

   15. lift-*.f32 N/A

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

   16. mul-1-neg N/A

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

   17. *-commutative N/A

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

   18. associate-+l+ N/A

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

8. Applied rewrites 98.7%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.6%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. difference-of-sqr-1 N/A

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

   10. difference-of-sqr--1-rev N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-*.f32 98.6

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

4. Applied rewrites 98.6%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.6%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. difference-of-sqr-1 N/A

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

   10. difference-of-sqr--1-rev N/A

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

   11. lower-fma.f32 N/A

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

   12. lower-*.f32 98.6

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

4. Applied rewrites 98.6%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. difference-of-sqr-1 N/A

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

   4. difference-of-sqr--1 N/A

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

   5. lift-fma.f32 98.6

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

6. Applied rewrites 98.6%

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

Alternative 5: 97.3% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in alpha around 0

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\log \alpha \cdot \left(1 + -1 \cdot {cosTheta}^{2}\right)\right)\right)}} \]
4. 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. metadata-eval N/A

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

   3. distribute-lft-neg-out N/A

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

   4. distribute-lft-neg-out N/A

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

   5. distribute-rgt-neg-out N/A

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

   6. distribute-lft-neg-out N/A

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

   7. log-rec N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. log-rec N/A

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

   12. distribute-lft-neg-out N/A

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

   13. *-commutative N/A

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

   14. distribute-lft-neg-in N/A

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

5. Applied rewrites 98.0%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\left(-2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\mathsf{fma}\left(cosTheta, cosTheta, -1\right) \cdot \log \alpha\right)}} \]
6. 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 98.6%

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

3. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\mathsf{PI}\left(\right) \cdot \log \left({\alpha}^{2}\right)}} \]
4. 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 96.2

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

5. Applied rewrites 96.2%

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

Alternative 7: 95.0% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\mathsf{PI}\left(\right) \cdot \log \left({\alpha}^{2}\right)}} \]
4. 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 96.2

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

5. Applied rewrites 96.2%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. difference-of-sqr-1 N/A

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

   4. difference-of-sqr--1 N/A

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

   5. lift-fma.f32 96.1

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

7. Applied rewrites 96.1%

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

Alternative 8: 65.4% 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 98.6%

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

3. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\mathsf{PI}\left(\right) \cdot \log \left({\alpha}^{2}\right)}} \]
4. 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 96.2

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

5. Applied rewrites 96.2%

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

6. Taylor expanded in alpha around 0

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

8. Applied rewrites 64.5%

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

Reproduce

herbie shell --seed 2025017 
(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)))))