GTR1 distribution

Percentage Accurate: 98.5% → 98.5%

Time: 7.6s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (fma alpha alpha -1.0)
  (*
   (* (fma (* cosTheta (fma alpha alpha -1.0)) cosTheta 1.0) (PI))
   (* 2.0 (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. 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 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in alpha around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-log.f32 98.4

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

7. Applied rewrites 98.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

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

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

   5. lift-fma.f32 98.6

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-fma.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-fma.f32 N/A

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

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

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

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

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

   13. lift-*.f32 N/A

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

   14. lift--.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. associate-*r* N/A

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

   17. lift-*.f32 N/A

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

   18. lift-*.f32 N/A

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

   19. lift-+.f32 N/A

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

   20. lift-*.f32 N/A

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

9. Applied rewrites 98.6%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

   1. lift-+.f32 N/A

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

   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 98.5

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

4. Applied rewrites 98.5%

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

5. Taylor expanded in alpha around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-log.f32 98.4

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

7. Applied rewrites 98.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

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

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

   5. lift-fma.f32 98.6

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

9. Applied rewrites 98.6%

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

Alternative 3: 97.4% 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. fp-cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 N/A

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

   5. unpow2 N/A

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

   6. 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. Add Preprocessing

Alternative 4: 97.4% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/
  (- (* alpha alpha) 1.0)
  (* (* 2.0 (PI)) (* (- 1.0 (* cosTheta cosTheta)) (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. 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. lift-PI.f32 N/A

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

   7. add-cube-cbrt N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-log.f32 N/A

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

   12. pow2 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   13. lower-pow.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   14. lift-PI.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   15. lower-cbrt.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   16. lift-PI.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   17. lower-cbrt.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   18. lower-*.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \color{blue}{\log \alpha \cdot \mathsf{PI}\left(\right)}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   19. lower-log.f32 98.5

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \color{blue}{\log \alpha} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

4. Applied rewrites 98.5%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

5. Taylor expanded in alpha around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

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

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

   8. metadata-eval N/A

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

   9. *-lft-identity N/A

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

   10. lower--.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-log.f32 97.4

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

7. Applied rewrites 97.4%

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

Alternative 5: 95.2% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   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. lift-PI.f32 N/A

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

   7. add-cube-cbrt N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-log.f32 N/A

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

   12. pow2 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   13. lower-pow.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   14. lift-PI.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   15. lower-cbrt.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   16. lift-PI.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   17. lower-cbrt.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   18. lower-*.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \color{blue}{\log \alpha \cdot \mathsf{PI}\left(\right)}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   19. lower-log.f32 98.5

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \color{blue}{\log \alpha} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

4. Applied rewrites 98.5%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

5. Taylor expanded in cosTheta around 0

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. times-frac N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-log.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower--.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 94.4

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

7. Applied rewrites 94.4%

   \[\leadsto \color{blue}{\frac{0.5}{\log \alpha} \cdot \frac{\alpha \cdot \alpha - 1}{\mathsf{PI}\left(\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 95.0%

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

Alternative 6: 66.5% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ (fma (* -0.5 cosTheta) cosTheta -0.5) (* (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. 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. lift-PI.f32 N/A

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

   7. add-cube-cbrt N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-log.f32 N/A

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

   12. pow2 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   13. lower-pow.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   14. lift-PI.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   15. lower-cbrt.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   16. lift-PI.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   17. lower-cbrt.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   18. lower-*.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \color{blue}{\log \alpha \cdot \mathsf{PI}\left(\right)}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   19. lower-log.f32 98.5

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \color{blue}{\log \alpha} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

4. Applied rewrites 98.5%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

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. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

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

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

   8. metadata-eval N/A

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

   9. *-lft-identity N/A

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

   10. lower--.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-log.f32 68.0

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

7. Applied rewrites 68.0%

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

8. Taylor expanded in cosTheta around 0

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

10. Applied rewrites 67.4%

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

Alternative 7: 65.8% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

   1. lift-*.f32 N/A

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

   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. lift-PI.f32 N/A

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

   7. add-cube-cbrt N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-log.f32 N/A

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

   12. pow2 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   13. lower-pow.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   14. lift-PI.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   15. lower-cbrt.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   16. lift-PI.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   17. lower-cbrt.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   18. lower-*.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \color{blue}{\log \alpha \cdot \mathsf{PI}\left(\right)}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   19. lower-log.f32 98.5

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \color{blue}{\log \alpha} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

4. Applied rewrites 98.5%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

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. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

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

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

   8. metadata-eval N/A

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

   9. *-lft-identity N/A

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

   10. lower--.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-log.f32 68.0

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

7. Applied rewrites 68.0%

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

8. Taylor expanded in cosTheta around 0

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

10. Applied rewrites 66.7%

    \[\leadsto \frac{-0.5}{\color{blue}{\mathsf{PI}\left(\right) \cdot \log \alpha}} \]
11. Step-by-step derivation

12. Applied rewrites 66.8%

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

Alternative 8: 65.8% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (cosTheta alpha) :precision binary32 (/ -0.5 (* (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. 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. lift-PI.f32 N/A

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

   7. add-cube-cbrt N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-log.f32 N/A

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

   12. pow2 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   13. lower-pow.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   14. lift-PI.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   15. lower-cbrt.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   16. lift-PI.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   17. lower-cbrt.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}}, \log \alpha \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   18. lower-*.f32 N/A

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \color{blue}{\log \alpha \cdot \mathsf{PI}\left(\right)}\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

   19. lower-log.f32 98.5

       \[\leadsto \frac{\alpha \cdot \alpha - 1}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \color{blue}{\log \alpha} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

4. Applied rewrites 98.5%

   \[\leadsto \frac{\alpha \cdot \alpha - 1}{\color{blue}{\mathsf{fma}\left(\log \alpha \cdot {\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)}, \log \alpha \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

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. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

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

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

   8. metadata-eval N/A

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

   9. *-lft-identity N/A

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

   10. lower--.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-log.f32 68.0

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

7. Applied rewrites 68.0%

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

8. Taylor expanded in cosTheta around 0

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

10. Applied rewrites 66.7%

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

Alternative 9: -0.0% accurate, 6.8× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ (fma alpha alpha -1.0) (* (NAN) (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. lift-*.f32 N/A

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

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

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

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

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

   5. lower-fma.f32 98.4

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f32 N/A

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

4. Applied rewrites -0.0%

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

5. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. lower-NAN.f32 N/A

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

   3. lower-PI.f32 -0.0

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

7. Applied rewrites -0.0%

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

Reproduce

herbie shell --seed 2024352 
(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)))))