GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 5.2s

Alternatives: 14

Speedup: 0.6×

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

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/
    t_0
    (*
     (log (pow (* alpha alpha) (PI)))
     (+ 1.0 (* (* t_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-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-log.f32 N/A

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

   4. log-pow-rev N/A

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

   5. lift-*.f32 N/A

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

   6. pow2 N/A

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

   7. lower-log.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 98.6

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

4. Applied rewrites 98.6%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

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

   3. lift-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. pow2 N/A

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

   7. lift-+.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift--.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 98.6%

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

   1. lift-*.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. pow2 N/A

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

   5. lift-fma.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. +-commutative N/A

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

   14. lower-*.f32 N/A

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

6. Applied rewrites 98.6%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

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

   3. lift-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. pow2 N/A

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

   7. lift-+.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift--.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 98.6%

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

   1. lift-*.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. pow2 N/A

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

   5. lift-fma.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. +-commutative N/A

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

   14. lower-*.f32 N/A

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

6. Applied rewrites 98.6%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

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

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

   5. lift-fma.f32 98.5

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

8. Applied rewrites 98.5%

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

Alternative 4: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

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

   3. lift-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. pow2 N/A

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

   7. lift-+.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift--.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 98.6%

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

   1. lift-*.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. pow2 N/A

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

   5. lift-fma.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. +-commutative N/A

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

   14. lower-*.f32 N/A

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

6. Applied rewrites 98.6%

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

7. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 97.6

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

9. Applied rewrites 97.6%

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

Alternative 5: 97.5% 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. 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-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. pow2 N/A

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

   7. lift-+.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift--.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 98.6%

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

   1. lift-*.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. pow2 N/A

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

   5. lift-fma.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. +-commutative N/A

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

   14. lower-*.f32 N/A

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

6. Applied rewrites 98.6%

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

7. Taylor expanded in alpha around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. pow2 N/A

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

   9. +-commutative N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f32 N/A

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

   13. mul-1-neg N/A

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

   14. lower-neg.f32 N/A

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

   15. lift-log.f32 97.5

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

9. Applied rewrites 97.5%

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

Alternative 6: 95.2% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.6%

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

3. Taylor expanded in cosTheta around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. pow2 N/A

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

   5. lift-*.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 95.7

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

5. Applied rewrites 95.7%

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

Alternative 7: 95.2% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (* 0.5 (/ (/ (fma alpha alpha -1.0) (PI)) (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. 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. pow2 N/A

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

   4. flip-- N/A

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

   5. lower-/.f32 N/A

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

   6. metadata-eval N/A

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

   7. lower--.f32 N/A

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

   8. pow-prod-up N/A

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

   9. metadata-eval N/A

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

   10. lower-pow.f32 N/A

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

   11. pow2 N/A

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

   12. lower-fma.f32 98.3

       \[\leadsto \frac{\frac{{\alpha}^{4} - 1}{\color{blue}{\mathsf{fma}\left(\alpha, \alpha, 1\right)}}}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

4. Applied rewrites 98.3%

   \[\leadsto \frac{\color{blue}{\frac{{\alpha}^{4} - 1}{\mathsf{fma}\left(\alpha, \alpha, 1\right)}}}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]

5. Taylor expanded in cosTheta around 0

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

   1. metadata-eval N/A

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

   2. pow-prod-up N/A

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

   3. metadata-eval N/A

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

   4. pow2 N/A

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

   5. flip-- N/A

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

   6. pow2 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lift-pow.f32 N/A

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

   11. lift--.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. *-commutative N/A

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

7. Applied rewrites 95.3%

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

8. Taylor expanded in alpha around 0

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

   1. distribute-lft-out-- N/A

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

   2. lower-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. associate-/r* N/A

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

   5. sub-div N/A

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

   6. sub-div N/A

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

   7. pow2 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

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

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

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

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

   12. lift-fma.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lift-log.f32 95.7

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

10. Applied rewrites 95.7%

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

Alternative 8: 95.2% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.6%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. pow2 N/A

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

   7. lift-+.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift--.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. associate-*l* N/A

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

   13. lower-*.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lower-*.f32 N/A

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in cosTheta around 0

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

   1. lower-log.f32 N/A

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

   2. pow2 N/A

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

   3. lift-*.f32 95.7

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

7. Applied rewrites 95.7%

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

Alternative 9: 95.2% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ (- (* alpha alpha) 1.0) (* (* 2.0 (PI)) (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. Step-by-step derivation

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-log.f32 N/A

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

   4. log-pow-rev N/A

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

   5. lift-*.f32 N/A

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

   6. pow2 N/A

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

   7. lower-log.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-PI.f32 98.6

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in cosTheta around 0

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

   1. log-pow-rev N/A

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

   2. pow-pow N/A

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

   3. log-pow-rev N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lower-log.f32 95.6

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

7. Applied rewrites 95.6%

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

8. Final simplification 95.6%

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

Alternative 10: 67.3% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ -0.5 (* (* (log alpha) (PI)) (fma 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. Taylor expanded in alpha around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. *-commutative N/A

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

   6. log-pow-rev N/A

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

   7. lower-log.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 65.8

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

5. Applied rewrites 65.8%

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

   1. lift-/.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. lift-pow.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-fma.f32 N/A

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

   8. log-pow N/A

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

   9. +-commutative N/A

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

   10. pow2 N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. associate-/r* N/A

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

   14. lower-/.f32 N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f32 N/A

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

7. Applied rewrites 65.8%

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

Alternative 11: 66.6% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ (- (* -0.5 (* cosTheta cosTheta)) 0.5) (* (log 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 alpha around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. *-commutative N/A

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

   6. log-pow-rev N/A

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

   7. lower-log.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 65.8

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

5. Applied rewrites 65.8%

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

6. Taylor expanded in cosTheta around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. lower-log.f32 N/A

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

   5. lift-PI.f32 65.2

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

8. Applied rewrites 65.2%

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

9. Taylor expanded in cosTheta around 0

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

    1. associate-*r/ N/A

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

    2. associate-*r/ N/A

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

    3. metadata-eval N/A

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

    4. sub-div N/A

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

    5. lower-/.f32 N/A

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

    6. lower--.f32 N/A

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

    7. lower-*.f32 N/A

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

    8. pow2 N/A

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

    9. lift-*.f32 N/A

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

    10. *-commutative N/A

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

    11. lift-log.f32 N/A

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

    12. lift-*.f32 N/A

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

    13. lift-PI.f32 65.7

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

11. Applied rewrites 65.7%

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

Alternative 12: 66.0% accurate, 1.3× speedup

Math

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

FPCore

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

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. *-commutative N/A

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

   6. log-pow-rev N/A

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

   7. lower-log.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 65.8

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

5. Applied rewrites 65.8%

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

6. Taylor expanded in cosTheta around 0

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

8. Applied rewrites 65.3%

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

Alternative 13: 65.9% 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 98.6%

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

3. Taylor expanded in alpha around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. *-commutative N/A

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

   6. log-pow-rev N/A

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

   7. lower-log.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 65.8

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

5. Applied rewrites 65.8%

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

6. Taylor expanded in cosTheta around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. lower-log.f32 N/A

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

   5. lift-PI.f32 65.2

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

8. Applied rewrites 65.2%

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

   1. lift-/.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lift-log.f32 N/A

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

   9. lift-PI.f32 65.3

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

10. Applied rewrites 65.3%

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

Alternative 14: 65.9% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (cosTheta alpha) :precision binary32 (/ -0.5 (* (log 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 alpha around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. *-commutative N/A

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

   6. log-pow-rev N/A

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

   7. lower-log.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. +-commutative N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 65.8

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

5. Applied rewrites 65.8%

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

6. Taylor expanded in cosTheta around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. lower-log.f32 N/A

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

   5. lift-PI.f32 65.2

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

8. Applied rewrites 65.2%

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

Reproduce

herbie shell --seed 2025037 
(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)))))