GTR1 distribution

Percentage Accurate: 98.5% → 98.6%

Time: 10.1s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.6% 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 98.5%

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

   1. log-pow-rev N/A

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

   2. pow2 N/A

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

   3. lower-log.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. pow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-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

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

FPCore

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

Derivation

1. Initial program 98.5%

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

   1. +-commutative N/A

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

   2. associate-*l* N/A

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

   3. pow2 N/A

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

   4. unpow2 N/A

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

   5. lower-fma.f32 N/A

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

   6. lower--.f32 N/A

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

   7. pow2 N/A

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

   8. lower-*.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 98.5

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

4. Applied rewrites 98.5%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

   1. pow2 N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-log.f32 N/A

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

   7. pow2 N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. associate-*l* N/A

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

   11. pow2 N/A

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

   12. unpow2 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f32 N/A

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

   17. lower--.f32 N/A

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

   18. pow2 N/A

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

   19. lower-*.f32 98.5

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

4. Applied rewrites 98.5%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

   1. log-pow-rev N/A

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

   2. pow2 N/A

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

   3. lower-log.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. pow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-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. Step-by-step derivation

   1. unpow-prod-down N/A

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

   2. pow-prod-up N/A

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

   3. lower-pow.f32 N/A

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

   4. lower-+.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-PI.f32 98.5

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

6. Applied rewrites 98.5%

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

   1. *-commutative N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. log-pow N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f32 N/A

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

8. Applied rewrites 98.3%

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

   1. count-2-rev N/A

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

   2. lower-+.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-PI.f32 98.3

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

10. Applied rewrites 98.3%

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

Alternative 5: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 98.0

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

5. Applied rewrites 98.0%

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

Alternative 6: 95.3% 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 98.5%

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

3. Taylor expanded in cosTheta around 0

   \[\leadsto \color{blue}{\frac{{\alpha}^{2} - 1}{\mathsf{PI}\left(\right) \cdot \log \left({\alpha}^{2}\right)}} \]
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. lower--.f32 N/A

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

   5. pow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-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. lower-*.f32 95.6

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

5. Applied rewrites 95.6%

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

Alternative 7: 95.4% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

   1. pow2 N/A

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

   2. associate-*l* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-log.f32 N/A

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

   7. pow2 N/A

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

   8. lower-*.f32 N/A

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

   9. +-commutative N/A

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

   10. associate-*l* N/A

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

   11. pow2 N/A

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

   12. unpow2 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f32 N/A

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

   17. lower--.f32 N/A

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

   18. pow2 N/A

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

   19. lower-*.f32 98.5

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

4. Applied rewrites 98.5%

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

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. lower-*.f32 95.6

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

7. Applied rewrites 95.6%

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

Alternative 8: 95.3% 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 98.5%

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

   1. log-pow-rev N/A

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

   2. pow2 N/A

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

   3. lower-log.f32 N/A

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

   4. lower-pow.f32 N/A

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

   5. pow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-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. pow2 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. +-commutative N/A

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

   8. pow2 N/A

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

   9. pow-pow N/A

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

7. Applied rewrites 95.3%

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

Alternative 9: 65.3% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in cosTheta around 0

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

5. Applied rewrites 95.6%

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

6. Taylor expanded in alpha around 0

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

8. Applied rewrites 65.5%

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

9. Taylor expanded in cosTheta around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. log-pow-rev N/A

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

    4. *-commutative N/A

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

    5. lower-*.f32 N/A

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

    6. lower-log.f32 N/A

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

    7. lower-PI.f32 65.5

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

11. Applied rewrites 65.5%

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

    1. *-commutative N/A

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

    2. log-pow-rev N/A

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

    3. lower-log.f32 N/A

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

    4. pow2 N/A

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

    5. lower-*.f32 65.5

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

13. Applied rewrites 65.5%

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

14. Final simplification 65.5%

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

Reproduce

herbie shell --seed 2025043 
(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)))))