GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 7.0s

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: 98.7% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift-*.f32 N/A

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

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

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

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

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

   11. lower-fma.f32 N/A

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

   12. lower-*.f32 98.7

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-log.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. lower--.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-PI.f32 N/A

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

   16. *-commutative N/A

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

7. Applied rewrites 98.7%

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

8. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. unpow2 N/A

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

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

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

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

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

   14. lower-fma.f32 N/A

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

   15. lower-log.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 N/A

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

10. Applied rewrites 98.7%

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

12. Applied rewrites 98.8%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift-*.f32 N/A

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

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

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

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

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

   11. lower-fma.f32 N/A

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

   12. lower-*.f32 98.7

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

4. Applied rewrites 98.7%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift-*.f32 N/A

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

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

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

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

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

   11. lower-fma.f32 N/A

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

   12. lower-*.f32 98.7

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-log.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. lower--.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-PI.f32 N/A

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

   16. *-commutative N/A

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

7. Applied rewrites 98.7%

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

8. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. unpow2 N/A

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

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

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

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

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

   14. lower-fma.f32 N/A

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

   15. lower-log.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 N/A

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

10. Applied rewrites 98.7%

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

12. Applied rewrites 98.6%

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

Alternative 4: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

3. Taylor expanded in alpha around 0

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 98.1

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

5. Applied rewrites 98.1%

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

Alternative 5: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-log-exp N/A

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

   5. log-pow-rev N/A

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

   6. lower-log.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. pow-exp N/A

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

   9. *-commutative N/A

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

   10. lift-log.f32 N/A

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

   11. pow-to-exp N/A

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

   12. lower-pow.f32 98.7

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in alpha around inf

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

   1. distribute-lft-out N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f32 N/A

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

7. Applied rewrites 98.2%

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

8. Taylor expanded in alpha around 0

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

   1. unpow2 N/A

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

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

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

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

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

   4. lower-fma.f32 98.3

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

10. Applied rewrites 98.3%

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

11. Taylor expanded in alpha around 0

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

13. Applied rewrites 98.0%

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

Alternative 6: 95.3% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift-*.f32 N/A

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

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

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

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

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

   11. lower-fma.f32 N/A

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

   12. lower-*.f32 98.7

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 96.1

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

7. Applied rewrites 96.1%

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

Alternative 7: 95.3% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-log-exp N/A

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

   5. log-pow-rev N/A

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

   6. lower-log.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. pow-exp N/A

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

   9. *-commutative N/A

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

   10. lift-log.f32 N/A

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

   11. pow-to-exp N/A

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

   12. lower-pow.f32 98.7

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in alpha around inf

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

   1. distribute-lft-out N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f32 N/A

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

7. Applied rewrites 98.2%

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

8. Taylor expanded in alpha around 0

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

   1. unpow2 N/A

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

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

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

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

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

   4. lower-fma.f32 98.3

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

10. Applied rewrites 98.3%

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

11. Taylor expanded in cosTheta around 0

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

13. Applied rewrites 96.0%

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

Alternative 8: 65.7% 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 98.7%

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

   1. lift-+.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lower-fma.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift-*.f32 N/A

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

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

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

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

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

   11. lower-fma.f32 N/A

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

   12. lower-*.f32 98.7

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

4. Applied rewrites 98.7%

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

5. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 96.1

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

7. Applied rewrites 96.1%

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

8. Taylor expanded in alpha around 0

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

10. Applied rewrites 66.6%

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

Alternative 9: -0.0% accurate, 6.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.7%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

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

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

   5. lower-fma.f32 98.6

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f32 N/A

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

4. Applied rewrites -0.0%

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

5. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. lower-NAN.f32 N/A

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

   3. lower-PI.f32 -0.0

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

7. Applied rewrites -0.0%

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

Reproduce

herbie shell --seed 2024360 
(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)))))