GTR1 distribution

Percentage Accurate: 98.5% → 98.6%

Time: 9.0s

Alternatives: 12

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 98.3%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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 98.6

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

4. Applied rewrites 98.6%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.3%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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. lower-fma.f32 98.3

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 98.3

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

   8. lift--.f32 N/A

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

   9. lift-*.f32 N/A

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

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

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

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

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

   12. lower-fma.f32 98.3

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

4. Applied rewrites 98.3%

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

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.3%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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. lower-fma.f32 98.3

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 98.3

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

   8. lift--.f32 N/A

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

   9. lift-*.f32 N/A

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

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

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

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

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

   12. lower-fma.f32 98.3

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

4. Applied rewrites 98.3%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

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

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

   5. lift-fma.f32 98.2

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-fma.f32 N/A

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

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

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

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

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

   11. lift-*.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lift--.f32 N/A

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

   16. lift-*.f32 N/A

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

   17. lower-fma.f32 N/A

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

   18. +-commutative N/A

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

   19. lift-*.f32 N/A

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

   20. associate-*r* N/A

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

6. Applied rewrites 98.2%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.3%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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. lower-fma.f32 98.3

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 98.3

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

   8. lift--.f32 N/A

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

   9. lift-*.f32 N/A

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

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

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

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

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

   12. lower-fma.f32 98.3

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

4. Applied rewrites 98.3%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

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

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

   5. lift-fma.f32 98.2

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

6. Applied rewrites 98.2%

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

Alternative 5: 97.3% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.3%

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

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. lower-fma.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower--.f32 98.2

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in alpha around 0

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

7. Applied rewrites 96.9%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-rgt-identity N/A

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

   4. *-commutative N/A

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

   5. distribute-rgt1-in N/A

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

   6. *-rgt-identity N/A

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

   7. lift--.f32 N/A

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

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

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

   9. lift-fma.f32 96.9

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. associate-*r* N/A

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

   14. lower-*.f32 N/A

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

9. Applied rewrites 97.0%

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

Alternative 6: 97.3% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.3%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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 98.6

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in alpha around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

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

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

   8. metadata-eval N/A

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

   9. *-lft-identity N/A

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

   10. lower--.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-log.f32 97.0

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

7. Applied rewrites 97.0%

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

Alternative 7: 95.1% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in alpha around 0

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

5. Applied rewrites 67.5%

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

6. Taylor expanded in cosTheta around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 94.3

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

8. Applied rewrites 94.3%

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

9. Taylor expanded in alpha around 0

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

11. Applied rewrites 65.9%

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

12. Taylor expanded in alpha around 0

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

14. Applied rewrites 94.3%

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

Alternative 8: 65.8% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in alpha around 0

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

   1. associate-*r* N/A

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

   2. associate-/r* N/A

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

   3. lower-/.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-log.f32 N/A

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

   8. lower-PI.f32 N/A

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

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

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

   10. metadata-eval N/A

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

   11. *-lft-identity N/A

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

   12. lower--.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 67.4

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

5. Applied rewrites 67.4%

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

6. Taylor expanded in cosTheta around 0

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

8. Applied rewrites 66.5%

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

Alternative 9: 65.2% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in alpha around 0

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

5. Applied rewrites 67.5%

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

6. Taylor expanded in cosTheta around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 94.3

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

8. Applied rewrites 94.3%

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

9. Taylor expanded in alpha around 0

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

11. Applied rewrites 65.9%

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

13. Applied rewrites 65.9%

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

Alternative 10: 65.1% 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 98.3%

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

3. Taylor expanded in alpha around 0

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

5. Applied rewrites 67.5%

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

6. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 65.9

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

8. Applied rewrites 65.9%

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

Alternative 11: 65.1% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.3%

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

3. Taylor expanded in alpha around 0

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

5. Applied rewrites 67.5%

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

6. Taylor expanded in cosTheta around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 94.3

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

8. Applied rewrites 94.3%

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

9. Taylor expanded in alpha around 0

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

11. Applied rewrites 65.9%

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

Alternative 12: -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 98.3%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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 98.2

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

   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 2024364 
(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)))))