GTR1 distribution

Percentage Accurate: 98.5% → 98.6%

Time: 6.8s

Alternatives: 13

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

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

FPCore

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

Derivation

1. Initial program 98.4%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lower-/.f32 N/A

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

   4. lower-neg.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift-*.f32 N/A

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

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

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

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

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

   9. lower-fma.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. distribute-lft-neg-in N/A

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

   13. lower-*.f32 N/A

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

4. Applied rewrites 98.5%

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

5. Final simplification 98.5%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

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))))))

Derivation

1. Initial program 98.4%

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

Alternative 3: 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 98.4%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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 98.4

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

4. Applied rewrites 98.4%

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

Alternative 4: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.4%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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 98.4

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

4. Applied rewrites 98.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

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

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

   5. lift-fma.f32 98.4

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

6. Applied rewrites 98.4%

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

Alternative 5: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

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

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

   5. lower-fma.f32 98.4

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f32 N/A

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. unpow2 N/A

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

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

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

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

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

   11. lower-*.f32 N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-PI.f32 N/A

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

   14. lower-PI.f32 98.3

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

7. Applied rewrites 98.3%

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

Alternative 6: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.4%

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 97.8

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

5. Applied rewrites 97.8%

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

   1. lift-/.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-/l/ N/A

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

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

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

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

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

   11. lift-fma.f32 N/A

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

   12. lift-/.f32 N/A

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

7. Applied rewrites 97.9%

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

Alternative 7: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.4%

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 97.8

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

5. Applied rewrites 97.8%

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

   1. lift-/.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. associate-/r* N/A

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

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

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

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

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

   10. lift-PI.f32 N/A

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

   11. lower-/.f32 N/A

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

7. Applied rewrites 97.8%

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

Alternative 8: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.4%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

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

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

   5. lower-fma.f32 98.4

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f32 N/A

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in alpha around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. cancel-sign-sub N/A

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

   8. metadata-eval N/A

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

   9. *-lft-identity N/A

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

   10. lower--.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-log.f32 97.8

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

7. Applied rewrites 97.8%

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

8. Final simplification 97.8%

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

Alternative 9: 95.2% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in cosTheta around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

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

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

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

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

   7. lower-fma.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-log.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 94.9

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

5. Applied rewrites 94.9%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-fma.f32 N/A

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

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

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

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

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

   6. lift-*.f32 N/A

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

   7. lift--.f32 N/A

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

   8. associate-/l/ N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f32 N/A

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

   12. lower-/.f32 95.1

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

   13. lift--.f32 N/A

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

   14. lift-*.f32 N/A

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

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

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

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

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

   17. lift-fma.f32 95.1

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

7. Applied rewrites 95.1%

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

Alternative 10: 95.2% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.4%

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

3. Taylor expanded in cosTheta around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

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

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

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

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

   7. lower-fma.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-log.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 94.9

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

5. Applied rewrites 94.9%

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

   1. lift-log.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. pow2 N/A

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

   4. log-pow N/A

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

   5. lower-*.f32 N/A

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

   6. lower-log.f32 95.0

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

7. Applied rewrites 95.0%

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

Alternative 11: 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 98.4%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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 98.4

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

4. Applied rewrites 98.4%

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

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

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

   2. lower-*.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 95.0

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

7. Applied rewrites 95.0%

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

Alternative 12: 95.3% 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 98.4%

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

3. Taylor expanded in cosTheta around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

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

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

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

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

   7. lower-fma.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-log.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 94.9

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

5. Applied rewrites 94.9%

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

6. Taylor expanded in alpha around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. lower-log.f32 N/A

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

   5. lower-PI.f32 65.4

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

8. Applied rewrites 65.4%

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

9. Taylor expanded in alpha around 0

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

11. Applied rewrites 95.0%

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

12. Final simplification 95.0%

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

Alternative 13: 65.2% 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 98.4%

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

3. Taylor expanded in cosTheta around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

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

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

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

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

   7. lower-fma.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-log.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 94.9

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

5. Applied rewrites 94.9%

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

6. Taylor expanded in alpha around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. lower-log.f32 N/A

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

   5. lower-PI.f32 65.4

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

8. Applied rewrites 65.4%

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

Reproduce

herbie shell --seed 2025026 
(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)))))