GTR1 distribution

Percentage Accurate: 98.5% → 98.6%

Time: 7.3s

Alternatives: 11

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 98.5%

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

   1. 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 98.6%

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

5. Final simplification 98.6%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing
3. 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 98.6%

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

6. Applied rewrites 98.5%

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

Alternative 3: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing
3. 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 98.6%

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

6. Applied rewrites 98.5%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. distribute-frac-neg2 N/A

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

   4. lift-/.f32 N/A

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

   5. distribute-neg-frac2 N/A

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

   6. lift-fma.f32 N/A

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

   7. distribute-neg-in N/A

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

   8. *-commutative N/A

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

   9. distribute-lft-neg-out N/A

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

   10. lift-neg.f32 N/A

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

   11. metadata-eval N/A

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

   12. lift-fma.f32 N/A

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

   13. associate-/r* N/A

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

8. Applied rewrites 98.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 98.5

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

   11. lift-fma.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f32 98.5

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 98.5

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

10. Applied rewrites 98.5%

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

Alternative 4: 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 98.5%

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

   1. 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 98.5

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

   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 98.5

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

   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 98.5

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

4. Applied rewrites 98.5%

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

Alternative 5: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

   1. 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 98.6%

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

6. Applied rewrites 98.5%

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

Alternative 6: 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 98.5%

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

   1. 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. Add Preprocessing

Alternative 7: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing
3. 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 98.6%

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

6. Applied rewrites 98.5%

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

7. Taylor expanded in alpha around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 97.3

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

9. Applied rewrites 97.3%

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

Alternative 8: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

   1. 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 98.6%

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

6. Applied rewrites 98.5%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. distribute-frac-neg2 N/A

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

   4. lift-/.f32 N/A

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

   5. distribute-neg-frac2 N/A

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

   6. lift-fma.f32 N/A

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

   7. distribute-neg-in N/A

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

   8. *-commutative N/A

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

   9. distribute-lft-neg-out N/A

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

   10. lift-neg.f32 N/A

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

   11. metadata-eval N/A

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

   12. lift-fma.f32 N/A

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

   13. associate-/r* N/A

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in alpha around 0

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

    1. mul-1-neg N/A

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

    2. lower-neg.f32 97.3

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

11. Applied rewrites 97.3%

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

12. Final simplification 97.3%

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

Alternative 9: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

   1. 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 98.6%

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

6. Applied rewrites 98.5%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. distribute-frac-neg2 N/A

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

   4. lift-/.f32 N/A

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

   5. distribute-neg-frac2 N/A

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

   6. lift-fma.f32 N/A

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

   7. distribute-neg-in N/A

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

   8. *-commutative N/A

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

   9. distribute-lft-neg-out N/A

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

   10. lift-neg.f32 N/A

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

   11. metadata-eval N/A

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

   12. lift-fma.f32 N/A

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

   13. associate-/r* N/A

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in alpha around 0

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

    1. lower--.f32 N/A

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

    2. unpow2 N/A

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

    3. lower-*.f32 97.3

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

11. Applied rewrites 97.3%

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

Alternative 10: 95.2% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
2. Add Preprocessing
3. 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 98.6%

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

6. Applied rewrites 98.5%

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

7. Taylor expanded in cosTheta around 0

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

   1. unpow2 N/A

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

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

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

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

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

   4. lower-fma.f32 95.8

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

9. Applied rewrites 95.8%

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

Alternative 11: 65.3% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in 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 68.8%

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

6. Taylor expanded in cosTheta around 0

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

8. Applied rewrites 67.7%

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

9. Taylor expanded in cosTheta around 0

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

    1. *-commutative N/A

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

    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 67.7

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

11. Applied rewrites 67.7%

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

Reproduce

herbie shell --seed 2025010 
(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)))))