GTR1 distribution

Percentage Accurate: 98.6% → 98.6%

Time: 7.9s

Alternatives: 15

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: 98.6% 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, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (- (* alpha alpha) 1.0)))
   (/
    t_0
    (* (+ (* (* cosTheta t_0) cosTheta) 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. Final simplification 98.5%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

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

   2. lift-log.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. log-prod N/A

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

   5. distribute-rgt-in N/A

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

   6. distribute-lft-out N/A

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

   7. lower-*.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. lower-+.f32 98.4

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

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

Alternative 3: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 97.6

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

5. Applied rewrites 97.6%

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

6. Final simplification 97.6%

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

Alternative 4: 97.4% accurate, 1.1× speedup

Math

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

FPCore

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

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

   2. lift-log.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. log-prod N/A

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

   5. distribute-rgt-in N/A

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

   6. distribute-lft-out N/A

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

   7. lower-*.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. lower-+.f32 98.4

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in alpha around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. mul-1-neg N/A

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

   8. unsub-neg N/A

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

   9. lower--.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-log.f32 97.5

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

7. Applied rewrites 97.5%

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

8. Final simplification 97.5%

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

Alternative 5: 95.2% 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 98.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lift-*.f32 N/A

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

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

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

   7. associate-*l* N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f32 N/A

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

   10. distribute-rgt-in N/A

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

   11. *-lft-identity N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower--.f32 75.7

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

4. Applied rewrites 76.4%

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

5. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 96.1

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

7. Applied rewrites 96.1%

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

Alternative 6: 65.5% accurate, 1.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

3. Taylor expanded in alpha around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 N/A

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

   8. mul-1-neg N/A

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

   9. unsub-neg N/A

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

   10. lower--.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-log.f32 64.1

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

5. Applied rewrites 64.1%

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

6. Taylor expanded in cosTheta around 0

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

8. Applied rewrites 63.2%

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

Alternative 7: 65.5% 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 98.5%

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

3. Taylor expanded in alpha around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 N/A

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

   8. mul-1-neg N/A

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

   9. unsub-neg N/A

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

   10. lower--.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-log.f32 64.1

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

5. Applied rewrites 64.1%

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

6. Taylor expanded in cosTheta around 0

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

8. Applied rewrites 63.1%

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

Alternative 8: 6.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta, cosTheta, 1\right) \cdot \mathsf{PI}\left(\right)\\ \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{t\_0}, \frac{0}{0}, \frac{-1}{t\_0 \cdot \frac{0}{0}}\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (let* ((t_0 (* (fma (* (fma alpha alpha -1.0) cosTheta) cosTheta 1.0) (PI))))
   (fma (/ (* alpha alpha) t_0) (/ 0.0 0.0) (/ -1.0 (* t_0 (/ 0.0 0.0))))))

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

4. Applied rewrites -0.0%

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

5. Final simplification -0.0%

   \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta, cosTheta, 1\right) \cdot \mathsf{PI}\left(\right)}, \frac{0}{0}, \frac{-1}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta, cosTheta, 1\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{0}{0}}\right) \]
6. Add Preprocessing

Alternative 9: 6.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\alpha \cdot \alpha, \frac{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta, cosTheta, 1\right)}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (fma
  (* alpha alpha)
  (/
   (/ 1.0 (fma (* (fma alpha alpha -1.0) cosTheta) cosTheta 1.0))
   (* (/ 0.0 0.0) (PI)))
  (/ -1.0 (* (PI) (/ 0.0 0.0)))))

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

4. Applied rewrites -0.0%

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

5. Taylor expanded in cosTheta around 0

   \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right)}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{0}{0}}\right) \]
6. Step-by-step derivation

   1. lower-PI.f32 -0.0

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

7. Applied rewrites -0.0%

   \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right)}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{0}{0}}\right) \]
8. Step-by-step derivation

9. Applied rewrites -0.0%

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

10. Final simplification -0.0%

    \[\leadsto \mathsf{fma}\left(\alpha \cdot \alpha, \frac{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta, cosTheta, 1\right)}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]
11. Add Preprocessing

Alternative 10: 6.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\alpha \cdot \alpha, \frac{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (fma
  (* alpha alpha)
  (/
   (/ 1.0 (fma (fma alpha alpha -1.0) (* cosTheta cosTheta) 1.0))
   (* (/ 0.0 0.0) (PI)))
  (/ -1.0 (* (PI) (/ 0.0 0.0)))))

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

4. Applied rewrites -0.0%

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

5. Taylor expanded in cosTheta around 0

   \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right)}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{0}{0}}\right) \]
6. Step-by-step derivation

   1. lower-PI.f32 -0.0

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

7. Applied rewrites -0.0%

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

8. Taylor expanded in cosTheta around 0

   \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\color{blue}{1 + {cosTheta}^{2} \cdot \left({\alpha}^{2} - 1\right)}}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\color{blue}{{cosTheta}^{2} \cdot \left({\alpha}^{2} - 1\right) + 1}}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\color{blue}{\left({\alpha}^{2} - 1\right) \cdot {cosTheta}^{2}} + 1}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]

   3. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\color{blue}{\mathsf{fma}\left({\alpha}^{2} - 1, {cosTheta}^{2}, 1\right)}}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\mathsf{fma}\left(\color{blue}{{\alpha}^{2} + \left(\mathsf{neg}\left(1\right)\right)}, {cosTheta}^{2}, 1\right)}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\mathsf{fma}\left(\color{blue}{\alpha \cdot \alpha} + \left(\mathsf{neg}\left(1\right)\right), {cosTheta}^{2}, 1\right)}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]

   6. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\mathsf{fma}\left(\alpha \cdot \alpha + \color{blue}{-1}, {cosTheta}^{2}, 1\right)}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]

   7. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\alpha, \alpha, -1\right)}, {cosTheta}^{2}, 1\right)}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]

   8. unpow2 N/A

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

   9. lower-*.f32 -0.0

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

10. Applied rewrites -0.0%

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

11. Final simplification -0.0%

    \[\leadsto \mathsf{fma}\left(\alpha \cdot \alpha, \frac{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta \cdot cosTheta, 1\right)}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]
12. Add Preprocessing

Alternative 11: 6.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\alpha \cdot \alpha, \frac{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta, cosTheta, 1\right)}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (fma
  (* alpha alpha)
  (/
   (/ 1.0 (fma (* (fma alpha alpha -1.0) cosTheta) cosTheta 1.0))
   (* (/ 0.0 0.0) (PI)))
  (/ -1.0 (* (PI) (/ 0.0 0.0)))))

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

4. Applied rewrites -0.0%

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

5. Taylor expanded in cosTheta around 0

   \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right)}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{0}{0}}\right) \]
6. Step-by-step derivation

   1. lower-PI.f32 -0.0

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

7. Applied rewrites -0.0%

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

8. Taylor expanded in alpha around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{{\alpha}^{2}}, \frac{\frac{1}{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right)}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]
9. Step-by-step derivation

   1. unpow2 N/A

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

   2. lower-*.f32 -0.0

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

10. Applied rewrites -0.0%

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

11. Final simplification -0.0%

    \[\leadsto \mathsf{fma}\left(\alpha \cdot \alpha, \frac{\frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta, cosTheta, 1\right)}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]
12. Add Preprocessing

Alternative 12: 6.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\alpha \cdot \alpha, \frac{\frac{1}{\left(\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta\right) \cdot cosTheta}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (fma
  (* alpha alpha)
  (/
   (/ 1.0 (* (* (fma alpha alpha -1.0) cosTheta) cosTheta))
   (* (/ 0.0 0.0) (PI)))
  (/ -1.0 (* (PI) (/ 0.0 0.0)))))

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

4. Applied rewrites -0.0%

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

5. Taylor expanded in cosTheta around 0

   \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right)}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{0}{0}}\right) \]
6. Step-by-step derivation

   1. lower-PI.f32 -0.0

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

7. Applied rewrites -0.0%

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

8. Taylor expanded in cosTheta around inf

   \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\color{blue}{{cosTheta}^{2} \cdot \left({\alpha}^{2} - 1\right)}}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\color{blue}{\left({\alpha}^{2} - 1\right) \cdot {cosTheta}^{2}}}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\left({\alpha}^{2} - 1\right) \cdot \color{blue}{\left(cosTheta \cdot cosTheta\right)}}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\color{blue}{\left(\left({\alpha}^{2} - 1\right) \cdot cosTheta\right) \cdot cosTheta}}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\color{blue}{\left(\left({\alpha}^{2} - 1\right) \cdot cosTheta\right) \cdot cosTheta}}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\color{blue}{\left(\left({\alpha}^{2} - 1\right) \cdot cosTheta\right)} \cdot cosTheta}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\left(\color{blue}{\left({\alpha}^{2} + \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot cosTheta\right) \cdot cosTheta}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]

   7. unpow2 N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f32 -0.0

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

10. Applied rewrites -0.0%

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

11. Final simplification -0.0%

    \[\leadsto \mathsf{fma}\left(\alpha \cdot \alpha, \frac{\frac{1}{\left(\mathsf{fma}\left(\alpha, \alpha, -1\right) \cdot cosTheta\right) \cdot cosTheta}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]
12. Add Preprocessing

Alternative 13: 6.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\alpha \cdot \alpha, \frac{\frac{1}{\left(cosTheta \cdot \left(\alpha \cdot \alpha\right)\right) \cdot cosTheta}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (fma
  (* alpha alpha)
  (/ (/ 1.0 (* (* cosTheta (* alpha alpha)) cosTheta)) (* (/ 0.0 0.0) (PI)))
  (/ -1.0 (* (PI) (/ 0.0 0.0)))))

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

4. Applied rewrites -0.0%

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

5. Taylor expanded in cosTheta around 0

   \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right)}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{0}{0}}\right) \]
6. Step-by-step derivation

   1. lower-PI.f32 -0.0

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

7. Applied rewrites -0.0%

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

8. Taylor expanded in alpha around inf

   \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\color{blue}{{\alpha}^{2} \cdot {cosTheta}^{2}}}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]
9. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{{\alpha}^{2} \cdot \color{blue}{\left(cosTheta \cdot cosTheta\right)}}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\color{blue}{\left({\alpha}^{2} \cdot cosTheta\right) \cdot cosTheta}}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\color{blue}{\left({\alpha}^{2} \cdot cosTheta\right) \cdot cosTheta}}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\color{blue}{\left({\alpha}^{2} \cdot cosTheta\right)} \cdot cosTheta}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]

   5. unpow2 N/A

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

   6. lower-*.f32 -0.0

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

10. Applied rewrites -0.0%

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

11. Final simplification -0.0%

    \[\leadsto \mathsf{fma}\left(\alpha \cdot \alpha, \frac{\frac{1}{\left(cosTheta \cdot \left(\alpha \cdot \alpha\right)\right) \cdot cosTheta}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]
12. Add Preprocessing

Alternative 14: 6.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\alpha \cdot \alpha, \frac{\frac{1}{1}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (fma
  (* alpha alpha)
  (/ (/ 1.0 1.0) (* (/ 0.0 0.0) (PI)))
  (/ -1.0 (* (PI) (/ 0.0 0.0)))))

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

4. Applied rewrites -0.0%

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

5. Taylor expanded in cosTheta around 0

   \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\mathsf{fma}\left(cosTheta \cdot \mathsf{fma}\left(\alpha, \alpha, -1\right), cosTheta, 1\right)}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{0}{0}}\right) \]
6. Step-by-step derivation

   1. lower-PI.f32 -0.0

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

7. Applied rewrites -0.0%

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

8. Taylor expanded in cosTheta around 0

   \[\leadsto \mathsf{fma}\left(\frac{\alpha \cdot \alpha}{1}, \frac{\frac{1}{\color{blue}{1}}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]
9. Step-by-step derivation

10. Applied rewrites 0.0%

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

11. Final simplification -0.0%

    \[\leadsto \mathsf{fma}\left(\alpha \cdot \alpha, \frac{\frac{1}{1}}{\frac{0}{0} \cdot \mathsf{PI}\left(\right)}, \frac{-1}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}}\right) \]
12. Add Preprocessing

Alternative 15: -0.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\mathsf{PI}\left(\right) \cdot \frac{0}{0}} \end{array} \]

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ (fma alpha alpha -1.0) (* (PI) (/ 0.0 0.0))))

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

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

   2. sub-neg N/A

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

   3. lift-*.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. metadata-eval 11.3

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f32 N/A

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

4. Applied rewrites -0.0%

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

5. Taylor expanded in cosTheta around 0

   \[\leadsto \frac{\mathsf{fma}\left(\alpha, \alpha, -1\right)}{\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{0}{0}} \]
6. Step-by-step derivation

   1. lower-PI.f32 -0.0

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

7. Applied rewrites -0.0%

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

Reproduce

herbie shell --seed 2024294 
(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)))))