GTR1 distribution

Percentage Accurate: 98.5% → 98.7%

Time: 10.6s

Alternatives: 14

Speedup: 1.0×

Specification

\[\left(0 \leq cosTheta \land cosTheta \leq 1\right) \land \left(0.0001 \leq \alpha \land \alpha \leq 1\right)\]

Math

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

FPCore

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

Initial Program: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.7% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

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

   3. lift-PI.f32 N/A

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

   4. add-log-exp N/A

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

   5. log-pow-rev N/A

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

   6. lower-log.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. pow-exp N/A

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

   9. *-commutative N/A

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

   10. lift-log.f32 N/A

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

   11. pow-to-exp N/A

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

   12. lower-pow.f32 98.6

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

4. Applied rewrites 98.6%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

Alternative 3: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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. fp-cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 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. Add Preprocessing

Alternative 4: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

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

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

   3. lift-PI.f32 N/A

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

   4. add-log-exp N/A

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

   5. log-pow-rev N/A

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

   6. lower-log.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. pow-exp N/A

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

   9. *-commutative N/A

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

   10. lift-log.f32 N/A

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

   11. pow-to-exp N/A

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

   12. lower-pow.f32 98.6

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

4. Applied rewrites 98.6%

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

5. Taylor expanded in alpha around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

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

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

   8. metadata-eval N/A

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

   9. *-lft-identity N/A

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

   10. lower--.f32 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-log.f32 97.6

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

7. Applied rewrites 97.6%

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

Alternative 5: 95.4% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ (/ (- (* alpha alpha) 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. Taylor expanded in cosTheta around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

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

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

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

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

   7. lower-fma.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-log.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 8.9

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

5. Applied rewrites 9.0%

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

6. Taylor expanded in cosTheta around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower--.f32 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 94.6

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

8. Applied rewrites 94.6%

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

Alternative 6: 95.4% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ (- (* alpha alpha) 1.0) (* (* (PI) (log (* alpha alpha))) 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. Taylor expanded in cosTheta around 0

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

5. Applied rewrites 94.5%

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

Alternative 7: 95.4% accurate, 1.2× speedup

Math

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

FPCore

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

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. pow2 N/A

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

   5. log-pow N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-log.f32 98.4

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in cosTheta around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-log.f32 N/A

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

   6. lower-PI.f32 94.5

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

7. Applied rewrites 94.5%

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

Alternative 8: 64.9% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (cosTheta alpha)
 :precision binary32
 (/ (/ -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. Taylor expanded in cosTheta around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

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

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

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

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

   7. lower-fma.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-log.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 10.7

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

5. Applied rewrites 9.6%

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

6. Taylor expanded in alpha around 0

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

8. Applied rewrites 65.3%

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

Alternative 9: 64.9% 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 cosTheta around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. unpow2 N/A

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

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

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

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

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

   7. lower-fma.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-log.f32 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f32 9.8

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

5. Applied rewrites 9.4%

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

6. Taylor expanded in alpha around 0

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

8. Applied rewrites 65.3%

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

Alternative 10: 6.6% accurate, 1.5× speedup

Math

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

FPCore

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

4. Applied rewrites -0.0%

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

5. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f32 N/A

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

7. Applied rewrites -0.0%

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

8. Taylor expanded in alpha around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 -0.0

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

10. Applied rewrites -0.0%

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

Alternative 11: 6.6% accurate, 2.0× speedup

Math

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

FPCore

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

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

5. Taylor expanded in cosTheta around 0

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

7. Applied rewrites -0.0%

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

8. Taylor expanded in cosTheta around 0

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

   1. lower-PI.f32 -0.0

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

10. Applied rewrites -0.0%

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

Alternative 12: 6.2% accurate, 2.9× speedup

Math

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

FPCore

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

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

   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 11.6

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f32 N/A

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

4. Applied rewrites -0.0%

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

5. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f32 N/A

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

7. Applied rewrites -0.0%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lower-+.f32 -0.0

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

9. Applied rewrites -0.0%

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

Alternative 13: 5.9% accurate, 3.0× speedup

Math

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

FPCore

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

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

   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 10.8

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f32 N/A

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

4. Applied rewrites -0.0%

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

5. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f32 N/A

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

7. Applied rewrites -0.0%

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

8. Taylor expanded in cosTheta around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*r* N/A

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

   5. lower-fma.f32 N/A

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

10. Applied rewrites -0.0%

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

Alternative 14: -0.0% accurate, 6.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.5%

   \[\frac{\alpha \cdot \alpha - 1}{\left(\mathsf{PI}\left(\right) \cdot \log \left(\alpha \cdot \alpha\right)\right) \cdot \left(1 + \left(\left(\alpha \cdot \alpha - 1\right) \cdot cosTheta\right) \cdot cosTheta\right)} \]
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 11.4

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

   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. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f32 N/A

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

4. Applied rewrites -0.0%

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

5. Taylor expanded in cosTheta around 0

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

   1. lower-*.f32 N/A

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

   2. lower-NAN.f32 N/A

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

   3. lower-PI.f32 -0.0

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

7. Applied rewrites -0.0%

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

Reproduce

herbie shell --seed 2024337 
(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)))))