Lanczos kernel

Percentage Accurate: 97.9% → 97.8%

Time: 9.6s

Alternatives: 11

Speedup: N/A×

Specification

\[\left(10^{-5} \leq x \land x \leq 1\right) \land \left(1 \leq tau \land tau \leq 5\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ t_2 := t\_1 \cdot tau\\ \frac{\sin t\_2}{t\_2} \cdot \frac{\sin t\_1}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI))) (t_2 (* t_1 tau)))
   (* (/ (sin t_2) t_2) (/ (sin t_1) t_1))))

Initial Program: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ t_2 := t\_1 \cdot tau\\ \frac{\sin t\_2}{t\_2} \cdot \frac{\sin t\_1}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI))) (t_2 (* t_1 tau)))
   (* (/ (sin t_2) t_2) (/ (sin t_1) t_1))))

Alternative 1: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ t_2 := \left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\\ \frac{\frac{\sin t\_1}{t\_1} \cdot \sin t\_2}{t\_2} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI))) (t_2 (* (* tau x) (PI))))
   (/ (* (/ (sin t_1) t_1) (sin t_2)) t_2)))

Derivation

1. Initial program 97.6%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{tau \cdot \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   6. lower-*.f32 97.1

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right)} \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

4. Applied rewrites 97.1%

   \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot tau\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\sin \color{blue}{\left(x \cdot \left(\mathsf{PI}\left(\right) \cdot tau\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot tau\right) \cdot x\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(tau \cdot x\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(tau \cdot x\right)}\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   8. lift-*.f32 97.6

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(tau \cdot x\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot tau\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   11. lower-*.f32 97.6

       \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot tau\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

6. Applied rewrites 97.6%

   \[\leadsto \frac{\sin \color{blue}{\left(\left(x \cdot tau\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

8. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\left(-\mathsf{PI}\left(\right)\right) \cdot x} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(tau \cdot x\right)\right)}{\left(\left(-tau\right) \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \]

9. Final simplification 97.6%

   \[\leadsto \frac{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \cdot \sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \]
10. Add Preprocessing

Alternative 2: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ t_2 := \left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\\ \frac{\sin t\_1 \cdot \sin t\_2}{t\_2 \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI))) (t_2 (* (* tau x) (PI))))
   (/ (* (sin t_1) (sin t_2)) (* t_2 t_1))))

Derivation

1. Initial program 97.6%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{tau \cdot \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   6. lower-*.f32 97.1

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right)} \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

4. Applied rewrites 97.1%

   \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot tau\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\sin \color{blue}{\left(x \cdot \left(\mathsf{PI}\left(\right) \cdot tau\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot tau\right) \cdot x\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(tau \cdot x\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(tau \cdot x\right)}\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   8. lift-*.f32 97.6

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(tau \cdot x\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot tau\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   11. lower-*.f32 97.6

       \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot tau\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

6. Applied rewrites 97.6%

   \[\leadsto \frac{\sin \color{blue}{\left(\left(x \cdot tau\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

8. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\left(-\mathsf{PI}\left(\right)\right) \cdot x} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(tau \cdot x\right)\right)}{\left(\left(-tau\right) \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \]
9. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\left(-\mathsf{PI}\left(\right)\right) \cdot x} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(tau \cdot x\right)\right)}{\left(\left(-tau\right) \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \]

10. Applied rewrites 97.6%

    \[\leadsto \color{blue}{\frac{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{\left(\left(\left(-tau\right) \cdot x\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right)}} \]

11. Final simplification 97.6%

    \[\leadsto \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)} \]
12. Add Preprocessing

Alternative 3: 97.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ \frac{\sin t\_1 \cdot \sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(\left(t\_1 \cdot \mathsf{PI}\left(\right)\right) \cdot x\right) \cdot tau} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI))))
   (/ (* (sin t_1) (sin (* (* tau x) (PI)))) (* (* (* t_1 (PI)) x) tau))))

Derivation

1. Initial program 97.6%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{tau \cdot \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   6. lower-*.f32 97.1

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right)} \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

4. Applied rewrites 97.1%

   \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot tau\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\sin \color{blue}{\left(x \cdot \left(\mathsf{PI}\left(\right) \cdot tau\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot tau\right) \cdot x\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(tau \cdot x\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(tau \cdot x\right)}\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   8. lift-*.f32 97.6

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(tau \cdot x\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot tau\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   11. lower-*.f32 97.6

       \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot tau\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

6. Applied rewrites 97.6%

   \[\leadsto \frac{\sin \color{blue}{\left(\left(x \cdot tau\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

8. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\left(-\mathsf{PI}\left(\right)\right) \cdot x} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(tau \cdot x\right)\right)}{\left(\left(-tau\right) \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \]

10. Applied rewrites 97.0%

    \[\leadsto \color{blue}{\frac{\left(-\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{\left(\left(\left(\left(-\mathsf{PI}\left(\right)\right) \cdot x\right) \cdot \mathsf{PI}\left(\right)\right) \cdot x\right) \cdot tau}} \]

11. Final simplification 97.0%

    \[\leadsto \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(\left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot x\right) \cdot tau} \]
12. Add Preprocessing

Alternative 4: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{\left(\left(\left(tau \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot x\right) \cdot x} \cdot \sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right) \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (*
  (/ (sin (* x (PI))) (* (* (* (* tau (PI)) (PI)) x) x))
  (sin (* (* tau x) (PI)))))

Derivation

1. Initial program 97.6%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{tau \cdot \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   6. lower-*.f32 97.1

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right)} \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

4. Applied rewrites 97.1%

   \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot tau\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\sin \color{blue}{\left(x \cdot \left(\mathsf{PI}\left(\right) \cdot tau\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot tau\right) \cdot x\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(tau \cdot x\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(tau \cdot x\right)}\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   8. lift-*.f32 97.6

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(tau \cdot x\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot tau\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   11. lower-*.f32 97.6

       \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot tau\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

6. Applied rewrites 97.6%

   \[\leadsto \frac{\sin \color{blue}{\left(\left(x \cdot tau\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

8. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\left(-\mathsf{PI}\left(\right)\right) \cdot x} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(tau \cdot x\right)\right)}{\left(\left(-tau\right) \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \]

10. Applied rewrites 96.7%

    \[\leadsto \color{blue}{\sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{\left(\left(\left(tau \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot x\right) \cdot x}} \]

11. Final simplification 96.7%

    \[\leadsto \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{\left(\left(\left(tau \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot x\right) \cdot x} \cdot \sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right) \]
12. Add Preprocessing

Alternative 5: 70.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ t_2 := tau \cdot t\_1\\ \frac{t\_1 \cdot \frac{\sin t\_2}{t\_1}}{t\_2} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI))) (t_2 (* tau t_1))) (/ (* t_1 (/ (sin t_2) t_1)) t_2)))

Derivation

1. Initial program 97.6%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \]

   4. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \]

4. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\mathsf{PI}\left(\right) \cdot x} \cdot \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\mathsf{PI}\left(\right) \cdot x} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\mathsf{PI}\left(\right) \cdot x} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   3. lower-PI.f32 71.5

      \[\leadsto \frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\mathsf{PI}\left(\right) \cdot x} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot x\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \]

7. Applied rewrites 71.5%

   \[\leadsto \frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\mathsf{PI}\left(\right) \cdot x} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \]

8. Final simplification 71.5%

   \[\leadsto \frac{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{\sin \left(tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)\right)}{x \cdot \mathsf{PI}\left(\right)}}{tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)} \]
9. Add Preprocessing

Alternative 6: 70.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1 \cdot \sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(\left(-tau\right) \cdot x\right) \cdot \mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (/ (* -1.0 (sin (* (* tau x) (PI)))) (* (* (- tau) x) (PI))))

Derivation

1. Initial program 97.6%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{tau \cdot \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   6. lower-*.f32 97.1

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right)} \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

4. Applied rewrites 97.1%

   \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot tau\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\sin \color{blue}{\left(x \cdot \left(\mathsf{PI}\left(\right) \cdot tau\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot tau\right) \cdot x\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(tau \cdot x\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(tau \cdot x\right)}\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   8. lift-*.f32 97.6

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(tau \cdot x\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot tau\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   11. lower-*.f32 97.6

       \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot tau\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

6. Applied rewrites 97.6%

   \[\leadsto \frac{\sin \color{blue}{\left(\left(x \cdot tau\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

8. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\left(-\mathsf{PI}\left(\right)\right) \cdot x} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(tau \cdot x\right)\right)}{\left(\left(-tau\right) \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 71.5%

    \[\leadsto \frac{\color{blue}{-1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(tau \cdot x\right)\right)}{\left(\left(-tau\right) \cdot x\right) \cdot \mathsf{PI}\left(\right)} \]

12. Final simplification 71.5%

    \[\leadsto \frac{-1 \cdot \sin \left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(\left(-tau\right) \cdot x\right) \cdot \mathsf{PI}\left(\right)} \]
13. Add Preprocessing

Alternative 7: 64.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ \frac{1}{\frac{t\_1}{\sin t\_1}} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI)))) (/ 1.0 (/ t_1 (sin t_1)))))

Derivation

1. Initial program 97.6%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \]

   3. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{x \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \]

   5. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x}}{\mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \]

   6. frac-2neg N/A

      \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x}\right)}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)}} \cdot \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \]

   7. lift-/.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x}\right)}{\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)} \cdot \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \]

   8. frac-times N/A

      \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x}\right)\right) \cdot \sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}} \]

   9. *-rgt-identity N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x}\right)\right) \cdot \sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(\mathsf{neg}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot 1}\right)\right) \cdot \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)} \]

   10. distribute-rgt-neg-in N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x}\right)\right) \cdot \sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{neg}\left(1\right)\right)\right)} \cdot \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x}\right)\right) \cdot \sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(\mathsf{PI}\left(\right) \cdot \color{blue}{-1}\right) \cdot \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)} \]

   12. associate-*r* N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x}\right)\right) \cdot \sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot \left(-1 \cdot \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)\right)}} \]

   13. neg-mul-1 N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x}\right)\right) \cdot \sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{neg}\left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)\right)}} \]

4. Applied rewrites 97.1%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{-x} \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot \left(-tau\right)}} \]

5. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(tau \cdot \left(x \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot \left(-tau\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(tau \cdot x\right) \cdot {\mathsf{PI}\left(\right)}^{2}}\right)}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot \left(-tau\right)} \]

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

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(tau \cdot x\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}}}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot \left(-tau\right)} \]

   4. lower-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(tau \cdot x\right)\right) \cdot {\mathsf{PI}\left(\right)}^{2}}}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot \left(-tau\right)} \]

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

      \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(tau\right)\right) \cdot x\right)} \cdot {\mathsf{PI}\left(\right)}^{2}}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot \left(-tau\right)} \]

   6. mul-1-neg N/A

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

   7. lower-*.f32 N/A

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

   8. mul-1-neg N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\mathsf{neg}\left(tau\right)\right)} \cdot x\right) \cdot {\mathsf{PI}\left(\right)}^{2}}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot \left(-tau\right)} \]

   9. lower-neg.f32 N/A

      \[\leadsto \frac{\left(\color{blue}{\left(-tau\right)} \cdot x\right) \cdot {\mathsf{PI}\left(\right)}^{2}}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot \left(-tau\right)} \]

   10. unpow2 N/A

       \[\leadsto \frac{\left(\left(-tau\right) \cdot x\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot \left(-tau\right)} \]

   11. lower-*.f32 N/A

       \[\leadsto \frac{\left(\left(-tau\right) \cdot x\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot \left(-tau\right)} \]

   12. lower-PI.f32 N/A

       \[\leadsto \frac{\left(\left(-tau\right) \cdot x\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot \left(-tau\right)} \]

   13. lower-PI.f32 64.0

       \[\leadsto \frac{\left(\left(-tau\right) \cdot x\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot \left(-tau\right)} \]

7. Applied rewrites 64.0%

   \[\leadsto \frac{\color{blue}{\left(\left(-tau\right) \cdot x\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot \left(-tau\right)} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\left(\left(-tau\right) \cdot x\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot \left(-tau\right)}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\left(\left(-tau\right) \cdot x\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot \left(-tau\right)}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\left(\left(-tau\right) \cdot x\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{\left(-tau\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}} \]

   4. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\left(\left(-tau\right) \cdot x\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{-tau}}{\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   5. clear-num N/A

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

   6. lower-/.f32 N/A

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

9. Applied rewrites 64.0%

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

10. Taylor expanded in tau around 0

    \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]
11. Step-by-step derivation

    1. lower-/.f32 N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

    2. *-commutative N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{PI}\left(\right) \cdot x}}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}} \]

    3. lower-*.f32 N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{PI}\left(\right) \cdot x}}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}} \]

    4. lower-PI.f32 N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{PI}\left(\right)} \cdot x}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}} \]

    5. lower-sin.f32 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\color{blue}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}} \]

    6. *-commutative N/A

       \[\leadsto \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

    7. lower-*.f32 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

    8. lower-PI.f32 65.0

       \[\leadsto \frac{1}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot x\right)}} \]

12. Applied rewrites 65.0%

    \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{PI}\left(\right) \cdot x}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}} \]

13. Final simplification 65.0%

    \[\leadsto \frac{1}{\frac{x \cdot \mathsf{PI}\left(\right)}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}} \]
14. Add Preprocessing

Alternative 8: 64.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ \frac{\mathsf{PI}\left(\right) \cdot \sin t\_1}{t\_1 \cdot \mathsf{PI}\left(\right)} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI)))) (/ (* (PI) (sin t_1)) (* t_1 (PI)))))

Derivation

1. Initial program 97.6%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \]

   4. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \]

4. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\mathsf{PI}\left(\right) \cdot x} \cdot \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\mathsf{PI}\left(\right) \cdot x}} \]

   5. lift-/.f32 N/A

      \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \color{blue}{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\mathsf{PI}\left(\right) \cdot x}} \]

   6. times-frac N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)}}{\left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right) \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   8. frac-times N/A

      \[\leadsto \color{blue}{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right) \cdot x}} \]

6. Applied rewrites 97.1%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{x \cdot tau} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}} \]

7. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\mathsf{PI}\left(\right)} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)} \]
8. Step-by-step derivation

   1. lower-PI.f32 65.0

      \[\leadsto \frac{\color{blue}{\mathsf{PI}\left(\right)} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)} \]

9. Applied rewrites 65.0%

   \[\leadsto \frac{\color{blue}{\mathsf{PI}\left(\right)} \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)} \]
10. Add Preprocessing

Alternative 9: 64.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ \frac{\sin t\_1 \cdot tau}{tau \cdot t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI)))) (/ (* (sin t_1) tau) (* tau t_1))))

Derivation

1. Initial program 97.6%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

   2. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   3. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \]

   4. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right) \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \]

4. Applied rewrites 97.5%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)\right)}{\mathsf{PI}\left(\right) \cdot x} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)}} \]

5. Taylor expanded in tau around 0

   \[\leadsto \frac{\color{blue}{tau \cdot \sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   3. lower-sin.f32 N/A

      \[\leadsto \frac{\color{blue}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot tau}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot tau}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)} \cdot tau}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \]

   6. lower-PI.f32 65.0

      \[\leadsto \frac{\sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot x\right) \cdot tau}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \]

7. Applied rewrites 65.0%

   \[\leadsto \frac{\color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot x\right) \cdot tau}}{tau \cdot \left(\mathsf{PI}\left(\right) \cdot x\right)} \]

8. Final simplification 65.0%

   \[\leadsto \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}{tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)} \]
9. Add Preprocessing

Alternative 10: 64.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \mathsf{PI}\left(\right)\\ \frac{\sin t\_1}{t\_1} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* x (PI)))) (/ (sin t_1) t_1)))

Derivation

1. Initial program 97.6%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{tau \cdot \left(x \cdot \mathsf{PI}\left(\right)\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{tau \cdot \color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   6. lower-*.f32 97.1

      \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right)} \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

4. Applied rewrites 97.1%

   \[\leadsto \frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\color{blue}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot \mathsf{PI}\left(\right)\right)} \cdot tau\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   3. associate-*l* N/A

      \[\leadsto \frac{\sin \color{blue}{\left(x \cdot \left(\mathsf{PI}\left(\right) \cdot tau\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot tau\right) \cdot x\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   5. associate-*r* N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(tau \cdot x\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(tau \cdot x\right)}\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   8. lift-*.f32 97.6

      \[\leadsto \frac{\sin \color{blue}{\left(\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\sin \left(\color{blue}{\left(tau \cdot x\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot tau\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

   11. lower-*.f32 97.6

       \[\leadsto \frac{\sin \left(\color{blue}{\left(x \cdot tau\right)} \cdot \mathsf{PI}\left(\right)\right)}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

6. Applied rewrites 97.6%

   \[\leadsto \frac{\sin \color{blue}{\left(\left(x \cdot tau\right) \cdot \mathsf{PI}\left(\right)\right)}}{\left(tau \cdot x\right) \cdot \mathsf{PI}\left(\right)} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]

8. Applied rewrites 97.6%

   \[\leadsto \color{blue}{\frac{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\left(-\mathsf{PI}\left(\right)\right) \cdot x} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(tau \cdot x\right)\right)}{\left(\left(-tau\right) \cdot x\right) \cdot \mathsf{PI}\left(\right)}} \]

9. Taylor expanded in tau around 0

   \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]
10. Step-by-step derivation

    1. lower-/.f32 N/A

       \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]

    2. lower-sin.f32 N/A

       \[\leadsto \frac{\color{blue}{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}}{x \cdot \mathsf{PI}\left(\right)} \]

    3. *-commutative N/A

       \[\leadsto \frac{\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)}}{x \cdot \mathsf{PI}\left(\right)} \]

    4. lower-*.f32 N/A

       \[\leadsto \frac{\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot x\right)}}{x \cdot \mathsf{PI}\left(\right)} \]

    5. lower-PI.f32 N/A

       \[\leadsto \frac{\sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot x\right)}{x \cdot \mathsf{PI}\left(\right)} \]

    6. *-commutative N/A

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \]

    7. lower-*.f32 N/A

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\color{blue}{\mathsf{PI}\left(\right) \cdot x}} \]

    8. lower-PI.f32 65.0

       \[\leadsto \frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\color{blue}{\mathsf{PI}\left(\right)} \cdot x} \]

11. Applied rewrites 65.0%

    \[\leadsto \color{blue}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot x\right)}{\mathsf{PI}\left(\right) \cdot x}} \]

12. Final simplification 65.0%

    \[\leadsto \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
13. Add Preprocessing

Alternative 11: 63.4% accurate, 258.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x tau) :precision binary32 1.0)

C

float code(float x, float tau) {
	return 1.0f;
}

Fortran

real(4) function code(x, tau)
    real(4), intent (in) :: x
    real(4), intent (in) :: tau
    code = 1.0e0
end function

Julia

function code(x, tau)
	return Float32(1.0)
end

MATLAB

function tmp = code(x, tau)
	tmp = single(1.0);
end

Derivation

1. Initial program 97.6%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 64.2%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024331 
(FPCore (x tau)
  :name "Lanczos kernel"
  :precision binary32
  :pre (and (and (<= 1e-5 x) (<= x 1.0)) (and (<= 1.0 tau) (<= tau 5.0)))
  (* (/ (sin (* (* x (PI)) tau)) (* (* x (PI)) tau)) (/ (sin (* x (PI))) (* x (PI)))))