Lanczos kernel

Percentage Accurate: 98.0% → 97.9%

Time: 11.7s

Alternatives: 11

Speedup: 1.0×

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: 98.0% 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: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.1%

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

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

   3. associate-*r/ N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. associate-*l* N/A

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

   7. associate-*l/ N/A

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

   8. clear-num N/A

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

4. Applied rewrites 98.1%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. lift-/.f32 N/A

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

   6. clear-num N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. *-lft-identity N/A

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

6. Applied rewrites 98.1%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f32 N/A

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

   5. associate-/r/ N/A

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

   6. associate-*l/ N/A

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

   7. associate-/r* N/A

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

   8. frac-times N/A

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

8. Applied rewrites 98.2%

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

9. Final simplification 98.2%

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

Alternative 2: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.1%

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

3. Final simplification 98.1%

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

Alternative 3: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.1%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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 97.6

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

4. Applied rewrites 97.6%

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

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 98.0

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

6. Applied rewrites 98.0%

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

Alternative 4: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.1%

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

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

   3. associate-*r/ N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. associate-*l* N/A

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

   7. associate-*l/ N/A

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

   8. clear-num N/A

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

4. Applied rewrites 98.1%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. lift-/.f32 N/A

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

   6. clear-num N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. *-lft-identity N/A

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

6. Applied rewrites 98.1%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f32 N/A

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

   5. associate-/r/ N/A

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

   6. associate-*l/ N/A

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

   7. associate-/r* N/A

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

   8. frac-times N/A

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

   9. frac-2neg N/A

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

8. Applied rewrites 97.9%

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

9. Final simplification 97.9%

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

Alternative 5: 97.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.1%

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

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

   3. associate-*r/ N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. associate-*l* N/A

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

   7. associate-*l/ N/A

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

   8. clear-num N/A

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

4. Applied rewrites 98.1%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. lift-/.f32 N/A

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

   6. clear-num N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. *-lft-identity N/A

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

6. Applied rewrites 98.1%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f32 N/A

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

   5. associate-/r/ N/A

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

   6. associate-*l/ N/A

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

   7. associate-/r* N/A

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

   8. *-commutative N/A

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

8. Applied rewrites 97.7%

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

9. Final simplification 97.7%

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

Alternative 6: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.1%

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

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

   3. associate-*r/ N/A

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

   4. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. associate-*l* N/A

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

   7. associate-*l/ N/A

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

   8. clear-num N/A

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

4. Applied rewrites 98.1%

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

   1. lift-/.f32 N/A

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

   2. div-inv N/A

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

   3. lift-/.f32 N/A

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

   4. clear-num N/A

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

   5. lift-/.f32 N/A

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

   6. clear-num N/A

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

   7. lift-/.f32 N/A

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

   8. frac-times N/A

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

   9. *-lft-identity N/A

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

6. Applied rewrites 98.1%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lift-/.f32 N/A

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

   5. associate-/r/ N/A

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

   6. associate-*l/ N/A

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

   7. associate-/r* N/A

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

   8. frac-times N/A

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

8. Applied rewrites 98.2%

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

10. Applied rewrites 97.2%

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

11. Final simplification 97.2%

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

Alternative 7: 79.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(1 + tau \cdot tau\right) \cdot \left({\left(x \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot -0.16666666666666666\right) + 1 \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (+ (* (+ 1.0 (* tau tau)) (* (pow (* x (PI)) 2.0) -0.16666666666666666)) 1.0))

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {x}^{2}} + 1 \]

   3. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}, {x}^{2}, 1\right)} \]

5. Applied rewrites 65.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -0.16666666666666666\right), x \cdot x, 1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 38.0%

   \[\leadsto \left(-0.16666666666666666 \cdot {\left(\mathsf{PI}\left(\right) \cdot x\right)}^{2}\right) \cdot \mathsf{fma}\left(tau, tau, 1\right) + \color{blue}{1} \]
8. Step-by-step derivation

9. Applied rewrites 78.9%

   \[\leadsto \left(-0.16666666666666666 \cdot {\left(\mathsf{PI}\left(\right) \cdot x\right)}^{2}\right) \cdot \left(tau \cdot tau + 1\right) + 1 \]

10. Final simplification 78.9%

    \[\leadsto \left(1 + tau \cdot tau\right) \cdot \left({\left(x \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot -0.16666666666666666\right) + 1 \]
11. Add Preprocessing

Alternative 8: 70.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \left(tau \cdot tau\right) \cdot \left({\left(x \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot -0.16666666666666666\right) + 1 \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (+ (* (* tau tau) (* (pow (* x (PI)) 2.0) -0.16666666666666666)) 1.0))

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {x}^{2}} + 1 \]

   3. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}, {x}^{2}, 1\right)} \]

5. Applied rewrites 65.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -0.16666666666666666\right), x \cdot x, 1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 38.0%

   \[\leadsto \left(-0.16666666666666666 \cdot {\left(\mathsf{PI}\left(\right) \cdot x\right)}^{2}\right) \cdot \mathsf{fma}\left(tau, tau, 1\right) + \color{blue}{1} \]

8. Taylor expanded in tau around inf

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

10. Applied rewrites 71.2%

    \[\leadsto \left(-0.16666666666666666 \cdot {\left(\mathsf{PI}\left(\right) \cdot x\right)}^{2}\right) \cdot \left(tau \cdot tau\right) + 1 \]

11. Final simplification 71.2%

    \[\leadsto \left(tau \cdot tau\right) \cdot \left({\left(x \cdot \mathsf{PI}\left(\right)\right)}^{2} \cdot -0.16666666666666666\right) + 1 \]
12. Add Preprocessing

Alternative 9: 70.3% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(x \cdot x\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(tau \cdot tau\right) \cdot -0.16666666666666666\right) + 1 \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (+ (* (* (* (* x x) (PI)) (PI)) (* (* tau tau) -0.16666666666666666)) 1.0))

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {x}^{2}} + 1 \]

   3. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}, {x}^{2}, 1\right)} \]

5. Applied rewrites 65.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -0.16666666666666666\right), x \cdot x, 1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 38.0%

   \[\leadsto \left(-0.16666666666666666 \cdot {\left(\mathsf{PI}\left(\right) \cdot x\right)}^{2}\right) \cdot \mathsf{fma}\left(tau, tau, 1\right) + \color{blue}{1} \]

8. Taylor expanded in tau around inf

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

10. Applied rewrites 71.2%

    \[\leadsto \left(\left(tau \cdot tau\right) \cdot -0.16666666666666666\right) \cdot \left(\left(\left(x \cdot x\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) + 1 \]

11. Final simplification 71.2%

    \[\leadsto \left(\left(\left(x \cdot x\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\left(tau \cdot tau\right) \cdot -0.16666666666666666\right) + 1 \]
12. Add Preprocessing

Alternative 10: 65.1% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right) + 1 \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (+ (* (* (* (* x x) -0.16666666666666666) (PI)) (PI)) 1.0))

Derivation

1. Initial program 98.1%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {x}^{2}} + 1 \]

   3. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2}, {x}^{2}, 1\right)} \]

5. Applied rewrites 65.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(tau, tau, 1\right) \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -0.16666666666666666\right), x \cdot x, 1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 38.0%

   \[\leadsto \left(-0.16666666666666666 \cdot {\left(\mathsf{PI}\left(\right) \cdot x\right)}^{2}\right) \cdot \mathsf{fma}\left(tau, tau, 1\right) + \color{blue}{1} \]

8. Taylor expanded in tau around 0

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

10. Applied rewrites 66.5%

    \[\leadsto \left(\left(\left(x \cdot x\right) \cdot -0.16666666666666666\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right) + 1 \]
11. Add Preprocessing

Alternative 11: 64.1% 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 98.1%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 65.7%

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

Reproduce

herbie shell --seed 2024266 
(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)))))