Lanczos kernel

Percentage Accurate: 98.0% → 98.0%

Time: 6.6s

Alternatives: 7

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: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.0%

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

3. Final simplification 98.0%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.0%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 97.7

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 97.7

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 97.7

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

4. Applied rewrites 97.7%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. associate-*l/ N/A

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

   6. lift-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. lift-*.f32 N/A

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

6. Applied rewrites 97.8%

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

7. Final simplification 97.8%

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

Alternative 3: 71.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{PI}\left(\right) \cdot x\\ t_2 := t\_1 \cdot tau\\ \frac{1}{\frac{\left(\frac{t\_2}{\sin t\_2} \cdot x\right) \cdot \mathsf{PI}\left(\right)}{t\_1}} \end{array} \end{array} \]

FPCore

(FPCore (x tau)
 :precision binary32
 (let* ((t_1 (* (PI) x)) (t_2 (* t_1 tau)))
   (/ 1.0 (/ (* (* (/ t_2 (sin t_2)) x) (PI)) t_1))))

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f32 N/A

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

4. Applied rewrites 97.8%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 69.2

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

7. Applied rewrites 69.2%

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

9. Applied rewrites 69.0%

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

    1. lift-/.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. times-frac N/A

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

    5. associate-*r/ N/A

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

    6. lower-/.f32 N/A

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

11. Applied rewrites 69.4%

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

12. Final simplification 69.4%

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

Alternative 4: 71.0% accurate, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.0%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 97.7

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 97.7

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 97.7

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

4. Applied rewrites 97.7%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. associate-*l/ N/A

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

   6. lift-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. lift-*.f32 N/A

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

6. Applied rewrites 97.8%

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

7. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 69.3

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

9. Applied rewrites 69.3%

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

10. Final simplification 69.3%

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

Alternative 5: 64.5% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f32 N/A

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

4. Applied rewrites 97.8%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 69.2

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

7. Applied rewrites 69.2%

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

9. Applied rewrites 69.0%

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

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 63.0

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

12. Applied rewrites 63.0%

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

Alternative 6: 64.4% accurate, 2.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 98.0%

   \[\frac{\sin \left(\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau\right)}{\left(x \cdot \mathsf{PI}\left(\right)\right) \cdot tau} \cdot \frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)} \]
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. lift-/.f32 N/A

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

   5. div-inv N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. times-frac N/A

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

   10. lower-*.f32 N/A

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

4. Applied rewrites 97.2%

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

5. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-PI.f32 61.8

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

7. Applied rewrites 61.8%

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

9. Applied rewrites 61.8%

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

10. Taylor expanded in tau around 0

    \[\leadsto \color{blue}{\frac{\sin \left(x \cdot \mathsf{PI}\left(\right)\right)}{x \cdot \mathsf{PI}\left(\right)}} \]
11. 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 63.0

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

12. Applied rewrites 63.0%

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

Alternative 7: 63.6% 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 98.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 62.2%

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

Reproduce

herbie shell --seed 2024332 
(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)))))