Lanczos kernel

Percentage Accurate: 97.9% → 97.9%

Time: 1.9s

Alternatives: 5

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: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 97.9% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.6%

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

Alternative 2: 70.5% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 68.1%

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

Alternative 3: 63.8% accurate, 2.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2} + {x}^{2} \cdot \left(\frac{1}{120} \cdot \left({tau}^{4} \cdot {\mathsf{PI}\left(\right)}^{4}\right) + \left(\frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{4} + \frac{1}{36} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)} \]

5. Applied rewrites 22.2%

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

6. Taylor expanded in x around 0

   \[\leadsto x \cdot \color{blue}{\left(tau \cdot \mathsf{PI}\left(\right) + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + {x}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({tau}^{7} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right)\right) + \frac{1}{120} \cdot \left({tau}^{5} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)} \]

8. Applied rewrites 18.7%

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

9. Taylor expanded in x around 0

   \[\leadsto x \cdot \color{blue}{\left(tau \cdot \mathsf{PI}\left(\right) + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{120} \cdot \left({tau}^{5} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)} \]

11. Applied rewrites 60.1%

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

Alternative 4: 63.0% accurate, 11.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \left(\frac{-1}{6} \cdot {\mathsf{PI}\left(\right)}^{2} + {x}^{2} \cdot \left(\frac{1}{120} \cdot \left({tau}^{4} \cdot {\mathsf{PI}\left(\right)}^{4}\right) + \left(\frac{1}{120} \cdot {\mathsf{PI}\left(\right)}^{4} + \frac{1}{36} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)\right)\right)} \]

5. Applied rewrites 22.2%

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

6. Taylor expanded in x around 0

   \[\leadsto x \cdot \color{blue}{\left(tau \cdot \mathsf{PI}\left(\right) + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + {x}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({tau}^{7} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right)\right) + \frac{1}{120} \cdot \left({tau}^{5} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)} \]

8. Applied rewrites 18.7%

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

9. Taylor expanded in x around 0

   \[\leadsto x \cdot \color{blue}{\left(tau \cdot \mathsf{PI}\left(\right) + {x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{3} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \frac{1}{120} \cdot \left({tau}^{5} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)} \]

11. Applied rewrites 60.1%

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

12. Taylor expanded in x around inf

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

14. Applied rewrites 59.4%

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

Alternative 5: 24.3% accurate, 258.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{PI}\left(\right) \end{array} \]

FPCore

(FPCore (x tau) :precision binary32 (PI))

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 68.1%

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

6. Taylor expanded in x around 0

   \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{-1}{6} \cdot \left({tau}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + {x}^{2} \cdot \left(\frac{-1}{5040} \cdot \left({tau}^{6} \cdot \left({x}^{2} \cdot {\mathsf{PI}\left(\right)}^{6}\right)\right) + \frac{1}{120} \cdot \left({tau}^{4} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)} \]

8. Applied rewrites 23.1%

   \[\leadsto \mathsf{PI}\left(\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(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)))))