碩士論文投影片

1. 分層的表格為主函數近似方法
Hierarchical Multipartite
Function Evaluation
Advisor : Prof. Shen-Fu Hsiao (蕭勝夫)
Student : Yi-Hau Chen (陳奕豪)

2. Outline
•Motivation
•Related Work
•Proposed
•Results & Comparison
•Conclusion
2

3. Outline
•Motivation
•Related Work
•Proposed
•Results & Comparison
•Conclusion
3

4. Motivation
• 特殊函數運算單元被廣泛應用於在數位訊號處理及多媒體
應用，如:圖像處理器(graphics processing unit)。
• 特殊函數運算單元(Special function unit)
• 三角函數(trigonometric)、倒數(reciprocal)、指數
(exponential) 與對數(logarithm)。
• 查表(lookup tables(LUT)) 與一些簡單的算數運算單元所
構成
• 主要分為兩類：
• piecewise polynomial approximation (PPA)
• table-lookup-and-addition (TA)
• 本論文主要探討如何有效地減少TA 的表格面積，仍然可以
保持TA 運算速度較快的優點。
4

5. Outline
• Motivation
• Related Work
• Category
• Piecewise Polynomial Approximation (PPA)
• Table-Lookup-and-Addition (TA)
• Bipartite Table Methods (BP)
• Symmetric Bipartite Table Methods (SBTM)
• Symmetric Table Addition Methods (STAM)
• Multipartite Table Methods (MP)
• Proposed
• Results & Comparison
• Conclusion
5

6. Category
6

7. Piecewise Polynomial
Approximation (PPA)-(1/2)
7
𝑓 𝑥 ≅ 𝑎0 𝑥 𝑚 + 𝑎1(𝑥 𝑚) ∙ 𝑥𝑙

8. PPA-(2/2) deg-2 Architecture
8
𝑓 𝑥 ≅ 𝑎0 𝑥 𝑚 + 𝑎1(𝑥 𝑚) ∙ 𝑥𝑙 + 𝑎2(𝑥 𝑚) ∙ 𝑥𝑙
2

9. Table-Lookup-and-Addition (TA)
• 主要分為兩類，add-table-add(ATA) 方法
以及bipartite/multipartite 方法。
• 而bipartite/multipartite 類的方法包含
• bipartite table methods (BP) [16]
• symmetric bipartite table methods (SBTM) [17]
• symmetric table addition methods (STAM) [18]
• multipartite table methods (MP) [1,19]
9

10. Bipartite Table Methods (BP)-
(1/5) 位元分區(bit partition)
10
在函數近似方法裡，為了近似一個的函數f(x)，n-bit 的輸
入 x 被分成兩個部分𝑥0以及𝑥1，其位元寬度分別為𝛼和𝛽且𝛼 +
𝛽 = 𝑛。我們假設初始輸入區間為0 ≤ 𝑥 < 1， 即
𝑥 = 𝑥0 + 𝑥1
0 ≤ 𝑥0 ≤ 1 − 2−𝛼
0 ≤ 𝑥1 ≤ 2−𝛼
− 2−𝑛
0 ≤ 𝑥0,1 ≤ 1 − 2−𝛾

11. BP-(2/5) 泰勒展開式
11
因此，函數f(x) 可以透過泰勒展開式的前兩項來近似：
𝑛=0
∞
𝑓 𝑛
(𝑎)
𝑛!
∙ 𝑥 − 𝑎 𝑛
(𝑎 = 𝑥0 and x = 𝑥0 + 𝑥1)
𝑓 𝑥 = 𝑓 𝑥0 + 𝑓′ 𝑥0 ∙ 𝑥1 + 𝜀𝑙𝑖𝑛
𝑓 𝑥 = 𝑓 𝑥0 + 𝑓′
𝑥0,1 ∙ 𝑥1 + 𝜀𝑙𝑖𝑛 + 𝜀 𝑠𝑙𝑝

12. BP-(3/5) 曲線放大示意圖
12

13. BP-(4/5) 架構(Architecture)
13
𝑓 𝑥 ≅ 𝑓 𝑥0 + 𝑓′
𝑥0,1 ∙ 𝑥1
≅ 𝑇𝐼 𝑥0 + 𝑇𝑂(𝑥0,1, 𝑥1)
𝑇𝐼 𝑥0 ≅ 𝑄[𝑓 𝑥0 ]
𝑇𝑂(𝑥0,1, 𝑥1) ≅ 𝑄[𝑓′ 𝑥0,1 ∙ 𝑥1]
Table of Initial Values
Table of Offset

14. BP-(5/5) 表格分割(table decomposition)
14

15. Symmetric Bipartite Table
Methods (SBTM)
15
0 ≤ 𝑥0 ≤ 1 − 2−𝛼
0 ≤ 𝑥1 ≤ 2−𝛼
− 2−𝑛
0 ≤ 𝑥0,1 ≤ 1 − 2−𝛾
𝛿1 = 2−𝛼
− 2−𝑛
𝛿0 = 2−𝛾
− 2−𝛼

16. 16
𝑓 𝑥 ≅ 𝑓 𝑥0 +
𝛿1
2
+ 𝑓′(𝑥0 +
𝛿1
2
) ∙ (𝑥1 −
𝛿1
2
)

17. 17
𝑓 𝑥 ≅ 𝑓 𝑥0 +
𝛿1
2
+ 𝑓′(𝑥0,1 +
𝛿0
2
+
𝛿1
2
) ∙ (𝑥1 −
𝛿1
2
)

18. 18
𝑓 𝑥 ≅ 𝑓 𝑥0 +
𝛿1
2
+ 𝑓′(𝑥0,1 +
𝛿0
2
+
𝛿1
2
) ∙ (𝑥1 −
𝛿1
2
)
𝑇𝐼 𝑥0 = 𝑄[𝑓 𝑥0 +
𝛿1
2
]
𝑇𝑂 𝑥0,1, 𝑥1 = 𝑄[𝑓′
𝑥0,1 +
𝛿0
2
+
𝛿1
2
∙ 𝑥1 −
𝛿1
2
]
Symmetric Bipartite Table
Methods (SBTM)

19. Symmetric Table Addition
Methods (STAM)
19
𝛿1 =
𝑖=1
𝑚
𝛿1,𝑖, 𝛿1,𝑖 = 2−𝑝 𝑖−1 − 2−𝑝 𝑖, 𝑖 = 1,2, … , 𝑚
𝑓 𝑥 ≅ 𝑓 𝑥0 +
𝛿1
2
+ 𝑓′(𝑥0,1 +
𝛿0
2
+
𝛿1
2
) ∙ (𝑥1 −
𝛿1
2
)
with
𝑝0 = 𝛼, 𝑝𝑖 = 𝑝𝑖−1 + 𝛽𝑖, 𝑖 = 1,2, … , 𝑚
𝑥1 =
𝑖=1
𝑚
𝑥1,𝑖
𝑓 𝑥 ≅ 𝑓 𝑥0 +
𝛿1
2
+ 𝑓′(𝑥0,1 +
𝛿0
2
+
𝛿1
2
) ∙ (
𝑖=1
𝑚
𝑥1,𝑖 −
𝑖=1
𝑚
𝛿1,𝑖
2
)
𝑓 𝑥 ≅ 𝑓 𝑥0 +
𝛿1
2
+ 𝑓′(𝑥0,1 +
𝛿0
2
+
𝛿1
2
) ∙
𝑖=1
𝑚
(𝑥1,𝑖 −
𝛿1,𝑖
2
)

20. Multipartite Table Methods (MP[1])-
(1/5) 位元分區(bit partition)
20

21. Multipartite Table Methods (MP[1])-
(2/5) 不同的初值以及斜率產生方式
21
𝑇𝐼 𝑥0 = 𝑄[
𝑓 𝑥0 + 𝑓 𝑥0 + 𝛿1
2
]
𝑇𝑂 𝑥0,1, 𝑥1 = 𝑄[𝑠 𝑥0,𝑖 ∙ 𝑥1,𝑖 −
𝛿1,𝑖
2
]
𝑇𝐼 𝑥0 = 𝑄[𝑓 𝑥0 +
𝛿1
2
]
𝑇𝑂 𝑥0,1, 𝑥1,𝑖 = 𝑄[𝑓′ 𝑥0,1 +
𝛿0
2
+
𝛿1
2
∙ 𝑥1,𝑖 −
𝛿1,𝑖
2
]
MP[1]:
STAM:

22. Multipartite Table Methods (MP[1])-
(3/5) 斜率s的算法
22
𝑠 𝑥0,𝑖 =
𝑓 𝜑2 − 𝑓 𝜑1 + 𝑓 𝜑4 − 𝑓 𝜑3
2 ∙ 𝛿1,𝑖

23. Multipartite Table Methods (MP[1])-
(4/5)架構(Architecture)
23

24. Multipartite Table Methods (MP[1])-
(5/5)表格分割(table decomposition)
24

25. Outline
• Motivation
• Related Work
• Proposed
• 函數的定義域(domain) 與值域(range)
• 取樣方法及誤差分配(Error Budget)
• HMP方法概述
• Lossless ROM Compression with Low Cost
• 整合誤差(Combined Error) 與窮舉搜尋(Exhaustive Search)
• 搜尋方法的加速
• Results & Comparison
• Conclusion
25

26. 函數的定義域(domain) 與值域(range)
26

27. 27
𝑇𝐼 𝑥0 = 𝑄[
𝑓 𝑥0 + 𝑓 𝑥0 + 𝛿1
2
]
𝜀 𝑞 = 𝑚 + 1 ∙ 2−𝑛−𝑔−1
取樣方法及誤差分配(Error Budget)
𝜀 𝑟𝑛𝑑 = 0.5 ∙ (2−𝑛
− 2−𝑔
)
𝜀 𝑎𝑝𝑥 +𝜀 𝑞 +𝜀 𝑟𝑛𝑑 = 𝜀𝑡𝑜𝑡𝑎𝑙 < 2−𝑛

28. 28
取樣方法及誤差分配(Error Budget)
|𝜀1| = 𝜀2 = |𝜀3| = |𝜀4|

29. HMP方法概述
29

30. HMP方法概述
30

31. HMP方法概述
31

32. HMP方法概述
32

33. 比較MP與HMP
33

34. Lossless ROM Compression
原理示意圖(s= 3)
34

35. Lossless ROM Compression with Low Cost-
表格分割(table decomposition)
35

36. 整合誤差(Combined Error) 與窮舉
搜尋(Exhaustive Search)
36

37. 搜尋方法的加速 流程圖
37

38. 搜尋方法的加速 驗證示意圖
38

39. Outline
•Motivation
•Related Work
•Proposed
•Results & Comparison
•Conclusion
39

40. 40

41. 41
表4.2: 24 位元
SIN 函數採用
MP [1] 及
HMP 之表格
分解

42. 42

43. 43

44. 比較 MP, HMP, HMP_TI
44

45. 45

46. 46

47. 47
使用版本為ISE10.1
編號為Xilinx Virtex-II XC2V1000-fg456-5

48. Outline
•Motivation
•Related Work
•Proposed
•Results & Comparison
•Conclusion
48

49. Conclusion
• 本論文提出之HMP能有效改良MP[1]的表格
面積。
• 本論文提出之Lossless ROM Compression不僅
有效降低表格面積，且delay增加得很少。
• 本論文一併提出的整合誤差(Combined Error)
與窮舉搜尋(Exhaustive Search)能加速到有效
時間內完成，相比過去有很大的進展。
• 未來展望：希望能將這些方法，拓展到更高
精確度上。(i.e.,32 bits)
49

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