隨著現代系統中浮點數運算應用的快速成長，使得浮點數運算單元已經成為這些系統主要的能量消耗來源。幸運地是許多浮點數應用可以忍受些許輸出資料的失真，而這些失真是人類感官可以忽略或是接受的。換句話說，我們可以利用多重模式浮點數運算單元，藉著調降浮點數運算單元的精確度(低於IEEE單精度浮點數指令)，犧牲整體應用輸出資料的準確度，以換取降低能量消耗。所以，如何在可容許的確實性損失下，能夠快速、有效率地指定每個浮點數指令一個合適的精確度模式並且達到最低能量消耗，已經成為非常重要的議題(稱為精確度指定問題，即PAP)。由於針對低能量、高效能或其他特殊目的，有些運算可以轉換成不同的指令，而如何選擇指令並且在多重模式運算單元上執行上述運算，則是一個關鍵的問題(我們稱為指令轉換問題，即ITP)。此外，指令排程問題(即ISP)對於追求高效能的系統而言，亦是非常重要的。因此，為了有效解決上述三個問題，本論文提出一個低能量指令精確度指定系統，其中包括多重模式浮點數運算單元的硬體實作以及誤差分析和指令精確度指定方法的軟體發展兩方面。首先，我們將介紹多重模式浮點數運算單元的設計與特色。我們利用堆疊和截斷技術實現多重精確度模式設計，其所有精確度模式皆可隨不同指令的需求動態調整，以達到降低更多能量消耗的目標。為了有效使用多重模式浮點數運算單元，並且確保應用程式的輸出資料確實性限制可以滿足，我們利用仿射算術(AA)建立一個區間分析之浮點數誤差模式，以便指出每一個浮點數指令的精確度和浮點數應用程式輸出資料確實性之間的關係。接著，我們將誤差模式所產生的輸出資料仿射算術格式儲存在確實性檢查函式，以便在PAP和ITP問題下，執行確實性限制檢查。此外，本論文採用簡化指令排程方法和應用程式的資料非循環圖(DAG)建立效能檢查函式，用來檢查是否滿足ISP的效能限制。基於多重模式浮點數運算單元的資訊和上述兩個檢查函式，我們發展出一個指令精確度指定方法，此方法結合了快速貪婪方法以及我們所改良的快速塔布搜尋(禁忌搜尋)演算法，能在應用程式的確實性和效能限制下，透過快速指定每個浮點指令的精確度模式並且重新排程所有指令，以同時解決PAP、ITP和ISP三個問題。從實際應用程式和人工隨機例子的實驗結果顯示，我們所提出的方法可以在有限的時間限制之下，找到比其他方法節省更多能量消耗的精確度指定解。

With the rapid growth in applying floating-point (FP) arithmetic to the modem systems, FP arithmetic units have become the main energy consumers in these systems. Fortunately, many FP applications allow a slight output distortion that human senses can neglect or tolerate. In other words, we can trade the energy consumption with output quality of FP applications by reducing the precision of FP instructions (less accurate than IEEE single-precision FP one) via multiple-mode FP arithmetic units. However, how to quickly and effectively assign each FP instruction to a suitable precision mode of these multiple-mode FP arithmetic units for maximizing the energy saving under acceptable accuracy constraints is an essential problem (called precision assignment problem, PAP). Because some operations can be transformed to different instructions for low energy, high performance or other special purposes, it is a critical problem that determines which instructions will be chosen to perform above operations in various multiple-mode FP arithmetic units (called instruction transform problem, ITP). Moreover, instruction scheduling problem (denoted to ISP) is also very important for many high performance systems. Thus, this dissertation proposes a low energy instruction precision assignment system that includes the hardware implementation of multiple-mode FP arithmetic units and the software development of error analysis model and instruction precision assignment methods for efficiently solving PAP, ITP and ISP. Firstly, we introduce the design and characteristics of our multiple-mode FP arithmetic units which utilize the iterative and truncated techniques to support multiple-modes with various errors and energy consumption. All precision modes of above arithmetic units can be dynamically changed when they perform different FP instructions to reduce more energy consumption. In order to effectively utilize above-mentioned multiple-mode FP arithmetic units and ensure that the accuracy constraints of application are satisfied, affine arithmetic (AA) is modified to build a FP error model in interval analysis that indicates the relationship between the accuracies of each FP instruction and the output data of the given FP applications. Afterward, we store the AA form of output data generated by above FP error model in accuracy check function for checking accuracy constraints in PAP and ITP. In addition, a simplified instruction scheduling and the DAG of application are used to build performance check function for checking performance constraints in ISP. Based on the information of multiple -mode arithmetic units and above two check functions, our proposed precision assignment method that integrates a fast greedy method with our modified fast Tabu search (TS) algorithm is then developed to quickly solve PAP, ITP and ISP by assigning the precision modes of each FP instruction and re-scheduling all instructions under the accuracy and performance constraints on the given application. Experimental results for real applications and artificial random cases show that our proposed method can efficiently find a precision assignment solution and the most energy saving within acceptable time when compared to previous methods on average.

Contents
CHAPTER 1. Introduction 1
CHAPTER 2. Related Works 5
2.1 Multiple-mode Floating-Point Arithmetic Unit 5
2.1.1 Multiplier 5
2.1.2 Multiply-add Fused Unit (MAF) 8
2.1.3 Special Function Interpolator (SFI) 10
2.2 Error Model 12
2.3 Precision Assignment Method 14
CHAPTER 3. Proposed Multiple-mode Floating-point Arithmetic Units 16
3.1 Multiple-mode Floating-point Multiplier 17
3.1.1 Floating-point Multiplication 17
3.1.2 Iterative Multiplication 21
3.1.3 Iterative and Truncated Multiplication 25
3.1.4 Error Analysis 33
3.2 Multiple-mode Floating-Point Multiply-add Fused Unit 45
3.2.1 Iterative Multiplication and Truncated Addition 47
3.2.2 The Operations of MMAF 50
3.2.3 Error Analysis 53
3.3 Multiple-mode Floating-point Function Interpolator 57
3.3.1 Fundamental Function Interpolator (FFI) 57
3.3.2 Row-column-based Multiple-mode Mechanism (RCBMM) 61
CHAPTER 4. Modified Affine Arithmetic Based Error Model 65
4.1 Interval Arithmetic (IA) 66
4.2 Affine Arithmetic (AA) 66
4.3 Proposed Affine Arithmetic Based Error Model 70
4.3.1 Multiplication (MUL) 72
4.3.2 Addition (ADD) 74
4.3.3 Multiply and Accumulate Operation (MAC) 74
4.3.4 Dot Product 3 (DP3) and Dot Product 4 (DP4) 75
4.3.5 Reciprocal operation (REC) 76
4.3.6 Reciprocal square root operation (RSQ) 81
4.3.7 Logarithm operation (LG2) 82
4.3.8 Exponential operation (EX2) 84
4.3.9 Accuracy Constraint 85
CHAPTER 5. Proposed Precision Assignment Method 90
5.1 PAP, ITP and ISP 90
5.2 Our Proposed Precision Assignment Method 96
5.3 Fast Greedy Method 96
5.4 Our Modified Fast Tabu Search 97
5.4.1 Initial Solution 99
5.4.2 Neighbor Structure 99
5.4.3 Fast Tabu List 100
5.4.4 Tabu Length Adjust Mechanism 103
5.4.5 Branch and Bound Strategies 104
5.4.6 Aspiration Rule 106
5.4.7 Diversification Strategies 106
5.4.8 Stop Rule 107
CHAPTER 6. Experimental Results 108
6.1 Experimental Environment 108
6.2 Compared Methods 109
6.3 The Results and Discussion of Real Applications 111
6.4 The Results and Discussion of Artificial Cases 118
CHAPTER 7. Conclusions 121
CHAPTER 8. Future Work 122
References 125
Publication List 136

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