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author Robert McIntyre <rlm@mit.edu>
date Fri, 28 Mar 2014 21:05:12 -0400
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rlm@421 1 %% This is an example first chapter. You should put chapter/appendix that you
rlm@421 2 %% write into a separate file, and add a line \include{yourfilename} to
rlm@421 3 %% main.tex, where `yourfilename.tex' is the name of the chapter/appendix file.
rlm@421 4 %% You can process specific files by typing their names in at the
rlm@421 5 %% \files=
rlm@421 6 %% prompt when you run the file main.tex through LaTeX.
rlm@421 7 \chapter{Introduction}
rlm@421 8
rlm@421 9 Micro-optimization is a technique to reduce the overall operation count of
rlm@421 10 floating point operations. In a standard floating point unit, floating
rlm@421 11 point operations are fairly high level, such as ``multiply'' and ``add'';
rlm@421 12 in a micro floating point unit ($\mu$FPU), these have been broken down into
rlm@421 13 their constituent low-level floating point operations on the mantissas and
rlm@421 14 exponents of the floating point numbers.
rlm@421 15
rlm@421 16 Chapter two describes the architecture of the $\mu$FPU unit, and the
rlm@421 17 motivations for the design decisions made.
rlm@421 18
rlm@421 19 Chapter three describes the design of the compiler, as well as how the
rlm@421 20 optimizations discussed in section~\ref{ch1:opts} were implemented.
rlm@421 21
rlm@421 22 Chapter four describes the purpose of test code that was compiled, and which
rlm@421 23 statistics were gathered by running it through the simulator. The purpose
rlm@421 24 is to measure what effect the micro-optimizations had, compared to
rlm@421 25 unoptimized code. Possible future expansions to the project are also
rlm@421 26 discussed.
rlm@421 27
rlm@421 28 \section{Motivations for micro-optimization}
rlm@421 29
rlm@421 30 The idea of micro-optimization is motivated by the recent trends in computer
rlm@421 31 architecture towards low-level parallelism and small, pipelineable
rlm@421 32 instruction sets \cite{patterson:risc,rad83}. By getting rid of more
rlm@421 33 complex instructions and concentrating on optimizing frequently used
rlm@421 34 instructions, substantial increases in performance were realized.
rlm@421 35
rlm@421 36 Another important motivation was the trend towards placing more of the
rlm@421 37 burden of performance on the compiler. Many of the new architectures depend
rlm@421 38 on an intelligent, optimizing compiler in order to realize anywhere near
rlm@421 39 their peak performance
rlm@421 40 \cite{ellis:bulldog,pet87,coutant:precision-compilers}. In these cases, the
rlm@421 41 compiler not only is responsible for faithfully generating native code to
rlm@421 42 match the source language, but also must be aware of instruction latencies,
rlm@421 43 delayed branches, pipeline stages, and a multitude of other factors in order
rlm@421 44 to generate fast code \cite{gib86}.
rlm@421 45
rlm@421 46 Taking these ideas one step further, it seems that the floating point
rlm@421 47 operations that are normally single, large instructions can be further broken
rlm@421 48 down into smaller, simpler, faster instructions, with more control in the
rlm@421 49 compiler and less in the hardware. This is the idea behind a
rlm@421 50 micro-optimizing FPU; break the floating point instructions down into their
rlm@421 51 basic components and use a small, fast implementation, with a large part of
rlm@421 52 the burden of hardware allocation and optimization shifted towards
rlm@421 53 compile-time.
rlm@421 54
rlm@421 55 Along with the hardware speedups possible by using a $\mu$FPU, there are
rlm@421 56 also optimizations that the compiler can perform on the code that is
rlm@421 57 generated. In a normal sequence of floating point operations, there are
rlm@421 58 many hidden redundancies that can be eliminated by allowing the compiler to
rlm@421 59 control the floating point operations down to their lowest level. These
rlm@421 60 optimizations are described in detail in section~\ref{ch1:opts}.
rlm@421 61
rlm@421 62 \section{Description of micro-optimization}\label{ch1:opts}
rlm@421 63
rlm@421 64 In order to perform a sequence of floating point operations, a normal FPU
rlm@421 65 performs many redundant internal shifts and normalizations in the process of
rlm@421 66 performing a sequence of operations. However, if a compiler can
rlm@421 67 decompose the floating point operations it needs down to the lowest level,
rlm@421 68 it then can optimize away many of these redundant operations.
rlm@421 69
rlm@421 70 If there is some additional hardware support specifically for
rlm@421 71 micro-optimization, there are additional optimizations that can be
rlm@421 72 performed. This hardware support entails extra ``guard bits'' on the
rlm@421 73 standard floating point formats, to allow several unnormalized operations to
rlm@421 74 be performed in a row without the loss information\footnote{A description of
rlm@421 75 the floating point format used is shown in figures~\ref{exponent-format}
rlm@421 76 and~\ref{mantissa-format}.}. A discussion of the mathematics behind
rlm@421 77 unnormalized arithmetic is in appendix~\ref{unnorm-math}.
rlm@421 78
rlm@421 79 The optimizations that the compiler can perform fall into several categories:
rlm@421 80
rlm@421 81 \subsection{Post Multiply Normalization}
rlm@421 82
rlm@421 83 When more than two multiplications are performed in a row, the intermediate
rlm@421 84 normalization of the results between multiplications can be eliminated.
rlm@421 85 This is because with each multiplication, the mantissa can become
rlm@421 86 denormalized by at most one bit. If there are guard bits on the mantissas
rlm@421 87 to prevent bits from ``falling off'' the end during multiplications, the
rlm@421 88 normalization can be postponed until after a sequence of several
rlm@421 89 multiplies\footnote{Using unnormalized numbers for math is not a new idea; a
rlm@421 90 good example of it is the Control Data CDC 6600, designed by Seymour Cray.
rlm@421 91 \cite{thornton:cdc6600} The CDC 6600 had all of its instructions performing
rlm@421 92 unnormalized arithmetic, with a separate {\tt NORMALIZE} instruction.}.
rlm@421 93
rlm@421 94 % This is an example of how you would use tgrind to include an example
rlm@421 95 % of source code; it is commented out in this template since the code
rlm@421 96 % example file does not exist. To use it, you need to remove the '%' on the
rlm@421 97 % beginning of the line, and insert your own information in the call.
rlm@421 98 %
rlm@421 99 %\tagrind[htbp]{code/pmn.s.tex}{Post Multiply Normalization}{opt:pmn}
rlm@421 100
rlm@421 101 As you can see, the intermediate results can be multiplied together, with no
rlm@421 102 need for intermediate normalizations due to the guard bit. It is only at
rlm@421 103 the end of the operation that the normalization must be performed, in order
rlm@421 104 to get it into a format suitable for storing in memory\footnote{Note that
rlm@421 105 for purposed of clarity, the pipeline delays were considered to be 0, and
rlm@421 106 the branches were not delayed.}.
rlm@421 107
rlm@421 108 \subsection{Block Exponent}
rlm@421 109
rlm@421 110 In a unoptimized sequence of additions, the sequence of operations is as
rlm@421 111 follows for each pair of numbers ($m_1$,$e_1$) and ($m_2$,$e_2$).
rlm@421 112 \begin{enumerate}
rlm@421 113 \item Compare $e_1$ and $e_2$.
rlm@421 114 \item Shift the mantissa associated with the smaller exponent $|e_1-e_2|$
rlm@421 115 places to the right.
rlm@421 116 \item Add $m_1$ and $m_2$.
rlm@421 117 \item Find the first one in the resulting mantissa.
rlm@421 118 \item Shift the resulting mantissa so that normalized
rlm@421 119 \item Adjust the exponent accordingly.
rlm@421 120 \end{enumerate}
rlm@421 121
rlm@421 122 Out of 6 steps, only one is the actual addition, and the rest are involved
rlm@421 123 in aligning the mantissas prior to the add, and then normalizing the result
rlm@421 124 afterward. In the block exponent optimization, the largest mantissa is
rlm@421 125 found to start with, and all the mantissa's shifted before any additions
rlm@421 126 take place. Once the mantissas have been shifted, the additions can take
rlm@421 127 place one after another\footnote{This requires that for n consecutive
rlm@421 128 additions, there are $\log_{2}n$ high guard bits to prevent overflow. In
rlm@421 129 the $\mu$FPU, there are 3 guard bits, making up to 8 consecutive additions
rlm@421 130 possible.}. An example of the Block Exponent optimization on the expression
rlm@421 131 X = A + B + C is given in figure~\ref{opt:be}.
rlm@421 132
rlm@421 133 % This is an example of how you would use tgrind to include an example
rlm@421 134 % of source code; it is commented out in this template since the code
rlm@421 135 % example file does not exist. To use it, you need to remove the '%' on the
rlm@421 136 % beginning of the line, and insert your own information in the call.
rlm@421 137 %
rlm@421 138 %\tgrind[htbp]{code/be.s.tex}{Block Exponent}{opt:be}
rlm@421 139
rlm@421 140 \section{Integer optimizations}
rlm@421 141
rlm@421 142 As well as the floating point optimizations described above, there are
rlm@421 143 also integer optimizations that can be used in the $\mu$FPU. In concert
rlm@421 144 with the floating point optimizations, these can provide a significant
rlm@421 145 speedup.
rlm@421 146
rlm@421 147 \subsection{Conversion to fixed point}
rlm@421 148
rlm@421 149 Integer operations are much faster than floating point operations; if it is
rlm@421 150 possible to replace floating point operations with fixed point operations,
rlm@421 151 this would provide a significant increase in speed.
rlm@421 152
rlm@421 153 This conversion can either take place automatically or or based on a
rlm@421 154 specific request from the programmer. To do this automatically, the
rlm@421 155 compiler must either be very smart, or play fast and loose with the accuracy
rlm@421 156 and precision of the programmer's variables. To be ``smart'', the computer
rlm@421 157 must track the ranges of all the floating point variables through the
rlm@421 158 program, and then see if there are any potential candidates for conversion
rlm@421 159 to floating point. This technique is discussed further in
rlm@421 160 section~\ref{range-tracking}, where it was implemented.
rlm@421 161
rlm@421 162 The other way to do this is to rely on specific hints from the programmer
rlm@421 163 that a certain value will only assume a specific range, and that only a
rlm@421 164 specific precision is desired. This is somewhat more taxing on the
rlm@421 165 programmer, in that he has to know the ranges that his values will take at
rlm@421 166 declaration time (something normally abstracted away), but it does provide
rlm@421 167 the opportunity for fine-tuning already working code.
rlm@421 168
rlm@421 169 Potential applications of this would be simulation programs, where the
rlm@421 170 variable represents some physical quantity; the constraints of the physical
rlm@421 171 system may provide bounds on the range the variable can take.
rlm@421 172 \subsection{Small Constant Multiplications}
rlm@421 173
rlm@421 174 One other class of optimizations that can be done is to replace
rlm@421 175 multiplications by small integer constants into some combination of
rlm@421 176 additions and shifts. Addition and shifting can be significantly faster
rlm@421 177 than multiplication. This is done by using some combination of
rlm@421 178 \begin{eqnarray*}
rlm@421 179 a_i & = & a_j + a_k \\
rlm@421 180 a_i & = & 2a_j + a_k \\
rlm@421 181 a_i & = & 4a_j + a_k \\
rlm@421 182 a_i & = & 8a_j + a_k \\
rlm@421 183 a_i & = & a_j - a_k \\
rlm@421 184 a_i & = & a_j \ll m \mbox{shift}
rlm@421 185 \end{eqnarray*}
rlm@421 186 instead of the multiplication. For example, to multiply $s$ by 10 and store
rlm@421 187 the result in $r$, you could use:
rlm@421 188 \begin{eqnarray*}
rlm@421 189 r & = & 4s + s\\
rlm@421 190 r & = & r + r
rlm@421 191 \end{eqnarray*}
rlm@421 192 Or by 59:
rlm@421 193 \begin{eqnarray*}
rlm@421 194 t & = & 2s + s \\
rlm@421 195 r & = & 2t + s \\
rlm@421 196 r & = & 8r + t
rlm@421 197 \end{eqnarray*}
rlm@421 198 Similar combinations can be found for almost all of the smaller
rlm@421 199 integers\footnote{This optimization is only an ``optimization'', of course,
rlm@421 200 when the amount of time spent on the shifts and adds is less than the time
rlm@421 201 that would be spent doing the multiplication. Since the time costs of these
rlm@421 202 operations are known to the compiler in order for it to do scheduling, it is
rlm@421 203 easy for the compiler to determine when this optimization is worth using.}.
rlm@421 204 \cite{magenheimer:precision}
rlm@421 205
rlm@421 206 \section{Other optimizations}
rlm@421 207
rlm@421 208 \subsection{Low-level parallelism}
rlm@421 209
rlm@421 210 The current trend is towards duplicating hardware at the lowest level to
rlm@421 211 provide parallelism\footnote{This can been seen in the i860; floating point
rlm@421 212 additions and multiplications can proceed at the same time, and the RISC
rlm@421 213 core be moving data in and out of the floating point registers and providing
rlm@421 214 flow control at the same time the floating point units are active. \cite{byte:i860}}
rlm@421 215
rlm@421 216 Conceptually, it is easy to take advantage to low-level parallelism in the
rlm@421 217 instruction stream by simply adding more functional units to the $\mu$FPU,
rlm@421 218 widening the instruction word to control them, and then scheduling as many
rlm@421 219 operations to take place at one time as possible.
rlm@421 220
rlm@421 221 However, simply adding more functional units can only be done so many times;
rlm@421 222 there is only a limited amount of parallelism directly available in the
rlm@421 223 instruction stream, and without it, much of the extra resources will go to
rlm@421 224 waste. One process used to make more instructions potentially schedulable
rlm@421 225 at any given time is ``trace scheduling''. This technique originated in the
rlm@421 226 Bulldog compiler for the original VLIW machine, the ELI-512.
rlm@421 227 \cite{ellis:bulldog,colwell:vliw} In trace scheduling, code can be
rlm@421 228 scheduled through many basic blocks at one time, following a single
rlm@421 229 potential ``trace'' of program execution. In this way, instructions that
rlm@421 230 {\em might\/} be executed depending on a conditional branch further down in
rlm@421 231 the instruction stream are scheduled, allowing an increase in the potential
rlm@421 232 parallelism. To account for the cases where the expected branch wasn't
rlm@421 233 taken, correction code is inserted after the branches to undo the effects of
rlm@421 234 any prematurely executed instructions.
rlm@421 235
rlm@421 236 \subsection{Pipeline optimizations}
rlm@421 237
rlm@421 238 In addition to having operations going on in parallel across functional
rlm@421 239 units, it is also typical to have several operations in various stages of
rlm@421 240 completion in each unit. This pipelining allows the throughput of the
rlm@421 241 functional units to be increased, with no increase in latency.
rlm@421 242
rlm@421 243 There are several ways pipelined operations can be optimized. On the
rlm@421 244 hardware side, support can be added to allow data to be recirculated back
rlm@421 245 into the beginning of the pipeline from the end, saving a trip through the
rlm@421 246 registers. On the software side, the compiler can utilize several tricks to
rlm@421 247 try to fill up as many of the pipeline delay slots as possible, as
rlm@421 248 seendescribed by Gibbons. \cite{gib86}
rlm@421 249
rlm@421 250