### Archive

Posts Tagged ‘optimize’

## Unique values generated by expressions of a given complexity

The majority of integer constants appearing in source code can be represented using a few bits. CPU designers use this characteristic when designing instruction sets, creating so called short-form or quick instructions that perform some operation involving small integer values, e.g., adding a value between 1 and 8 to a register. Writers of code optimizers are always looking for sequences of short-form instructions that are faster/smaller than the longer forms (the INMOS Transputer only had a short form for load immediate).

I have recently been looking at optimizing expressions written for a virtual machine that only supports immediate loads of decimal values between 1 an 9, and binary add/subtract/multiply/divide, e.g., optimizing an expression containing four operators, `((2*7)+9)*4+9` which evaluates to 101, to one containing three operators, `(8+9)*6-1` also evaluates to 101. Intermediate results can have fractional values, but I am only interested in expressions whose final result has an integer value (i.e., zero fractional part).

A little thought shows that the value of an expression containing a subexpression whose value includes a fractional value (e.g., 1/3) can always be generated by an expression containing the same or a fewer number of operators and no intermediate fractional value intermediate results (e.g., 9/(1/5) can be generated using 9*5, i.e., the result of any divide operation always has to be an integer if the final result is to be a unique integer. Enumerating the unique set of values generated by expressions containing a given number of operators shows that divide is redundant for expressions containing six of fewer operators and only adds 11 unique values for seven operators (379,073 possibilities without divide)

Removing support for the minus operator only reduces the size of the result set by around 10%. Possibly being worthwhile time saving for expressions containing many operators or searching for an expression whose result value is very large.

There does not appear to be a straightforward (and fast) algorithm that returns the minimal operation expression for a given constant.

I wrote an R program to exhaustively generate all integer values returned by expressions containing up to seven operators. To find out how many different values, integer/real, could be calculated I wrote a maxima program (this represents fractional values using a rational number representation and exceeds 4G byte of storage for expressions containing more than five operators).

The following figure shows the number of different values that can be generated by an expression containing a given number of operators (blue), the number of integer values (black), the number of positive integer values (red), the smallest positive integer that cannot be calculated by an expression containing the given number of operators (green) (circles are for add/subtract/multiply/divide, squares for add/multiply). Any value below a green line is guaranteed to have a solution in the in the given number of operators (or fewer). The blue diamond line is the mean value of a random expression containing the given number of operators.

Limiting the operators to just add/multiply reduces the number of unique value possibilities. The difference increases linearly’ish to around 35% for seven operators.

The following uses colors to show the minimum number of operators needed to generate the given value, 1 is in the bottom left, 100,000 in the top right; red for one operator, yellow for two, green for three and so on.

Knowing that `N` can be calculated using `p` operators does not mean that `N-1` can also be generated using `p` operators; it is possible to generate 729 using two operators (i.e., `9*9*9`), three operators are required to generate 92 and four to generate 417.

Values under the green line (first figure) are known to have solutions in the given number of operators; quickly obtaining the solution is another matter. There is at least a 50/50 chance that a randomly generated expression containing the given number of operators, and producing an integer value, will calculate a value on or below the diamond blue line. The overhead of storing precomputed minimal operator expressions is not that great for small numbers of operators.

Suggestions for a fast/low storage algorithm (random generation + modification through a cost function performs quite well) for large integer values welcome.

Update. Values from the first figure have been accepted by the On-Line Encyclopedia of Integer Sequences as entries: A181898, A181957, A181958, A181959 and A181960.

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## Compiling to reduce the impact of soft errors on program output

Optimizing compilers have traditionally made code faster and smaller (sometimes a choice has to be made between faster/larger and slower/smaller). The huge growth in the use of battery power devices has created a new attribute for writers of optimizers to target, finding code sequences that minimise power consumption (I previously listed this as a major growth area in the next decade). Radiation (e.g., from cosmic rays) can cause a memory or processor bit to flip, known as a soft error, and I have recently been reading about how code can be optimized to reduce the probability that soft errors will alter the external behavior of a running program.

The soft error rate is usually quoted in FITs (Failure in Time), with 1 FIT corresponding to 1 error per hours per megabit, or errors per bit-hour. A PC with 4 GB of DRAM (say 1000 FIT/Mb which increases with altitude and is 10 times greater in Denver Colorado) has a MTBF (mean time between failure) of hours, around once every 33 hours. Calculating the FIT for processors is complicated.

Uncorrected soft errors place a limit on the maximum number of computing nodes that can be usefully used by one application. At around 50,000 nodes a system will be spending half its time saving checkpoints and restarting from previous checkpoints after an error occurs.

Why not rely on error correcting memory? Super computers containing terrabytes are built containing error correcting memory, but this does not make the problem go away, it ‘only’ reduces it by around two orders of magnitude. Builders of commodity processors don’t use much error correction circuitry because it would increase costs/power consumption/etc for an increased level of reliability that the commodity market is not interested in; vendors of high-end processors add significant amounts of error correction circuitry.

Most of the compiler research I am aware of involves soft errors occurring on the processor and this topic is discussed below; there has been some work on assigning variables deemed to be critical to a subset of memory that is protected with error correcting hardware. Pointers to other compiler research involving memory soft errors welcome.

A commonly used technique for handling hardware faults is redundancy, usually redundant hardware (e.g., three processors performing the same calculating and a majority vote used to decide which of the outputs to accept). Software only approaches include the compiler generating two or more independent machine code sequences for each source code sequence whose computed values are compared at various check points and running multiple copies of a program in different threads and comparing outputs. The
Shoestring compiler (based on llvm) takes a lightweight approach to redundancy by not duplicating those code sequences that are less affected by register bit flips (e.g., the value obtained from a bitwise AND that extracts 8 bits from a 32-bit register is 75% less likely to deliver an incorrect result than an operation that depends on all 32 bits).

The reliability of single ‘thread’ generated code can be improved by optimizing register lifetimes for this purpose. A value is loaded into a register and sometime later it is used one or more times. A soft error corrupting register contents after the last use of the value it contains has no impact on program execution, the soft error has to occur between the load and last use of the value for it to possibly influence program output. One group of researchers modified a compiler (Trimaran) to order register usage such that the average interval between load and last usage was reduced by 10%, compared to the default behavior.

Developers don’t have to wait for compiler or hardware support, they can improve reliability by using algorithms that are robust in the presence of ‘faulty’ hardware. For instance, the traditional algorithms for two-process mutual exclusion are not fault tolerant; a fault tolerant mutual exclusion algorithm using variables, where a single fault may occur in up to variables is available.

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## Searching for the source line implementing 3n+1

I have been doing some research on the variety of ways that different developers write code to implement the same specification and have been lucky enough to obtain the source code of approximately 6,000 implementations of a problem based on the 3n+1 algorithm. At some point this algorithm requires multiplying a value by three and adding one, e.g., `n=3*n+1;`.

While I expected some variation in the coding of many parts of the algorithm I did not expect to see much variation in the 3n+1 part, perhaps somebody might write `n=n*3+1;`. I was in for a surprise, the following are some of the different implementations I have seen so far:

n = n + n + n + 1 ;
n += n + n + 1;
n = (n << 1) + n + 1; n += (n << 1) + 1; n *= 3; n++; t = (n << 1) ; n = t + n + 1; n = (n << 2) - n + 1; I was already manually annotating the source and it was easy for me to locate the line implementing 3n+1 to annotate it. But what if I wanted to automate the search for the line of code containing this calculation, what tool could I use? Would I have to write down every possible ways in which 3n+1 could be implemented, with/without parenthesis and all possible orderings of operands? I am not aware of any automatic tool that could be told to locate expressions that calculated 3n+1. What is needed is abstract interpretation over short sequences of statements.

I mentioned this search problem over drinks after a talk I gave at the Oxford branch of the ACCU last week and somebody (Huw ???) suggested that perhaps the code generated by gcc would be the same no matter how 3n+1 was implemented. I could see lots of reasons why this would not be the case, but the idea was interesting and worth investigation.

At the default optimization level the generated x86 code is different for different implemenetations, but optimizing at the “-O 3″ level results in all but one of the above expressions generating the same evaluation code:

` leal 1(%rax,%rax,2), %eax`

The exception is `(n << 2) - n + 1` which results in shift/subtract/add. Perhaps I should report this as a bug in gcc :-)

I was surprised that gcc exhibited this characteristic and I plan to carry out more tests to trace out the envelope of this apparent "same generated code for equivalent expressions" behavior of gcc.

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