Communications - April 2009

from 1.0 Flops/Byte in the Opteron X2 to 4. 4 in the Opteron X4. Hence, to realize a performance gain using the X4, kernels need an operational intensity greater than 1.0 Flops/Byte.

figure 2: Roofline model with ceilings for opteron X2.

(a) Computational Ceilings

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Adding ceilings to the model

The Roofline model provides an upper bound to performance. Suppose a program performs far below its Roofline. What optimizations should one implement and in what order? Another advantage of bound-and-bottleneck analysis is that “a number of alternatives can be treated together, with a single bounding analysis providing useful information about them all.” ²⁰

We leverage this insight to add multiple ceilings to the Roofline model to guide which optimizations to implement. It is similar to the guidelines loop balance gives the compiler. We can think of each optimization as a “performance ceiling” below the appropriate Roofline, meaning you cannot break through a ceiling without first performing the associated optimization.

For example, to reduce computational bottlenecks on the Opteron X2, almost any kernel can be helped with two optimizations:

Improve instruction-level parallelism (ILP) and apply SIMD. For superscalar architectures, the highest performance comes when fetching, executing, and committing the maximum number of instructions per clock cycle. The goal is to improve the code from the compiler to increase ILP. The highest performance comes from completely covering the functional unit latency. One way to hide instruction latency is by unrolling loops. For x86-based architectures, another way is using floating-point SIMD instructions whenever possible, since a SIMD instruction operates on pairs of adjacent operands; and

Balance floating-point operation mix. The best performance requires that a significant fraction of the instruction mix be floating-point operations (discussed later). Peak floating-point performance typically also requires an equal number of simultaneous floating-point additions and multiplications, since many computers have multiply-add instructions or an equal number of adders and multipliers.