Accurate table methods allow for very accurate and efficient evaluation of elementary functions. We present new single-table approaches to logarithm and exponential evaluation, by which we mean that a single table of values works for both log(
x
) and log(1 +
x
), and a single table for
e
x
and
e
x
− 1. This approach eliminates special cases normally required to evaluate log(1 +
x
) and
e
x
− 1 accurately near zero, which will significantly improve performance on architectures which use SIMD parallelism, or on which data-dependent branching is expensive.
We have implemented it on the Cell/B.E. SPU (SIMD compute engine) and found the resulting functions to be up to twice as fast as the conventional implementations distributed in the IBM Mathematical Acceleration Subsystem (MASS). We include the literate code used to generate all the variants of exponential and log functions in the article, and discuss relevant language and hardware features.
We propose a Domain-Specific Architecture for elementary function computation to improve throughput while reducing power consumption as a model for more general applications: support fine-grained parallelism by eliminating branches, and eliminate the duplication required by coprocessors by decomposing computation into instructions which fit existing pipelined execution models and standard register files. Our example instruction architecture (ISA) extension supports scalar and vector/SIMD implementations of table-based methods of calculating all common special functions, with the aim of improving throughput by (1) eliminating the need for tables in memory, (2) eliminating all branches for special cases, and (3) reducing the total number of instructions. Two new instructions are required, a table lookup instruction and an extended-precision floating-point multiply-add instruction with special treatment for exceptional inputs. To estimate the performance impact of these instructions, we implemented them in a modified Cell/B.E. SPU simulator and observed an average throughput improvement of 2.5 times for optimized loops mapping single functions over long vectors.
A simple experiment is described that uses a CCD video camera, a display monitor and a laser- printed bar pattern. The imaging process is used as an example to illustrate the concepts involved in the correct sampling of an analogue signal. The analysis uses basic results from optics such as the lens equation and magnification, as well as more advanced topics from Fourier theory such as convolution and the sampling theorem. The experiment is suitable for use in undergraduate courses in imaging, signal processing, engineering and physics.
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