Vector Math Library (VML) Notes

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Document Number: 310712-003US

This document describes Vector Math Library (VML), which is designed to compute elementary functions on vector arguments. VML is an integral part of the Intel® Math Kernel Library and the VML terminology is used here for simplicity in discussing this group of functions.

VML includes a set of highly optimized implementations of certain computationally expensive core mathematical functions (power, trigonometric, exponential, hyperbolic, etc.) that operate on vectors. VML may significantly improve performance for such applications as nonlinear software, computations of integrals, and many others.

Each vector function from VML (for each data format) can work in two modes: High Accuracy (HA) and Low Accuracy (LA). For many functions, using the LA version improves performance at the cost of accuracy. For some cases, the advantage of relaxing the accuracy improves performance very little so the same function is employed for both versions. Error and special value behavior does not depend either on the accuracy mode (HA versus LA) or on the processor on which the software runs. Accuracy behavior however is processor specific, so results might slightly differ across different processor families and even within a processor family, for example, between some processor models of the family, or between 64-bit and 32-bit libraries. Also results might slightly differ from release to release. Nevertheless these differences are within specified error bounds.

This document refers to a more detailed description of performance and accuracy properties of VML functions, which you can find at the product web page. There are several issues considered (performance, accuracy, special values processing) and two levels of details (brief information for all functions in one table and more detailed information for every function on a separate page).

Performance issues: Performance numbers in the respective tables are shown for so-called "working" intervals arguments. Performance behavior may be different for other intervals. For example, it is quite expensive to compute trigonometric functions on "huge" arguments. Therefore, to obtain needed accuracy, performance is sacrificed. Each function lists the working interval over which performance is measured. The same page contains graphs that demonstrate how the performance behavior depends on vector length. There are two extreme cases: so-called "short" and "long" vectors (logarithmic scale is used to show both cases). For short vectors there are cycle organization and initialization overheads. The cost of such overheads is amortized with increasing vector length, and for vectors longer than a few dozens of elements the performance remains quite flat until the L2 cache size is exceeded with the length of the vector.

Data prefetching with the Intel® Pentium® III processor (explicit data prefetch in software) and Pentium 4 processor (implicit data prefetch in hardware) greatly reduce the out-of-cache problem.

See a comprehensive table for the performance data on all VML functions.

Accuracy issues: The design requirement for the HA functions is less than 1.0 ulp error with all special values being processed correctly. For the LA version some of these requirements are not so strict. For more details see the web-placed accuracy table with ulp errors for all functions.

Special Values processing issues: Special Values are processed in accordance with C9X standard. For full lists of special values, see the corresponding tables for real and complex functions.

For more details on individual functions see the list of VML functions at the product web page.

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