XLNSCPP: a C++ package for Logarithmic Number System eXperimentation
This repository provides xlns16.cpp and xlns32.cpp along with a few programs that illustrate their use. They are based on similar math foundation (Gaussian logs, sb and db) as the Python xlns repository, but unlike the Python they use different internal storage format.
Unlike the Python xlns, here internal representation is not twos complement; it is offset by a constant (xlns16_logsignmask or xlns32_logsignmask). With the 16-bit format of xlns16.cpp, this is roughly similar to bfloat16 (1 sign bit, 8 int(log2) bits, 7 frac(log2) bits). With the 32-bit format of xlns32.cpp, this is roughly similar to float (1 sign bit, 8 int(log2) bits, 23 frac(log2) bits). There is an exact representation of 0.0, but no subnormals or NaNs.
There are two ways to use this library: function calls (like xlns16_add or xlns32_add) that operate on integer representations (typedef as xlns_16 or xlns_32) that represent the LNS value; or C++ overloaded operators that operate on an LNS class (either xlns16_float or xlns32_float). The functions are a bit faster but overloading is easier.
All of the global symbols used begin with either xlns16 and xlns32. There are several compile-time options indicated by defining macros before including xlns16.cpp and xlns32.cpp in the main program. Defining xlns16_ideal or xlns32_ideal causes the Gaussian Log computation to occur as accurately as possible by doing it in floating point. Omitting this gives a cotransformation/interpolation approximation for 32-bit (at a cost of a few 100K bytes) and a LPVIP approximation for 16-bit (when xlns16_altopt is also defined, a less accurate LPVIP algorithm is used). Defining xlns16_alt or xlns_32alt uses an addition algorithm that reduces branching (in the sometimes false hope of improved performance on modern architectures). Omitting this defaults to an equivalent algorithm that runs better on older architectures (like say 8086). Defining xlns16_table causes conversion to/from float to occur from tables. When xlns16_alt is also defined, Gaussian Log computation comes from tables. Both of these improve speed at the cost of less than one megabyte. The file xlns16testcase.h itemizes the meaningful combinations of these options to help automate the regression testing of these options.
The Python and C++ code that begin with sb and db work together to test whether ideal and LPVIP Gaussian Log computations in xlns16.cpp match what the Python xlns library provides.
There is a sister repository, xlnscuda, which has related routines that work on CUDA devices.
M. G. Arnold, et al. “Arithmetic cotransformations in the Real and Complex Logarithmic Number Systems,” IEEE Trans. Comput., vol. 47, no. 7, pp.777–786, July 1998.
M. G. Arnold, "LPVIP: A Low-power ROM-Less ALU for Low-Precision LNS," 14th International Workshop on Power and Timing Modeling, Optimization and Simulation, LNCS 3254, pp.675-684, Santorini, Greece, Sept. 2004.