Linear interpolation requires a single multiplication but is significantly less accurate than quadratic interpolation. The latter requires two multiplications. Two novel quadratic interpolation schemes are shown that approximate the functions required by the logarithmic number system (LNS) with more accuracy than linear interpolation using only a single multiplication. One method uses two ROMs to give the accuracy of quadratic interpolation, whilst the other uses one ROM to give four- to six-bits better accuracy than linear interpolation. These techniques save four- to eight-fold on memory compared to linear interpolation for the same accuracy. We illustrate the usefulness of these techniques for serial implementation with a clone of the ARM/sup TM/ microprocessor (known as AWE) that we developed with LNS instructions. We also show a novel technique for decreasing the propagation delay in both linear and quadratic interpolation that stores the logarithm of the derivative of the function in a ROM, rather than the function itself.

The real logarithmic number system, which represents a value with a sign bit and a quantized logarithm, can be generalized to create the complex logarithmic number system, which replaces the sign bit with a quantized angle in a log/polar coordinate system. Although multiplication and related operations are easy in both real and complex systems, addition and subtraction are hard, especially when interpolation is used to implement the system. Both real and complex logarithmic arithmetic benefit from the use of co-transformation, which converts an addition or subtraction from a region where interpolation is expensive to a region where it is easier. Two co-transformations that accomplish this goal are introduced. The first is an approximation based on real analysis of the subtraction logarithm. The second is based on simple algebra that applies for both real and complex values and that works for both addition and subtraction.

The real logarithmic number system, which represents a value with a sign bit and a quantized logarithm, can be generalized to create the complex logarithmic number system, which replaces the sign bit with a quantized angle in a log/polar coordinate system. Although multiplication and related operations are easy in both real and complex systems, addition and subtraction are hard, especially when interpolation is used to implement the system. Both real and complex logarithmic arithmetic benefit from the use of co-transformation, which converts an addition or subtraction from a region where interpolation is expensive to a region where it is easier. Two co-transformations that accomplish this goal are introduced. The first is an approximation based on real analysis of the subtraction logarithm. The second is based on simple algebra that applies for both real and complex values and that works for both addition and subtraction.

Various features found in standard floating point arithmetic (IEEE 754) are examined in light of their appropriateness for sign/logarithm arithmetic. The emphasis is on a 32-b word size comparable to IEEE 754 single precision, although other word sizes are possible. A multilayer sign/logarithm format is considered. The lowest layer, similar to previous implementations, would provide only normalized representations but would not provide representations for zero, denormalized values, infinities, and NaNs. The highest layer would provide most of the features found in IEEE 754, including zeros, denormalized values, infinities, and NaNs. Novel algorithms for implementing logarithmic denormalized arithmetic are presented. Simulation results show that the error characteristics of the proposed logarithmic denormalized arithmetic algorithms are similar to those of the denormalized floating point arithmetic in IEEE 754.>