Piecewise Linear Log Circuits for Single-Cycle Hardware Math

The patent divides the logarithmic computation domain into multiple segments or intervals. Each segment is approximated using a simple linear function with pre-computed coefficients. This segmentation allows the complex logarithmic function to be broken down into multiple simple linear segments, achieving both accuracy (by using enough segments) and speed (by using simple linear calculations instead of complex LUT/multiplier operations).

The patent transforms the computational parameters by changing from direct logarithmic computation to piecewise linear approximation. By representing the logarithm function as a series of linear segments with different slopes and intercepts, the computation shifts from complex multiplication-based operations to simpler addition and subtraction operations, dramatically improving processing speed while maintaining acceptable accuracy.

The patent extracts only the essential characteristics of the logarithmic function needed for accurate computation, which are the linear approximation coefficients for each segment. Instead of storing complete LUT entries for all possible input values, the method extracts and stores only the minimal set of coefficients (slopes and intercepts) required to reconstruct the logarithmic values through linear equations, dramatically reducing memory requirements.

The patent replaces expensive, large LUT structures with cheaper, compact linear coefficient representations. The linear coefficients require minimal storage space compared to full LUT entries, and the computation using these coefficients is much faster and uses less hardware resources, effectively using a 'cheaper' approximation method that achieves the same functional goal.