Table-Based Logarithm Computation for Correct Rounding and Low Latency
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Solution Overview
Problem
Existing methods for computing logarithms of numbers are inefficient in terms of computing resources and time, particularly in computing logarithms for numbers, as they fail to efficiently compute logarithm values correctly and efficiently, especially in terms of throughput and latency.
Innovation Solution
The method involves receiving a number and computing its exponent and significand, selecting a breakpoint value from a range of significand values, computing a multiplication of the exponent and a logarithm value of two, and summing the logarithm values of the significand and the breakpoint value to compute the logarithm of the number.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If traditional methods are used to compute logarithm values of significand, then accuracy can be maintained, but computing time and resource utilization increase significantly
Solution Approach 1:
The patent precomputes logarithm values for breakpoint values (powers of 1.5) and stores them in lookup tables before runtime. During actual computation, the system retrieves these precomputed values instead of calculating them, significantly reducing computing time while maintaining accuracy through the use of precalculated precise values.
Solution Approach 2:
The patent divides the significand range into multiple segments using breakpoint values (1.5^0, 1.5^1, 1.5^2, etc.). By selecting the appropriate breakpoint based on the significand's magnitude and computing the logarithm as a sum of precomputed breakpoint logarithms and a remainder term, the system achieves efficient calculation without sacrificing precision.
2Productivity
If computational resources are increased to improve logarithm computation speed, then throughput increases, but device complexity and resource utilization increase
Solution Approach 1:
The patent moves computational work to preprocessing time by calculating and storing logarithm values of breakpoint values in lookup tables. The processing circuit only needs to perform simple operations like table lookup, addition, and a final adjustment calculation, dramatically reducing the computational resources needed during runtime while maintaining high throughput.
Solution Approach 2:
The patent creates lookup tables that store copies of precomputed logarithm values for breakpoint values. Instead of requiring complex real-time computation, the system retrieves these copied values from memory, reducing the processing circuit's computational burden and complexity while enabling faster throughput.
3Loss of time
If approximation methods are used to reduce computing time, then speed improves, but accuracy and precision of logarithm values may deteriorate
Solution Approach 1:
The patent segments the logarithm computation into two parts: a sum of precomputed logarithm values of breakpoint values (which are exact) and a small remainder term that accounts for the difference between the actual significand and the nearest breakpoint. This segmentation allows the use of precise precomputed values while handling the approximation needs only for a small correction term.
Solution Approach 2:
The patent transforms the original logarithm computation problem into an equivalent form using the identity log2(x) = log2(1.5^k * (x/1.5^k)) = k*log2(1.5) + log2(1 + δ), where δ is a small remainder. This parameter transformation allows the use of precomputed values for the dominant term while the small remainder can be handled with minimal computational effort and high precision.
Data Source
AI summary
A method of computing logarithms, comprising receiving a number, computing an exponent and significand of the received number, selecting a breakpoint value from a plurality of breakpoint values segmenting a range of the significand wherein the selected breakpoint value is the significand's greatest lower bound or lowest upper bound, computing a multiplication of the exponent and a logarithm value of two, computing a first intermediate value based on a least significant portion of the significand and an inverse value of the selected breakpoint value, computing an approximated logarithm value of a second intermediate value derived from the first intermediate value, computing a logarithm value of the significand by summing the approximated logarithm value and a logarithm value of the selected breakpoint value, computing a logarithm value of the received number by summing the logarithm value of the significand and the multiplication of the exponent and the logarithm value of two.