Large-Radix Segmentation for Decimal Arithmetic Precision

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Solution Overview

Problem

Modern computers struggle to accurately represent and perform calculations with repeating binary fractions due to design limitations, leading to rounding errors and discrepancies between decimal and binary representations, especially in monetary and exchange calculations.

Innovation Solution

Implementing large-radix computer arithmetic by segmenting decimal numbers into larger radix segments, allowing for precise placement of the decimal point and performing numeric operations on these segments, which can include unary, arithmetic, and logic operations, and storing the results in various data structures.

Engineering Contradictions & Design Principles

VSEngineering Contradiction Analysis

1Productivity

If binary number system (radix-2) is used for computer processing, then calculations can be performed rapidly using transistor-based logic, but repeating binary fractions cannot be accurately represented leading to rounding errors

Engineering Contradiction:

Improvecalculation speedVSAvoidrepresentation accuracy

Core Design Contradiction:

ProductivityVSMeasurement precision

Solution Approach 1:

The patent segments decimal numbers into large-radix segments (e.g., radix-1000 or radix-10000) where each segment fits within standard computer data types. This segmentation allows decimal numbers to be processed in chunks while maintaining exact representation, avoiding the rounding errors of binary fractions while still enabling efficient computation through segment-based operations

Inventive Principle:

Principle #1Segmentation

Solution Approach 2:

The patent changes the numerical base parameter from binary (radix-2) to large decimal radix (radix-1000 or radix-10000). This parameter change allows exact representation of decimal fractions while maintaining computational efficiency, as the large-radix segments can be processed using standard integer and floating-point arithmetic operations

Inventive Principle:

Principle #35Parameter changes

2Ease of operation

If small decimal digit approach is used (one-half to one byte per decimal digit), then decimal point placement becomes easier, but calculation speed decreases significantly

Engineering Contradiction:

Improvedecimal point placementVSAvoidcalculation speed

Core Design Contradiction:

Ease of operationVSProductivity

Solution Approach 1:

Instead of processing one decimal digit at a time, the patent segments decimal numbers into large-radix segments (e.g., 3-4 decimal digits per segment). This reduces the number of segments that need to be processed while still maintaining easy decimal point placement, as the decimal point can be positioned between segments based on the known segment radix

Inventive Principle:

Principle #1Segmentation

Solution Approach 2:

The patent uses standard computer data types (integers, single-precision floating point, double-precision floating point) to represent large-radix segments. This allows the same hardware and software infrastructure to handle decimal arithmetic efficiently, combining the ease of decimal operations with the speed of binary processing

Inventive Principle:

Principle #6Universality (Multi-functionality)

3Productivity

If large binary integer approach is used, then calculations can be done rapidly, but it is very difficult to determine decimal point placement

Engineering Contradiction:

Improvecalculation speedVSAvoiddecimal point placement difficulty

Core Design Contradiction:

ProductivityVSDifficulty of detecting and measuring

Solution Approach 1:

The patent segments large binary integers into groups that correspond to large-radix decimal segments. By organizing the binary representation into segments that map to decimal groups (e.g., groups of bits representing radix-1000 segments), the decimal point placement becomes deterministic based on the segment structure rather than requiring complex analysis of the entire large integer

Inventive Principle:

Principle #1Segmentation

Solution Approach 2:

The patent changes from treating the number as a single large binary integer to treating it as a sequence of large-radix segments. This parameter change in how the number is structured and interpreted makes decimal point placement straightforward, as it simply matters which segment boundary the decimal point falls on, not the individual bit patterns

Inventive Principle:

Principle #35Parameter changes

4Ease of operation

If small decimal digit approach is used, then decimal point placement is easier, but storage efficiency is significantly lost

Engineering Contradiction:

Improvedecimal point placementVSAvoidstorage efficiency

Core Design Contradiction:

Ease of operationVSQuantity of substance

Solution Approach 1:

The patent segments decimal numbers into large-radix segments that can be stored efficiently using standard computer data types. Each segment uses the full capacity of its storage representation, avoiding the waste of using multiple small bytes to store what could be stored in fewer, larger units. This maintains storage efficiency while still enabling easy decimal point placement between segments

Inventive Principle:

Principle #1Segmentation

Data Source

PatentUS7644115B2System and methods for large-radix computer processing

Publication Date: 2010.01.05 SAS INSTITUTE INC

AI summary

Systems and methods for performing large-radix numeric operations. A first number may be segmented into large-radix segments, wherein numbers of the segments are generated such that radix of the segment is greater than radix of the first number. As a result, a plurality of disparate processor-based computing systems may be configured to perform various numeric operations on the large-radix segments of the number and output results of a numeric operation as a number whose radix is equal to the radix of the first number.