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    Intel Math CoProcessor Technical Overview

    Contents:

    • INTRODUCTION TO SOFTWARE: DOC. # 3000
    • HOW THE INTEL MATH COPROCESSOR WORKS
    • FLOATING POINT EXPERT
    • THE INTEL MATH COPROCESSOR AND YOUR APPLICATIONS
    • GRAPHICS AND DESKTOP PUBLISHING
    • CHARTS & DIAGRAMS
    • FONT GENERATION
    • POSTSCRIPT EMULATION
    • COMPUTER-AIDED DESIGN
    • SPREADSHEETS AND STATISTICS
    • SPREADSHEETS
    • STATISTICS
    • DATABASES
    • WE MEAN BUSINESS
    
    

    INTRODUCTION TO SOFTWARE: DOC. # 3000

    
       As computer application programs get more sophisticated, they also get
    
       larger and slower.  Even if your application does not involve math
    
       explicitly, it may use math behind the scenes to do its work.  For
    
       example, graphics and font manipulation, spreadsheet calculations, even
    
       chart creation, all involve mathematical operations. 
    
    
    
    

    HOW THE INTEL MATH COPROCESSOR WORKS

    
       Your computer's CPU - Intel's 8086(TM), 8088(TM), 286(TM), i386(TM)
    
       SX(TM), i386(TM) SL, i386(TM) DX or i486(TM) SX microprocessors - can,
    
       of course, do basic integer arithmetic (addition, subtraction,
    
       multiplication, division of whole numbers).  However, mathematical
    
       operations involving very large numbers, floating point numbers (numbers
    
       with decimals), or more complex calculations put a substantial drag on
    
       the CPU.
    
    
    
       The CPU tackles such complex mathematical operations by using software
    
       subroutines.  Depending on the complexity of the operation, one of these
    
       subroutines can take hundreds of times longer than a basic arithmetic
    
       function. 
    
    
    
       The Intel Math CoProcessor (i8087(TM), 287(TM), i387(TM), i487(TM) Math
    
       coprocessor) takes math-intensive operations from the CPU and handles in
    
       one instruction what would take hundreds of instructions on the
    
       general-purpose CPU.  For example, a floating point division takes 24.4
    
       microseconds for an 8086/8087 combination and 2,000 microseconds with
    
       the 8086 alone.
    
    
    
    

    FLOATING POINT EXPERT

    
       Specifically, the Intel Math CoProcessor is an expert at floating point
    
       calculations.  Its instruction set includes 68 numeric functions for
    
       extended precision, floating point, trigonometric, logarithmic, and
    
       exponential functions. 
    
    
    
       While a program is running, the CPU continues to control overall program
    
       execution.  But when it encounters a floating point operation, it
    
       generates an ESCAPE instruction to the math coprocessor which tells it
    
       to perform the calculation.  The math coprocessor operates independently
    
       from the time it receives the instruction to the time it is ready to
    
       pass the result back to the CPU.  While the math coprocessor is working,
    
       the CPU can either be waiting for the result, or processing other tasks.
    
       
    
    
    
       In addition to performing many calculations considerably faster than the
    
       CPU, the Intel Math CoProcessor can often provide much more accurate
    
       answers than software subroutines. The math coprocessor can perform
    
       arithmetic on integers with 64-bit precision in the range of +/- 10E18,
    
       and can process decimal numbers up to 18 digits without round-off
    
       errors.  It holds and manipulates all numbers in a format called
    
       "temporary real," which has a precision of 64 bits and a range of +/-
    
       10E4932.
    
    
    
    

    THE INTEL MATH COPROCESSOR AND YOUR APPLICATIONS

    
       The Intel Math CoProcessor can speed up all kinds of business
    
       application software -- probably a program you're using now. The
    
       following examples give you a good idea of how the Intel Math
    
       CoProcessor works with a wide variety of application software to get
    
       work done faster.
    
    
    
    

    GRAPHICS AND DESKTOP PUBLISHING

    
       There are two types of graphics software: bit-mapped and
    
       vector-oriented.  Bit-mapped graphics use patterns of dots to make up
    
       lines, letters, boxes, and other shapes and does not take advantage of
    
       Intel Math CoProcessors.  Z-SOFT's PC Paintbrush is an example of a
    
       bit-mapped graphics package. 
    
    
    
       Vector-oriented graphics, however, uses formulas to represent what the
    
       user draws or writes and can benefit from the Intel Math CoProcessor. 
    
       For example, when the user draws a circle, the software actually calls
    
       up the formula for a circle. When the user saves the circle to disk, the
    
       formula gets stored.  And when the user changes the size or location of
    
       the circle, the software simply changes the parameters of its formula
    
       and recalculates.  The Intel Math CoProcessor makes these recalculations
    
       considerably faster, speeding up changes and other redraws.
    
    
    
    

    CHARTS & DIAGRAMS

    
       Picture Perfect and Diagraph/2000 are menu-driven business graphics
    
       programs from Computer Support Corporation.  Picture Perfect is designed
    
       to turn data into presentation-quality charts and graphs.  Diagraph/2000
    
       creates charts, diagrams, maps, symbols, forms, and much more. 
    
    
    
       Both programs have been optimized for speed with the Intel Math
    
       CoProcessor.  They are vector-oriented and use floating point data and
    
       operations.  The math coprocessor helps most on diagrams with bold
    
       lines, shading, and shadowing.  Tests show a time savings of 65 to 80
    
       percent for previewing a chart, and 63 to 75 percent for previewing text
    
       when the 287-10 Intel Math CoProcessor was added to a 286 PC. 
    
    
    
    

    FONT GENERATION

    
       Font generation is an excellent example of using math behind the scenes.
    
        Users define the typefaces they want with a font package such as
    
       Bitstream's Fontware or SoftCraft, Inc.'s Spinfont or WYSIfonts!  The
    
       software applies a formula to produce the size and typeface requested. 
    
       The Bitstream Fontware benchmarks show, on average, a 75 percent time
    
       savings with an Intel Math CoProcessor.
    
    
    
    

    POSTSCRIPT EMULATION

    
       POSTSCRIPT EMULATION
    
    
    
       Freedom of Press from Custom Applications, Inc., allows you to print
    
       PostScript files on several non-postscript laser and 24-pin dot matrix
    
       printers.  The program uses the computer's memory to interpret the
    
       PostScript code and makes a customized bit map for your printer.  The
    
    
    
       customized bit map is generated using formulas.  Intel Math CoProcessors
    
       speed up these formulas considerably. 
    
    
    
       Tests show that the 287 math coprocessor yielded a 44 percent time
    
       savings for an Epson LQ-850, and a 57 percent time savings for a
    
       Hewlett-Packard LaserJet Series II. 
    
    
    
    

    COMPUTER-AIDED DESIGN

    
       Computer-aided design (CAD) is a classic graphics application for Intel
    
       Math CoProcessors.  Many CAD drawings are created by specifying
    
       coefficients for formulas.  Math coprocessors are used to generate and
    
       change drawings and are essential for image rotations and 3D diagrams. 
    
    
    
       Many CAD packages today would be so slow without a math coprocessor that
    
       they require one.  Benchmarks were done with an older version (2.5) of
    
       Autodesk's AutoCAD to show the dramatic performance increases with Intel
    
       Math CoProcessors (newer releases of AutoCAD will not run without an
    
       Intel Math CoProcessor).  They show a 50 percent time savings for hidden
    
       line removal and a 77 percent time savings for drawing regeneration. 
    
    
    
    

    SPREADSHEETS AND STATISTICS

    
       Intel Math CoProcessors speed up many common spreadsheet functions,
    
       though some operations are not accelerated at all. Since basic integer
    
       arithmetic (addition, subtraction, multiplication, division of whole
    
       numbers) is performed easily by the main CPU, the math coprocessor will
    
       not accelerate those functions.  However, floating point operations are
    
       laborious for the CPU and run considerably faster with an Intel Math
    
       CoProcessor. 
    
    
    
       Floating point math includes basic arithmetic of numbers with decimals,
    
       trigonometric and exponential functions, and any operations that build
    
       on trigonometry or exponentiation. Many common financial and statistical
    
       functions are based on exponents and work much faster with an Intel Math
    
       CoProcessor. 
    
    
    
       Consider the net present value formula, the function for the periodic
    
       interest rate, and the formula for standard deviation.  Because these
    
       functions contain exponents, they take advantage of the Intel Math
    
       CoProcessor's specialties. The Intel Math CoProcessor can speed up
    
       recalculation time for these and many other common financial and
    
       statistical operations.
    
    
    
    

    SPREADSHEETS

    
       The Lotus 1-2-3 Release 2.01 benchmarks show a sample of different
    
       functions.  Each function was tested for math coprocessor sensitivity by
    
       filling all of the cells in a large spreadsheet (200K to 400K) with that
    
       formula.  While this does not represent a realistic spreadsheet, it does
    
       give a precise indication of the Intel Math CoProcessor's effect on each
    
       function. 
    
    
    
       Tests show a range of time savings from 16 percent for multiplication
    
       and division to 80 percent for periodic interest rate (@rate).  This
    
       could add up to a savings of several hours a week for many spreadsheet
    
    
    
       users. 
    
    
    
    

    STATISTICS

    
       Statistical analysis programs take advantage of Intel Math CoProcessors
    
       primarily when working with several data sets and doing floating point
    
       math.  The many statistical formulas which are based on exponents run
    
       much faster with an Intel Math CoProcessor installed. 
    
    
    
       The benchmarks show statistical operations with STSC, Inc.'s
    
       STATGRAPHICS, a fully integrated statistical graphics system for PCs. 
    
       The time savings with the Intel Math CoProcessor range from 8 percent
    
       for forecasting to 50 percent for correlation analysis. 
    
    
    
    

    DATABASES

    
       Most database applications, including Ashton-Tate's dBASE IV, Fox
    
       Software's FoxBASE +, Borland's Paradox, and Nantucket's Clipper, make
    
       use of Intel's Math CoProcessors for math calculations included in the
    
       database program.  The DBASE IV benchmarks show a 65 percent savings for
    
       doing square root and logarithmic calculations.  This translates into
    
       similar time savings for any function that uses exponents or logarithms
    
       (net present value, standard deviation, variance, square root). 
    
    
    
    

    WE MEAN BUSINESS

    
       So you see, math coprocessors are good for more than just speeding up
    
       engineering and math software.  They'll speed up your spreadsheets,
    
       graphics and font packages, chart programs, CAD software, and over other
    
       business programs. 
    
    
    
       Installation is easy; on most systems it's as easy as plugging it in and
    
       turning your system on.  You don't have to worry about compatibility,
    
       because virtually every IBM PC and compatible on the market provides a
    
       socket for an Intel Math CoProcessor.  And you don't have to worry about
    
       reliability, because we back every Intel Math CoProcessor with an
    
       industry-leading limited-lifetime warranty. 
    
    
    
       Visit your PC dealer today, or contact us at one of the numbers below
    
       for information.  You may have an application (or two or three) that an
    
       Intel Math CoProcessor could really send screaming. 
    
    
    
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