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IEEE Fast Square Root PowerPoint PPT Presentation

IEEE Fast Square Root. Ref: Graphics Gems III; 2 Spring 2003. Motivation. Square root operations are frequently used in many applications (e.g., computer graphics) Usually speed is more important than accuracy Is there any way faster than sqrt( )? Idea: tabulated sqrt!.

IEEE Fast Square Root

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IEEE Fast Square Root

Ref: Graphics Gems III; 2

Spring 2003

Motivation

• Square root operations are frequently used in many applications (e.g., computer graphics)

• Usually speed is more important than accuracy

• Is there any way faster than sqrt( )?

• Idea: tabulated sqrt!

Negative exponents: same!

Math Background

For 52-bit mantissa (double), only limited cases need to be computed: 2252 entries; each entry with 52 bits

Abridged Table

• Sacrifice accuracy for smaller tables

• Indexed by first 13 bits of mantissa only

• Only 2213 entries; each entry with 20 significant binary bits

• Further accuracy, if required, can be obtained by one or two Newton iterations, using the tabulated value as initial guess

Try this yourself!

B7

B6

B5

B4

B3

B2

B1

B0

How Numbers are Stored in Memory

Conceptually:

SEEEEEEEEEEEMMMM MMMMMMMMMMMMMMMM MMMMMMMMMMMMMMMM MMMMMMMMMMMMMMMM

B0

B1

B2

B3

B4

B5

B6

B7

Stored (on PC): byte swapping; Least Significant Byte first

Byte swapping:

Cautious when exchanging binary files and direct data access;

But when we read/operate as the declared data, do not need to worry (it reads backward)

This is why the examiner works

Byte Swapping (cont)

short 1029: 0x 0405

int 218+5: 0x 0040 0005

float 3.5: 0x 4060 0000

double 3.5: 0x 400c 0000 0000 0000

Use this program to see for yourself

Setup Table

Evaluation

B7

B6

B5

B4

B3

B2

B1

B0

Details: access M.S.Byte

f

fi

13

7

MMMMMMMMMMMMMMMM MMMMMMMMMMMMMMMM SEEEEEEEEEEEMMMM MMMMMMMMMMMMMMMM

f

fi

Example

Twice faster; but note the overhead for building up the tables