Lecture（Ⅲ）：Floating Point

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目录

Background: Fractional binary numbers

IEEE floating point standard: Definition

Example and properties

Rounding, addition, multiplication

Floating point in C

Summary

Additional Slides

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Background: Fractional binary numbers

p3	---What is 1011.101(2)?
p4	---Representation
p5	---Value Representation ---Observations
p6	---Limitation #1 ---Limitation #2

IEEE floating point standard: Definition

p8	IEEE Floating Point ---IEEE Standard 754 ---Driven by numerical concerns
p9	---Numerical Form ---Encoding
p10	Precision options ---Single precision: 32 bits ---Double precision: 64 bits ---Extended precision: 80 bits (Intel only)
p11	“Normalized” Values
p12	Normalized Encoding Example ---Value: float F = 15213.0; ---Significand ---Exponent ---Result:
p13	Denormalized Values
p14	Special Values
p15	Visualization: Floating Point Encodings

Example and properties

p17	Tiny Floating Point Example ---8-bit Floating Point Representation ---Same general form as IEEE Format
p18	Dynamic Range (Positive Only)
p19	Distribution of Values ---6-bit IEEE-like format ---Notice how the distribution gets denser toward zero.
p20	Distribution of Values (close-up view) --6-bit IEEE-like format
p21	Special Properties of the IEEE Encoding ---FP Zero Same as Integer Zero ---Can (Almost) Use Unsigned Integer Comparison

Rounding, addition, multiplication

p23	Floating Point Opera.ons: Basic Idea
p24	Rounding Modes (illustrate with $ rounding)
p25	Closer Look at Round-To-Even ---Default Rounding Mode ---Applying to Other Decimal Places / Bit Positions
p26	Rounding Binary Numbers ---Binary Fractional Numbers ---Examples
p27	FP Multiplication
p28	Floating Point Addition
p29	Mathematical Properties of FP Add ---Compare to those of Abelian Group ---Monotonicity
p30	Mathematical Properties of FP Mult ---Compare to Commutative Ring ---Monotonicity

Floating point in C

p32	Floating Point in C ---C Guarantees Two Levels ---Conversions/Casting
p33	Floating Point Puzzles ---For each of the following C expressions, either:

Summary

p34	---IEEE Floating Point has clear mathematical properties ---Represents numbers of form M x 2E ---One can reason about operations independent of ---Not the same as real arithmetic

Additional Slides

p36	Creating Floating Point Number ---Steps ---Case Study
p37	Normalize ---Requirement
p38	Rounding ---Round up conditions
p39	Postnormalize ---Issue
p40	Interesting Numbers