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CDA 3101 Summer 2007 Introduction to Computer Organization. Floating Point Theory, Notation, MIPS 19,21 June 2007. Overview. Floating point numbers Scientific notation Decimal scientific notation Binary scientific notation

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## Floating Point Theory, Notation, MIPS 19,21 June 2007

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**CDA 3101**Summer 2007Introduction to Computer Organization Floating Point Theory, Notation, MIPS 19,21 June 2007**Overview**• Floating point numbers • Scientific notation • Decimal scientific notation • Binary scientific notation • IEEE 754 FP Standard • Floating point representation inside a computer • Greater range vs. precision • Decimal to Floating Point conversion • Type is not associated with data • MIPS floating point instructions, registers**Computer Numbers**• Computers are made to deal with numbers • What can we represent in n bits? • Unsigned integers: 0 to 2n - 1 • Signed integers: -2(n-1) to 2(n-1) - 1 • What about other numbers? • Very large numbers? (seconds/century) 3,155,760,00010 (3.1557610 x 109) • Very small numbers? (atomic diameter) 0.0000000110 (1.010 x 10-8) • Rationals (repeating pattern) 2/3 (0.666666666. . .) • Irrationals: 21/2 (1.414213562373. . .) • Transcendentals : e (2.718...), (3.141...)**exponent**mantissa radix (base) decimal point Scientific Notation 6.02 x 1023 • Normalized form: no leadings 0s (exactly one digit to left of decimal point) • Alternatives to representing 1/1,000,000,000 • Normalized: 1.0 x 10-9 • Not normalized: 0.1 x 10-8, 10.0 x 10-10**Exponent**radix (base) “binary point” Binary Scientific Notation Mantissa • Floating point arithmetic • Binary point is not fixed (as it is for integers) • Declare such variable in C as float 1.0two x 2-1**31**30 23 22 0 S Exponent Significand 1 bit 8 bits 23 bits Floating Point Representation • Normal format:+1.xxxxxxxxxxtwo*2yyyytwo • Multiple of Word Size (32 bits) • S represents SignExponent represents y’s Significand represents x’s • Represent numbers as small as 2.0 x 10-38 to as large as 2.0 x 1038**Overflow and Underflow**• Overflow • Result is too large (> 2.0x1038 ) • Exponent larger than represented in 8-bit Exponent field • Underflow • Result is too small • >0, < 2.0x10-38 • Negative exponent larger than represented in 8-bit Exponent field • How to reduce chances of overflow or underflow?**Double Precision FP**31 30 20 19 0 • Use two words (64 bits) S Exponent Significand 1 bit 11 bits 20 bits Significand (cont’d) 32 bits • C variable declared as double • Represent numbers almost as small as 2.0 x 10-308 to almost as large as 2.0 x 10308 • Primary advantage is greater accuracy (52 bits)**Floating Point Representation**31 30 23 22 0 Single Precision Exponent Significand S 1 bit 8 bits 23 bits 31 30 20 19 0 Significand S Exponent Double Precision 1 bit 11 bits 20 bits Significand (cont’d) 32 bits 127 (SP) 1023 (DP) Bias Normalized scientific notation: +1.xxxxtwo*2yyyytwo Exponent: biased notation Significand: sign – magnitude notation**IEEE 754 FP Standard**• Used in almost all computers (since 1980) • Porting of FP programs • Quality of FP computer arithmetic • Sign bit: • Significand: • Leading 1 implicit for normalized numbers • 1 + 23 bits single, 1 + 52 bits double • always true: 0 < Significand < 1 • 0 has no leading 1 • Reserve exponent value 0 just for number 0 1 means negative 0 means positive (-1)S * (1 + Significand) * 2Exp**IEEE 754 Exponent**• Use FP numbers even without FP hardware • Sort records with FP numbers using integer compares • Break FP number into 3 parts: compare signs, then compare exponents, then compare significands • Faster (single comparison, ideally) • Highest order bit is sign ( negative < positive) • Exponent next, so big exponent => bigger # • Significand last: exponents same => bigger #**Biased Notation for Exponents**• Two’s complement does not work for exponent • Most negative exponent: 00000001two • Most positive exponent: 11111110two • Bias: number added to real exponent • 127 for single precision • 1023 for double precision • 1.0 * 2-1 0 0111 1110 0000 0000 0000 0000 0000 000 (-1)S * (1 + Significand) * 2(Exponent - Bias)**Significand**• Method 1 (Fractions): • In decimal: 0.34010 => 34010/100010 => 3410/10010 • In binary: 0.1102 => 1102/10002 (610/810) => 112/1002 (310/410) • Helps understand the meaning of the significand • Method 2 (Place Values): • Convert from scientific notation • In decimal: 1.6732 = (1x100) + (6x10-1) + (7x10-2) + (3x10-3) + (2x10-4) • In binary: 1.1001 = (1x20) + (1x2-1) + (0x2-2) + (0x2-3) + (1x2-4) • Interpretation of value in each position extends beyond the decimal/binary point • Good for quickly calculating significand value • Use this method for translating FP numbers**0**0110 1000 101 0101 0100 0011 0100 0010 Binary to Decimal FP • Sign: 0 => positive • Exponent: • 0110 1000two = 104ten • Bias adjustment: 104 - 127 = -23 • Significand: • 1 + 1x2-1+ 0x2-2 + 1x2-3 + 0x2-4 + 1x2-5 +...=1+2-1+2-3 +2-5 +2-7 +2-9 +2-14 +2-15 +2-17 +2-22= 1.0 + 0.666115 • Represents: 1.666115*2-23 ~ 1.986*10-7**1**0111 1110 100 0000 0000 0000 0000 0000 Decimal to Binary FP (1/2) • Simple Case: If denominator is a power of 2 (2, 4, 8, 16, etc.), then it’s easy. • Example: Binary FP representation of -0.75 • -0.75 = -3/4 • -11two/100two = -0.11two • Normalized to -1.1two x 2-1 • (-1)S x (1 + Significand) x 2(Exponent-127) • (-1)1 x (1 + .100 0000 ... 0000) x 2(126-127)**Decimal to Binary FP (2/2)**• Denominator is not an exponent of 2 • Number can not be represented precisely • Lots of significand bits for precision • Difficult part: get the significand • Rational numbers have a repeating pattern • Conversion • Write out binary number with repeating pattern. • Cut it off after correct number of bits (different for single vs. double precision). • Derive sign, exponent and significand fields.**Decimal to Binary**1 1000 0000 101 0101 0101 0101 0101 0101 0.33333333 x 2 0 .66666666 0.66666666 x 2 1 .33333332 0.33333332 x 2 0 .66666664 - 1 1 . 0 1 0 1 0 1 0 . . . - 3 . 3 3 3 3 3 3… => - 1.1010101.. x 21 • Significand: 101 0101 0101 0101 0101 0101 • Sign: negative => 1 • Exponent: 1+ 127 = 128ten= 1000 0000two**Types and Data**0011 0100 0101 0101 0100 0011 0100 0010 • 1.986 *10-7 • 878,003,010 • “4UCB” ori $s5, $v0, 17218 • Data can be anything; operation of instruction that accesses operand determines its type! • Power/danger of unrestricted addresses/pointers: • Use ASCII as FP, instructions as data, integers as instructions, ... • Security holes in programs**IEEE 754 Special Values**Negative Underflow Positive Underflow Negative Overflow Expressible Negative Numbers Expressible Positive Numbers Positive Overflow -(1-2-24)*2128 -.5*2-127 0 .5*2-127 (1-2-24)*2128**Value: Not a Number**• What is the result of: sqrt(-4.0)or 0/0? • If infinity is not an error, these shouldn’t be either. • Called Not a Number (NaN) • Exponent = 255, Significand nonzero • Applications • NaNs help with debugging • They contaminate: op(NaN, X) = NaN • Don’t use NaN • Ask math majors**b**Gap! - + 0 a - + 0 Value: Denorms • Problem: There’s a gap among representable FP numbers around 0 • Smallest pos num: a = 1.0… 2 * 2-126 = 2-126 • 2nd smallest pos num: b = 1.001 2 * 2-126 = 2-126 + 2-150 • a - 0 = 2-126 • b - a = 2-150 • Solution: • Denormalized numbers: no leading 1 • Smallest pos num: a = 2-150 • 2nd smallest num: b = 2-149**Rounding**• Math on real numbers => rounding • Rounding also occurs when converting types • Double single precision integer • Round towards +infinity • ALWAYS round “up”: 2.001 => 3; -2.001 => -2 • Round towards -infinity • ALWAYS round “down”: 1.999 => 1; -1.999 => -2 • Truncate • Just drop the last bits (round towards 0) • Round to (nearest) even (default) • 2.5 => 2; 3.5 => 4**FP Fallacy**• FP Add, subtract associative: FALSE! • x = – 1.5 x 1038, y = 1.5 x 1038, and z = 1.0 • x + (y + z) = –1.5x1038 + (1.5x1038 + 1.0) = –1.5x1038 + (1.5x1038) = 0.0 • (x + y) + z = (–1.5x1038 + 1.5x1038) + 1.0 = (0.0) + 1.0 = 1.0 • Floating Point add, subtract are not associative! • Why? FP result approximates real result • 1.5 x 1038 is so much larger than 1.0 that 1.5 x 1038 + 1.0 in floating point representation is still 1.5 x 1038**FP Addition / Subtraction**• Much more difficult than with integers • Can’t just add significands • Algorithm • De-normalize to match exponents • Add (subtract) significands to get resulting one • Keep the same exponent • Normalize (possibly changing exponent) • Note: If signs differ, just perform a subtract instead.**MIPS FP Architecture (1/2)**• Separate floating point instructions: • Single Precision: add.s, sub.s, mul.s, div.s • Double Precision: add.d, sub.d, mul.d, div.d • These instructions are far more complicated than their integer counterparts • Problems: • It’s inefficient to have different instructions take vastly differing amounts of time. • Generally, a particular piece of data will not change from FP to int, or vice versa, within a program. • Some programs do not do floating point calculations • It takes lots of hardware relative to integers to do FP fast**MIPS FP Architecture (2/2)**• 1990 Solution: separate chip that handles only FP. • Coprocessor 1: FP chip • Contains 32 32-bit registers: $f0, $f1, … • Most registers specified in .s and .d instructions ($f) • Separate load and store: lwc1 and swc1(“load word coprocessor 1”, “store …”) • Double Precision: by convention, even/odd pair contain one DP FP number: $f0/$f1, … , $f30/$f31 • 1990 Computer contains multiple separate chips: • Processor: handles all the normal stuff • Coprocessor 1: handles FP and only FP; • Move data between main processor and coprocessors: • mfc0, mtc0, mfc1, mtc1, etc.**C => MIPS**float f2c (float fahr) { return ((5.0 / 9.0) * (fahr – 32.0)); } F2c: lwc1 $f16, const5($gp) # $f16 = 5.0 lwc1 $f18, const9($gp) # $f18 = 9.0 div.s $f16, $f16, $f18 # $f16 = 5.0/9.0 lwc1 $f20, const32($gp) # $f20 = 32.0 sub.s $f20, $f12, $f20 # $f20 = fahr – 32.0 mul.s $f0, $f16, $f20 # $f0 = (5/9)*(fahr-32) jr $ra # return**Conclusion**• Floating Point numbers approximate values that we want to use. • IEEE 754 Floating Point Standard is most widely accepted attempt to standardize FP arithmetic • New MIPS architectural elements • Registers ($f0-$f31) • Single Precision (32 bits, 2x10-38… 2x1038) • add.s, sub.s, mul.s, div.s • Double Precision (64 bits , 2x10-308…2x10308) • add.d, sub.d, mul.d, div.d • Type is not associated with data, bits have no meaning unless given in context (e.g., int vs. float)

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