System Programming CISC 360. Floating Point September 16, 2008

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1 System Programming CISC 360 Floating Point September 16, 2008 Topics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties Powerpoint Lecture Notes for Computer Systems: A Programmer's Perspective, R. Bryant and D. O'Hallaron, Prentice Hall, 2003

2 IEEE Floating Point IEEE Standard 754 Established in 1985 as uniform standard for floating point arithmetic Before that, many idiosyncratic formats Supported by all major CPUs Driven by Numerical Concerns Nice standards for rounding, overflow, underflow Hard to make go fast Numerical analysts predominated over hardware types in defining standard 2

3 Fractional Binary Numbers Representation b i b i 1 b 2 b 1 b 0. b 1 b 2 b 3 b j 1/2 1/4 1/8 2 j 2 i 1 Bits to right of binary point represent fractional powers of 2 Represents rational number: 2 i i # k =" j b k!2 k 3

4 Frac. Binary Number Examples Value Representation 5-3/ / / Observations Divide by 2 by shifting right Multiply by 2 by shifting left Numbers of form just below 1.0 1/2 + 1/4 + 1/ /2 i Use notation 1.0 ε 4

5 Representable Numbers Limitation Can only exactly represent numbers of the form x/2 k Other numbers have repeating bit representations Value Representation 1/ [01] 2 1/ [0011] 2 1/ [0011] 2 5

6 Floating Point Representation Numerical Form 1 s M 2 E Sign bit s determines whether number is negative or positive Significand M normally a fractional value in range [1.0,2.0). Exponent E weights value by power of two Encoding s exp frac MSB is sign bit exp field encodes E frac field encodes M 6

7 Floating Point Precisions Encoding MSB is sign bit exp field encodes E frac field encodes M Sizes s exp frac Single precision: 8 exp bits, 23 frac bits 32 bits total Double precision: 11 exp bits, 52 frac bits 64 bits total Extended precision: 15 exp bits, 63 frac bits Only found in Intel-compatible machines Stored in 80 bits» 1 bit wasted 7

8 Normalized Numeric Values Condition exp and exp Exponent coded as biased value E = Exp Bias Exp : unsigned value denoted by exp Bias : Bias value» Single precision: 127 (Exp: 1 254, E: )» Double precision: 1023 (Exp: , E: )» in general: Bias = 2 e-1-1, where e is number of exponent bits Significand coded with implied leading 1 M = 1.xxx x 2 xxx x: bits of frac Minimum when (M = 1.0) Maximum when (M = 2.0 ε) Get extra leading bit for free 8

9 Normalized Encoding Example Value Float F = ; = = X 2 13 Significand M = frac= Exponent E = 13 Bias = 127 Exp = 140 = Floating Point Representation (Class 02): Hex: D B Binary: : :

10 Denormalized Values Condition Value exp = Exponent value E = Bias + 1 Significand value M = 0.xxx x 2 xxx x: bits of frac Cases 10 exp = 000 0, frac = Represents value 0 Note that have distinct values +0 and 0 exp = 000 0, frac Numbers very close to 0.0 Lose precision as get smaller Gradual underflow

11 Special Values Condition Cases exp = exp = 111 1, frac = Represents value (infinity) Operation that overflows Both positive and negative E.g., 1.0/0.0 = 1.0/ 0.0 = +, 1.0/ 0.0 = exp = 111 1, frac Not-a-Number (NaN) Represents case when no numeric value can be determined E.g., sqrt( 1), 11

12 Summary of Floating Point Real Number Encodings -Normalized -Denorm +Denorm +Normalized + NaN 0 +0 NaN 12

13 Tiny Floating Point Example 8-bit Floating Point Representation the sign bit is in the most significant bit. the next four bits are the exponent, with a bias of 7. the last three bits are the frac Same General Form as IEEE Format normalized, denormalized representation of 0, NaN, infinity s exp frac 0 13

14 Values Related to the Exponent Exp exp E 2 E /64 (denorms) / / / / / / n/a (inf, NaN) 14

15 Dynamic Range s exp frac E Value Denormalized numbers Normalized numbers /8*1/64 = 1/ /8*1/64 = 2/ /8*1/64 = 6/ /8*1/64 = 7/ /8*1/64 = 8/ /8*1/64 = 9/ /8*1/2 = 14/ /8*1/2 = 15/ /8*1 = /8*1 = 9/ /8*1 = 10/ /8*128 = /8*128 = n/a inf closest to zero largest denorm smallest norm closest to 1 below closest to 1 above largest norm

16 Distribution of Values 6-bit IEEE-like format e = 3 exponent bits f = 2 fraction bits Bias is 3 Notice how the distribution gets denser toward zero Denormalized Normalized Infinity 16

17 Distribution of Values (close-up view) 6-bit IEEE-like format e = 3 exponent bits f = 2 fraction bits Bias is Denormalized Normalized Infinity 17

18 Interesting Numbers Description 18 exp frac Zero Numeric Value Smallest Pos. Denorm {23,52} X 2 {126,1022} Single 1.4 X Double 4.9 X Largest Denormalized (1.0 ε) ) X 2 {126,1022} Single 1.18 X Double 2.2 X Smallest Pos. Normalized X 2 {126,1022} Just larger than largest denormalized One Largest Normalized (2.0 ε) ) X 2 Single 3.4 X Double 1.8 X

19 Special Properties of Encoding FP Zero Same as Integer Zero All bits = 0 Can (Almost) Use Unsigned Integer Comparison Must first compare sign bits Must consider -0 = 0 NaNs problematic Will be greater than any other values What should comparison yield? Otherwise OK Denorm vs. normalized Normalized vs. infinity 19

20 Floating Point Operations Conceptual View 20 First compute exact result Make it fit into desired precision Possibly overflow if exponent too large Possibly round to fit into frac Rounding Modes (illustrate with \$ rounding) \$1.40 \$1.60 \$1.50 \$2.50 \$1.50 Zero \$1 \$1 \$1 \$2 \$1 Round down (- ) \$1 \$1 \$1 \$2 \$2 Round up (+ ) \$2 \$2 \$2 \$3 \$1 Nearest Even (default) \$1 \$2 \$2 \$2 \$2 Note: 1. Round down: rounded result is close to but no greater than true result. 2. Round up: rounded result is close to but no less than true result.

21 Closer Look at Round-To-Even Default Rounding Mode Hard to get any other kind without dropping into assembly All others are statistically biased Sum of set of positive numbers will consistently be over- or underestimated Applying to Other Decimal Places / Bit Positions When exactly halfway between two possible values Round so that least significant digit is even E.g., round to nearest hundredth (Less than half way) (Greater than half way) (Half way round up) (Half way round down) 21

22 Rounding Binary Numbers Binary Fractional Numbers Even when least significant bit is 0 Half way when bits to right of rounding position = Examples Round to nearest 1/4 (2 bits right of binary point) Value Binary Rounded Action Rounded Value 2 3/ (<1/2 down) 2 2 3/ (>1/2 up) 2 1/4 2 7/ (1/2 up) 3 2 5/ (1/2 down) 2 1/2 22

23 FP Multiplication Operands ( 1) s1 M1 2 E1 * ( 1) s2 M2 2 E2 Exact Result ( 1) s M 2 E Sign s: s1 ^ s2 Significand M: Exponent E: M1 * M2 E1 + E2 Fixing If M 2, shift M right, increment E If E out of range, overflow Round M to fit frac precision Implementation Biggest chore is multiplying significands 23

24 FP Addition Operands ( 1) s1 M1 2 E1 E1 E2 ( 1) s2 M2 2 E2 ( 1) s1 M1 Assume E1 > E2 Exact Result ( 1) s M 2 E + ( 1) s M ( 1) s2 M2 Sign s, significand M: Result of signed align & add Exponent E: E1 Fixing If M 2, shift M right, increment E if M < 1, shift M left k positions, decrement E by k Overflow if E out of range 24 Round M to fit frac precision

25 Mathematical Properties of FP Add Compare to those of Abelian Group Closed under addition? But may generate infinity or NaN Commutative? YES YES Associative? NO Overflow and inexactness of rounding 0 is additive identity? Every element has additive inverse Except for infinities & NaNs Monotonicity a b a+c b+c? Except for infinities & NaNs YES ALMOST ALMOST 25

26 Math. Properties of FP Mult Compare to Commutative Ring Closed under multiplication? But may generate infinity or NaN Multiplication Commutative? YES YES Multiplication is Associative? NO Possibility of overflow, inexactness of rounding 1 is multiplicative identity? YES Multiplication distributes over addition? NO Possibility of overflow, inexactness of rounding Monotonicity a b & c 0 a *c b *c? Except for infinities & NaNs ALMOST 26

27 Floating Point in C C Guarantees Two Levels 27 float double Conversions single precision double precision Casting between int, float, and double changes numeric values Double or float to int Truncates fractional part Like rounding toward zero Not defined when out of range» Generally saturates to TMin or TMax int to double Exact conversion, as long as int has 53 bit word size int to float Will round according to rounding mode

28 Summary IEEE Floating Point Has Clear Mathematical Properties Represents numbers of form M X 2 E Can reason about operations independent of implementation As if computed with perfect precision and then rounded Not the same as real arithmetic Violates associativity/distributivity Makes life difficult for compilers & serious numerical applications programmers 28

29 Next Deadlines Sep 16 Lec04 Floating Point Quiz 2 on chapter 2 New reading assignment: sections Sep 18 Lec05 - Machine-Level Programming I - Introduction Sep 23 Lec06 Machine-Level Programming II - Introduction Lab1 is due today at 11:59PM Sep 25 Lec07 Machine-Level Programming III - Control Flow Quiz 3 on sections New reading assignment: sections

30 Floating Point Puzzles For each of the following C expressions, either: Argue that it is true for all argument values Explain why not true x == (int)(float) x 30 int x = ; float f = ; double d = ; Assume neither d nor f is NaN x == (int)(double) x f == (float)(double) f d == (float) d f == -(-f); 2/3 == 2/3.0 d < 0.0 ((d*2) < 0.0) d > f -f > -d d * d >= 0.0 (d+f)-d == f

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