Lecture 6: Floating Points

Transcription

1 CSCI-UA Computer Systems Organization Lecture 6: Floating Points Mohamed Zahran (aka Z)

2 Background: Fractional binary numbers What is ? Carnegie Mellon

3 Background: Fractional Binary Carnegie Mellon Numbers 2 i 2 i bi bi-1 b2 b1 b0 b-1 b-2 b-3 b-j 1/2 1/4 1/8 2 -j Value:

4 Value Fractional Binary Numbers: Examples Representation Carnegie Mellon 5 3/ / Observations Divide by 2 by shifting right Multiply by 2 by shifting left is just below 1.0 1/2 + 1/4 + 1/ /2 i + 1.0

5 Why not fractional binary Not efficient numbers? 3 * Carnegie Mellon 100 zeros Given a finite length (e.g. 32-bits), cannot represent very large nor very small numbers (ε 0)

6 IEEE Floating Point IEEE Standard 754 Supported by all major CPUs Driven by numerical concerns Standards for rounding, overflow, underflow Hard to make fast in hardware Numerical analysts predominated over hardware designers in defining standard

7 Floating Point Representation Numerical Form: ( 1) s M 2 E Sign bit s determines whether number is negative or positive Significand M a fractional value in range [1.0,2.0) or [0,1.0) Exponent E weights value by power of two Encoding MSB s is sign bit s exp field encodes E (but is not equal to E) frac field encodes M (but is not equal to M) s exp frac

8 Precisions Single precision: 32 bits s exp frac 1 8-bits 23-bits Double precision: 64 bits s exp frac 1 11-bits 52-bits Extended precision: 80 bits (Intel only) s exp frac 1 15-bits 63 or 64-bits

9 1. Normalized Encoding Condition: exp and exp referred to as Bias Exponent is: E = Exp (2 k-1 1), k is the # of exponent bits Single precision: E = Exp 127 Range(E)=?? Double precision: E = Exp 1023 Range(E)=?? frac Significand is: M = 1.xxx x 2 Range(M) = [1.0, 2.0-ε) Get extra leading bit for free Range(E)=[-126,127] Range(E)=[-1022,1023]

10 Normalized Encoding Example Value: Float F = ; = = x 2 13 Significand M = frac = Exponent E = Exp Bias = Exp = 13 Exp = 140 = Result: s exp frac

11 2. Denormalized Encoding Condition: exp = Exponent value: E = 1 Bias (instead of E = 0 Bias) Significand is: M = 0.xxx x 2 (instead of M=1.xxx 2 ) frac Cases exp = 000 0, frac = Represents zero Note distinct values: +0 and 0 exp = 000 0, frac Numbers very close to 0.0 Equi-spaced lose precision as get smaller

12 3. Special Values Encoding Condition: exp = Case: exp = 111 1, frac = Represents value (infinity) Operation that overflows E.g., 1.0/0.0 = 1.0/ 0.0 = +, 1.0/ 0.0 = Case: exp = 111 1, frac Not-a-Number (NaN) Represents case when no numeric value can be determined E.g., sqrt( 1),, 0

13 Visualization: Floating Point Encodings Normalized Denorm +Denorm +Normalized + NaN 0 +0 NaN

14 Tiny Floating Point Example s exp frac 1 3-bits 2-bits Toy example: 6-bit Floating Point Representation Bias? Normalized E = exp ( ) = exp 3 Denormalized E = 1 3 = -2

15 Distribution of Values 8 values Denormalized Normalized Infinity Denormalized Normalized Infinity

16 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, greater than any other values Otherwise OK Denorm vs. normalized Normalized vs. infinity

17 Floating Point Operations x +f y = Round(x + y) x f y = Round(x y) Basic idea: compute exact result, round to fit (possibly overflow) Rounding Modes $1.40 $1.60 $1.50 $2.50 $1.50 Towards 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

18 Round to nearest even 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

19 Mathematical Properties of FP Add Compare to those of Integer add in Abelian Group Yes Closed under addition? But may generate infinity or NaN Commutative? Yes Associative? i.e. (a+b)+c == a+(b+c)? 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 Almost No Yes Almost Carnegie Mellon

20 Mathematical Properties of FP Mult Compare to integer multiplication in Commutative Ring Closed under multiplication? But may generate infinity or NaN Multiplication Commutative? Multiplication is Associative? Possibility of overflow, inexactness of rounding 1 is multiplicative identity? Yes Yes No Yes Carnegie Mellon Monotonicity a b & c 0 a * c b *c? Except for infinities & NaNs Almost

21 Floating Point in C C : float double single precision double precision Conversions/Casting Casting between int, float, and double changes bit representation double/float int Truncates fractional part Like rounding toward zero Not defined when out of range or NaN: Generally sets to TMin int double Exact conversion, as long as int has 53 bit word size int float Will round according to rounding mode

22 Conclusions IEEE Floating Point has clear mathematical properties Represents numbers of form M x 2 E One can reason about operations independent of implementation As if computed with perfect precision and then rounded