CS 33. Data Representation (Part 3) CS33 Intro to Computer Systems VIII 1 Copyright 2018 Thomas W. Doeppner. All rights reserved.

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1 CS 33 Data Representation (Part 3) CS33 Intro to Computer Systems VIII 1 Copyright 2018 Thomas W. Doeppner. All rights reserved.

2 Byte-Oriented Memory Organization 00 0 FF F Programs refer to data by address conceptually, envision it as a very large array of bytes» in reality, it s not, but can think of it that way an address is like an index into that array» pointer variables contain addresses Note: system provides private address spaces to each process think of a process as a program being executed so, a program can clobber its own data, but not that of others CS33 Intro to Computer Systems VIII 2 Copyright 2018 Thomas W. Doeppner. All rights reserved.

3 Machine Words Any given computer has a word size nominal size of integer-valued data» and of addresses until a decade or so ago, most machines used 32 bits (4 bytes) as word size» limits addresses to 4GB (2 32 bytes)» became too small for memory-intensive applications leading to emergence of computers with 64-bit word size machines still support multiple data formats» fractions or multiples of word size» always integral number of bytes CS33 Intro to Computer Systems VIII 3 Copyright 2018 Thomas W. Doeppner. All rights reserved.

4 Word-Oriented Memory Organization 32-bit Words 64-bit Words Bytes Addr. Addresses specify byte locations address of first byte in word addresses of successive words differ by 4 (32-bit) or 8 (64-bit) Addr = 0000?? Addr = 0004?? Addr = 0008?? Addr = 0012?? Addr = 0000?? Addr = 0008?? CS33 Intro to Computer Systems VIII 4 Copyright 2018 Thomas W. Doeppner. All rights reserved.

5 Byte Ordering Four-byte integer 0x Stored at location 0x100 which byte is at 0x100? which byte is at 0x103? Little-endian? 0x100 0x101 0x102 0x Big-endian 0x100 0x101 0x102 0x103 CS33 Intro to Computer Systems VIII 5 Copyright 2018 Thomas W. Doeppner. All rights reserved.

6 Byte Ordering (2) Big Endian Little Endian CS33 Intro to Computer Systems VIII 6 Copyright 2018 Thomas W. Doeppner. All rights reserved.

7 Quiz 1 int main() { long x=1; func((int *)&x); return 0; } void func(int *arg) { printf("%d\n", *arg); } What value is printed on a big-endian 64-bit computer? a) 0 b) 1 c) 2 32 d) CS33 Intro to Computer Systems VIII 7 Copyright 2018 Thomas W. Doeppner. All rights reserved.

8 Which Byte Ordering Do We Use? char title[] = "This is where it begins!!!"; int array[] = {0x , 0x , 0x , 0x }; int main() { for (int i=0; i<4; i++) { printf("%x\n", array[i]); } abort(); return 0; } CS33 Intro to Computer Systems VIII 8 Copyright 2018 Thomas W. Doeppner. All rights reserved.

9 Fractional binary numbers What is ? CS33 Intro to Computer Systems VIII 9 Copyright 2018 Thomas W. Doeppner. All rights reserved.

10 Fractional Binary 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 Representation 2 -j bits to right of binary point represent fractional powers of 2 represents rational number: CS33 Intro to Computer Systems VIII 10 Copyright 2018 Thomas W. Doeppner. All rights reserved.

11 Representable Numbers Limitation #1 can exactly represent only numbers of the form n/2 k» other rational numbers have repeating bit representations value representation» 1/ [01] 2» 1/ [0011] 2» 1/ [0011] 2 Limitation #2 just one setting of decimal point within the w bits» limited range of numbers (very small values? very large?) CS33 Intro to Computer Systems VIII 11 Copyright 2018 Thomas W. Doeppner. All rights reserved.

12 IEEE Floating Point IEEE Standard 754 established in 1985 as uniform standard for floating point arithmetic» before that, many idiosyncratic formats supported on all major CPUs Driven by numerical concerns nice standards for rounding, overflow, underflow hard to make fast in hardware» numerical analysts predominated over hardware designers in defining standard CS33 Intro to Computer Systems VIII 12 Copyright 2018 Thomas W. Doeppner. All rights reserved.

13 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 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 CS33 Intro to Computer Systems VIII 13 Copyright 2018 Thomas W. Doeppner. All rights reserved.

14 Precision options 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 64-bits CS33 Intro to Computer Systems VIII 14 Copyright 2018 Thomas W. Doeppner. All rights reserved.

15 Normalized Values When: exp and exp Exponent coded as biased value: E = Exp Bias exp: unsigned value exp bias = 2 k-1-1, where k is number of exponent bits» single precision: 127 (Exp: 1 254, E: )» double precision: 1023 (Exp: , E: ) Significand coded with implied leading 1: M = 1.xxx x2 xxx x: bits of frac minimum when frac=000 0 (M = 1.0) maximum when frac=111 1 (M = 2.0 ε) get extra leading bit for free CS33 Intro to Computer Systems VIII 15 Copyright 2018 Thomas W. Doeppner. All rights reserved.

16 Normalized Encoding Example Value: float F = ; = = x 2 13 Significand M = frac = Exponent E = 13 bias = 127 exp = 140 = Result: s exp frac CS33 Intro to Computer Systems VIII 16 Copyright 2018 Thomas W. Doeppner. All rights reserved.

17 Denormalized Values Condition: exp = Exponent value: E = Bias + 1 (instead of E = 0 Bias) Significand coded with implied leading 0: M = 0.xxx x2 xxx x: bits of frac Cases exp = 000 0, frac = 000 0» represents zero value» note distinct values: +0 and 0 (why?) exp = 000 0, frac 000 0» numbers closest to 0.0» equispaced CS33 Intro to Computer Systems VIII 17 Copyright 2018 Thomas W. Doeppner. All rights reserved.

18 Special Values Condition: exp = Case: 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 = Case: exp = 111 1, frac not-a-number (NaN) represents case when no numeric value can be determined e.g., sqrt( 1),, 0 CS33 Intro to Computer Systems VIII 18 Copyright 2018 Thomas W. Doeppner. All rights reserved.

19 Visualization: Floating-Point Encodings Normalized Denorm +Denorm +Normalized + NaN 0 +0 NaN CS33 Intro to Computer Systems VIII 19 Copyright 2018 Thomas W. Doeppner. All rights reserved.

20 Tiny Floating-Point Example s exp frac 1 4-bits 3-bits 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 CS33 Intro to Computer Systems VIII 20 Copyright 2018 Thomas W. Doeppner. All rights reserved.

21 Dynamic Range (Positive Only) 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 CS33 Intro to Computer Systems VIII 21 Copyright 2018 Thomas W. Doeppner. All rights reserved.

22 Distribution of Values 6-bit IEEE-like format e = 3 exponent bits f = 2 fraction bits s exp frac bias is = bits 2-bits Notice how the distribution gets denser toward zero. 8 values Denormalized Normalized Infinity CS33 Intro to Computer Systems VIII 22 Copyright 2018 Thomas W. Doeppner. All rights reserved.

23 Distribution of Values (close-up view) 6-bit IEEE-like format e = 3 exponent bits f = 2 fraction bits s exp frac bias is bits 2-bits Denormalized Normalized Infinity CS33 Intro to Computer Systems VIII 23 Copyright 2018 Thomas W. Doeppner. All rights reserved.

24 Quiz 2 6-bit IEEE-like format e = 3 exponent bits f = 2 fraction bits s exp frac bias is bits 2-bits What number is represented by ? a) 12 b) 1.5 c).5 d) none of the above CS33 Intro to Computer Systems VIII 24 Copyright 2018 Thomas W. Doeppner. All rights reserved.

25 Floating-Point Operations: Basic Idea x +f y = Round(x + y) x f y = Round(x y) Basic idea first compute exact result make it fit into desired precision» possibly overflow if exponent too large» possibly round to fit into frac CS33 Intro to Computer Systems VIII 25 Copyright 2018 Thomas W. Doeppner. All rights reserved.

26 Rounding Rounding modes (illustrated with $ rounding) $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 integer $1 $2??? nearest even (default) $1 $2 $2 $2 $2 CS33 Intro to Computer Systems VIII 26 Copyright 2018 Thomas W. Doeppner. All rights reserved.

27 Creating a Floating Point Number Steps s exp frac normalize to have leading 1 round to fit within fraction 1 4-bits 3-bits postnormalize to deal with effects of rounding Case study convert 8-bit unsigned numbers to tiny floating-point format example numbers CS33 Intro to Computer Systems VIII 27 Copyright 2018 Thomas W. Doeppner. All rights reserved.

28 Normalize s exp frac Requirement set binary point so that numbers of form 1.xxxxx adjust all to have leading one 1 4-bits 3-bits» decrement exponent as shift left Value Binary Fraction Exponent CS33 Intro to Computer Systems VIII 28 Copyright 2018 Thomas W. Doeppner. All rights reserved.

29 Rounding 1.BBGRXXX Guard bit: LSB of result Round bit: 1 st bit removed Sticky bit: OR of remaining bits Round-up conditions round = 1, sticky = 1 Þ > 0.5 guard = 1, round = 1, sticky = 0 Þ round up to even Value Fraction GRS Incr? Rounded N N N Y Y Y CS33 Intro to Computer Systems VIII 29 Copyright 2018 Thomas W. Doeppner. All rights reserved.

30 Postnormalize Issue rounding may have caused overflow handle by shifting right once & incrementing exponent Value Rounded Exp Adjusted Result * CS33 Intro to Computer Systems VIII 30 Copyright 2018 Thomas W. Doeppner. All rights reserved.

31 Floating-Point Multiplication ( 1) s1 M1 2 E1 x ( 1) s2 M2 2 E2 Exact result: ( 1) s M 2 E sign s: s1 ^ s2 significand M: M1 x M2 exponent E: E1 + E2 Fixing if M 2, shift M right, increment E if E out of range, overflow (or underflow) round M to fit frac precision Implementation biggest chore is multiplying significands CS33 Intro to Computer Systems VIII 31 Copyright 2018 Thomas W. Doeppner. All rights reserved.

32 Floating-Point Addition ( 1) s1 M1 2 E1 assume E1 > E2 + ( 1) s2 M2 2 E2 E1 E2 Exact result: ( 1) s M 2 E sign s, significand M: + ( 1) s1 M1 ( 1) s2 M2» result of signed align & add exponent E: E1 ( 1) s M 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 round M to fit frac precision CS33 Intro to Computer Systems VIII 32 Copyright 2018 Thomas W. Doeppner. All rights reserved.

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