CS 33. Data Representation, Part 2. CS33 Intro to Computer Systems VII 1 Copyright 2018 Thomas W. Doeppner. All rights reserved.

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1 CS 33 Data Representation, Part 2 CS33 Intro to Computer Systems VII 1 Copyright 2018 Thomas W. Doeppner. All rights reserved.

2 Signed Integers Two s complement b w-1 = 0 Þ non-negative number value = b w-1 = 1 Þ negative number one zero! value = CS33 Intro to Computer Systems VII 2 Copyright 2018 Thomas W. Doeppner. All rights reserved.

3 Signed Integers Negating two s complement value = b w 1 2 w 1 + how to compute value? (~value)+1 w 2 i=0 b i 2 i CS33 Intro to Computer Systems VII 3 Copyright 2018 Thomas W. Doeppner. All rights reserved.

4 Signed Integers Negating two s complement (continued) value + (~value + 1) = (value + ~value) + 1 = (2 w 1) + 1 = 2 w w = 1 CS33 Intro to Computer Systems VII 4 Copyright 2018 Thomas W. Doeppner. All rights reserved.

5 Quiz 1 We have a computer with 4-bit words that uses two s complement to represent negative numbers. What is the result of subtracting 0010 (2) from 0001 (1)? a) 0111 b) 1001 c) 1110 d) 1111 CS33 Intro to Computer Systems VII 5 Copyright 2018 Thomas W. Doeppner. All rights reserved.

6 Signed vs. Unsigned in C char, short, int, and long signed integer types right shift (>>) is arithmetic unsigned char, unsigned short, unsigned int, unsigned long unsigned integer types right shift (>>) is logical CS33 Intro to Computer Systems VII 6 Copyright 2018 Thomas W. Doeppner. All rights reserved.

7 Numeric Ranges Unsigned Values UMin = UMax = 2 w Two s Complement Values TMin = 2 w TMax = 2 w Other Values Minus Values for W = 16 Decimal Hex Binary UMax FF FF TMax F FF TMin FF FF CS33 Intro to Computer Systems VII 7 Copyright 2018 Thomas W. Doeppner. All rights reserved.

8 Values for Different Word Sizes W UMax ,535 4,294,967,295 18,446,744,073,709,551,615 TMax ,767 2,147,483,647 9,223,372,036,854,775,807 TMin ,768-2,147,483,648-9,223,372,036,854,775,808 Observations TMin = TMax + 1» Asymmetric range UMax = 2 * TMax + 1 C Programming #include <limits.h> declares constants, e.g., ULONG_MAX LONG_MAX LONG_MIN values platform-specific CS33 Intro to Computer Systems VII 8 Copyright 2018 Thomas W. Doeppner. All rights reserved.

9 Quiz 2 What is TMin (assuming two s complement signed integers)? a) TMin b) TMax c) 0 d) 1 CS33 Intro to Computer Systems VII 9 Copyright 2018 Thomas W. Doeppner. All rights reserved.

10 4-Bit Computer Arithmetic CS33 Intro to Computer Systems VII 10 Copyright 2018 Thomas W. Doeppner. All rights reserved.

11 Signed vs. Unsigned in C Constants by default are considered to be signed integers unsigned if have U as suffix 0U, U Casting explicit casting between signed & unsigned int tx, ty; unsigned ux, uy; // unsigned means unsigned int tx = (int) ux; uy = (unsigned int) ty; implicit casting also occurs via assignments and procedure calls tx = ux; uy = ty; CS33 Intro to Computer Systems VII 11 Copyright 2018 Thomas W. Doeppner. All rights reserved.

12 Casting Surprises Expression evaluation if there is a mix of unsigned and signed in single expression, signed values implicitly cast to unsigned including comparison operations <, >, ==, <=, >= examples for W = 32: TMIN = -2,147,483,648, TMAX = 2,147,483,647 Constant 1 Constant 2 Relation Evaluation 0 0U == unsigned -1 0 < signed -1 0U > unsigned > signed U < unsigned -1-2 > signed (unsigned)-1-2 > unsigned U < unsigned (int) u > signed CS33 Intro to Computer Systems VII 12 Copyright 2018 Thomas W. Doeppner. All rights reserved.

13 Quiz 3 What is the value of (long)ulong_max - (unsigned long)-1??? a) -1 b) 0 c) 1 d) ULONG_MAX CS33 Intro to Computer Systems VII 13 Copyright 2018 Thomas W. Doeppner. All rights reserved.

14 Sign Extension Task: given w-bit signed integer x convert it to w+k-bit integer with same value Rule: make k copies of sign bit: X = x w 1,, x w 1, x w 1, x w 2,, x 0 k copies of MSB X w X k w CS33 Intro to Computer Systems VII 14 Copyright 2018 Thomas W. Doeppner. All rights reserved.

15 Sign Extension Example short int x = 15213; int ix = (int) x; short int y = ; int iy = (int) y; Decimal Hex Binary x B 6D ix B 6D y C iy FF FF C Converting from smaller to larger integer data type C automatically performs sign extension CS33 Intro to Computer Systems VII 15 Copyright 2018 Thomas W. Doeppner. All rights reserved.

16 Does it Work? val w = 2 w 1 + w 2 b i 2 i i=0 w 2 i=0 val w+1 = 2 w + 2 w 1 + b i 2 i w 2 = 2 w 1 + b i 2 i i=0 w 2 i=0 w 2 i=0 val w+2 = 2 w w + 2 w 1 + b i 2 i = 2 w + 2 w 1 + b i 2 i w 2 = 2 w 1 + b i 2 i i=0 CS33 Intro to Computer Systems VII 16 Copyright 2018 Thomas W. Doeppner. All rights reserved.

17 Unsigned Multiplication Operands: w bits True Product: 2*w bits Discard w bits: w bits u * v u * v UMult w (u, v) Standard multiplication function ignores high order w bits Implements modular arithmetic UMult w (u, v) = u v mod 2 w CS33 Intro to Computer Systems VII 17 Copyright 2018 Thomas W. Doeppner. All rights reserved.

18 Signed Multiplication Operands: w bits True Product: 2*w bits Discard w bits: w bits u * v u * v TMult w (u, v) Standard multiplication function ignores high order w bits some of which are different from those of unsigned multiplication lower bits are the same CS33 Intro to Computer Systems VII 18 Copyright 2018 Thomas W. Doeppner. All rights reserved.

19 Power-of-2 Multiply with Shift Operation u << k gives u * 2 k both signed and unsigned operands: w bits true product: w+k bits Examples u * 2 k u << 3 == u * 8 u << 5 - u << 3 == u * 24 most machines shift and add faster than multiply» compiler generates this code automatically * u 2 k discard k bits: w bits UMult w (u, 2 k ) TMult w (u, 2 k ) k CS33 Intro to Computer Systems VII 19 Copyright 2018 Thomas W. Doeppner. All rights reserved.

20 Unsigned Power-of-2 Divide with Shift Quotient of unsigned by power of 2 u >> k gives ë u / 2 k û uses logical shift operands: division: u 2 k / u / 2 k k binary point result: ë u / 2 k û Division Computed Hex Binary x B 6D x >> D B x >> B x >> B CS33 Intro to Computer Systems VII 20 Copyright 2018 Thomas W. Doeppner. All rights reserved.

21 Signed Power-of-2 Divide with Shift Quotient of signed by power of 2 x >> k gives ë x / 2 k û uses arithmetic shift rounds wrong direction when x < 0 operands: division: result: x 2 k / x / 2 k RoundDown(x / 2 k ) CS33 Intro to Computer Systems VII 21 Copyright 2018 Thomas W. Doeppner. All rights reserved. k 0. 0 binary point Division Computed Hex Binary y C y >> E y >> FC y >> FF C

22 Correct Power-of-2 Divide Quotient of negative number by power of 2 want é x / 2 k ù (round toward 0) compute as ë (x+2 k -1)/ 2 k û» in C: (x + (1<<k)-1) >> k» biases dividend toward 0 Case 1: no rounding dividend: divisor: u +2 k 1 / 2 k é u / 2 k ù k binary point Biasing has no effect CS33 Intro to Computer Systems VII 22 Copyright 2018 Thomas W. Doeppner. All rights reserved.

23 Correct Power-of-2 Divide (Cont.) Case 2: rounding dividend: k 1 x +2 k incremented by 1 binary point divisor: 2 k / é x / 2 k ù incremented by 1 Biasing adds 1 to final result CS33 Intro to Computer Systems VII 23 Copyright 2018 Thomas W. Doeppner. All rights reserved.

24 Why Should I Use Unsigned? Don t use just because number nonnegative easy to make mistakes unsigned i; for (i = cnt-2; i >= 0; i--) a[i] += a[i+1]; can be very subtle #define DELTA sizeof(int) int i; for (i = CNT; i-delta >= 0; i-= DELTA)... Do use when performing modular arithmetic multiprecision arithmetic Do use when using bits to represent sets logical right shift, no sign extension CS33 Intro to Computer Systems VII 24 Copyright 2018 Thomas W. Doeppner. All rights reserved.