CS 213 Class 03: Numeric Encodings and Integer Representations, Slides of Software Engineering

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“The course that gives CMU its Zip!”

Topics

• Numeric Encodings
  • Unsigned & Two’s complement
• Programming Implications
  • C promotion rules

class03.ppt CS 213 F’

Integer Representations

Sept. 1, 1998

Notation

W : Number of Bits in “Word”

C Data Type Sun, etc. Alpha long int 32 64 int 32 32 short 16 16 char 8 8

Integers

• Lower case
• E.g., x , y , z

Bit Vectors

• Upper Case
• E.g., X , Y , Z
• Write individual bits as integers with value 0 or 1
• E.g., X = xw –1 , xw –2 ,… x 0
  • Most significant bit on left

Encoding Example (Cont.)

Weight 15213 - 1 1 1 1 1 2 0 0 1 2 4 1 4 0 0 8 1 8 0 0 16 0 0 1 16 32 1 32 0 0 64 1 64 0 0 128 0 0 1 128 256 1 256 0 0 512 1 512 0 0 1024 0 0 1 1024 2048 1 2048 0 0 4096 1 4096 0 0 8192 1 8192 0 0 16384 0 0 1 16384 32768 0 0 1 - Sum 15213 -

x = 15213: 00111011 01101101 y = -15213: 11000100 10010011

Numeric Ranges

Unsigned Values

• UMin = 0 000…
• UMax = 2 w^ – 1 111…

Two’s Complement Values

• TMin = –2 w – 100…
• TMax = 2 w –1^ – 1 011…

Other Values

• Minus 1 111…

Decimal Hex Binary UMax 65535 FF FF 11111111 11111111 TMax 32767 7F FF 01111111 11111111 TMin (^) -32768 80 00 10000000 00000000 -1 (^) -1 FF FF 11111111 11111111 0 0 00 00 00000000 00000000

Values for W = 16

Unsigned & Signed Numeric Values

Example Values

• W = 4

Equivalence

• Same encodings for nonnegative values

Uniqueness

• Every bit pattern represents unique integer value
• Each representable integer has unique bit encoding

⇒ Can Invert Mappings

• U2B( x ) = B2U -1 ( x )
  • Bit pattern for unsigned integer
• T2B( x ) = B2T -1 ( x )
  • Bit pattern for two’s comp integer

X B2U( X ) B2T( X )

short int x = 15213; unsigned short int ux = (unsigned short) x; short int y = -15213; unsigned short int uy = (unsigned short) y;

Casting Signed to Unsigned

C Allows Conversions from Signed to Unsigned

Resulting Value

• No change in bit representation
• Nonnegative values unchanged
  • ux = 15213
• Negative values change into (large) positive values
  • uy = 50323

Relation Between Signed & Unsigned

• uy = y + 2 * 32768 = y + 65536

Weight -15213 50323 1 1 1 1 1 2 1 2 1 2 4 0 0 0 0 8 0 0 0 0 16 1 16 1 16 32 0 0 0 0 64 0 0 0 0 128 1 128 1 128 256 0 0 0 0 512 0 0 0 0 1024 1 1024 1 1024 2048 0 0 0 0 4096 0 0 0 0 8192 0 0 0 0 16384 1 16384 1 16384 32768 1 -32768 1 32768 Sum -15213^50323

From Two’s Complement to Unsigned

• T2U( x ) = B2U(T2B( x )) = x + xw –1 2 w
• What you get in C: int t2u(int x) { return (unsigned) x; }

+ 16

X B2U( X ) B2T( X )

T2U( x )

0000

0001 0010

0011 0100

0101 0110

0111 1000

1001 1010

1011 1100

1101 1110

1111

2 w

2 w –

–2 w –1^0 2 w –

0

Identity

Signed vs. Unsigned in C

Constants

• By default are considered to be signed integers
• Unsigned if have “U” as suffix 0U, 4294967259U

Casting

• Explicit casting between signed & unsigned same as U2T and T2U int tx, ty; unsigned ux, uy; tx = (int) ux; uy = (unsigned) ty;
• Implicit casting also occurs via assignments and procedure calls tx = ux; uy = ty;

Casting Surprises

Expression Evaluation

• If mix unsigned and signed in single expression, signed values implicitly cast to unsigned
• Including comparison operations <, >, ==, <=, >=
• Examples for W = 32

Constant 1 Constant 2 Relation Evaluation

0 0U == unsigned -1 0 < signed -1 0U > unsigned 2147483647 -2147483648 > signed 2147483647U -2147483648 < unsigned -1 -2 > signed (unsigned) -1 -2 > unsigned 2147483647 2147483648U < unsigned 2147483647 (int) 2147483648U > signed

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 ′ = xw –1 ,…, xw –1 , xw –1 , xw –2 ,…, x 0

k copies of MSB

X • • •

X ′ • • • • • •

• • •

w

k w

Sign Extension Example

• Converting from smaller to larger integer data type
• C automatically performs sign extension

Decimal Hex Binary x 15213 3B 6D 00111011 01101101

ix 15213 00 00 C4 92 00000000 00000000 00111011 01101101

y (^) -15213 C4 93 11000100 10010011

iy (^) -15213 FF FF C4 93 11111111 11111111 11000100 10010011

short int x = 15213; int ix = (int) x; short int y = -15213; int iy = (int) y;

Casting Order Dependencies

iux = 15213: 00000000 00000000 00111011 01101101

iuy = 50323: 00000000 00000000 11000100 10010011

uix = 15213: 00000000 00000000 00111011 01101101 uiy = 4294952083: 11111111 11111111 11000100 10010011

uuy = 4294952083: 11111111 11111111 11000100 10010011

short int x = 15213; short int y = -15213; unsigned iux = (unsigned)(unsigned short) x; unsigned iuy = (unsigned)(unsigned short) y; unsigned uix = (unsigned) (int) x; unsigned uiy = (unsigned) (int) y; unsigned uuy = y;

y

unsigned Fill 0’s

Sign Extend unsigned

iuy

uiy

(unsigned short) (unsigned)

(int) (unsigned)

Why Should I Use Unsigned?

Don’t Use Just Because Number Nonzero

• C compiler on Alpha generates less efficient code unsigned i; for (i = 1; i < cnt; i++) a[i] += a[i-1];
• Easy to make mistakes for (i = cnt-2; i >= 0; i--) a[i] += a[i+1];

Do Use When Performing Modular Arithmetic

• Multiprecision arithmetic
• Other esoteric stuff

Do Use When Need Extra Bit’s Worth of Range

• Working right up to limit of word size