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Signed ArithmeticLogical Operations

Ellen SpertusMCS 111

October 1, 2002

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Homework 5: 1

a[0] = a[0] + a[1] - a[2];

lw $a0, __($t0)lw $a1, __($t0)lw $a2, __($t0)add $t1, ,sub $t2, $t1, sw $t2, __($t0)

byte offset contents 0 00000000 1 00000000 2 00000000 3 11110010 4 00000000 5 00000000 6 00000000 7 00000110 8 00000000 9 00000000

10 00000000 11 00001110

Diagram assumes Big-Endian

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Homework 5: 2foo(int i, int j) { int temp;

temp = a[i];

a[i] = a[j];

a[j] = temp;

return;}

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From representation to value

1000

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Today

• Review: unsigned numbers• Signed numbers• MIPS instructions for signed and

unsigned instructions• Character encoding• Logical operations

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Unsigned binary arithmetic

10012= 1x2 3 + 0x2 2 + 0x2 1 + 1x2 0

Eights (2 3) columnFours (2 2) columnTwos (2 1) columnOnes (2 0) column

• Biggest number?• Smallest number?

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Signed binary arithmetic (two’s complement)

10012= 1x-23 + 0x22 + 0x21 + 1x20

Minus Eights (-23) columnFours (22) columnTwos (21) columnOnes (20) column

• Sign bit• Biggest number?• Smallest number?

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Signed binary to decimal

-8 4 2 1 1 0 0 1 = 1 1 1 1 = 0 1 1 1 = 1 0 0 0 =

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Shortcut

• To negate a number– Invert each bit– Add 1

• Practice– 1001 0110 0111 (710), so the original was

-7– 1111– 1000

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Relative sizes

Which 4-bit number is bigger: 1000 or 0111?

– Signed comparison

– Unsigned comparison

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Sign extension

The number 710

0111 000001110000000000000111

The number -110

1111 111111111111111111111111

Extend the sign bit all the way to the left.

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A closer look at load-byte (lb)

• load byte signed (lb)– Treat the byte as a signed number– Sign extend it to word length– lb $t0, 400($zero)

• load byte unsigned (lbu)– Treat the byte as an unsigned number– Zero extend it to word length– lbu $t0, 400($zero)

400 10000001 404 408 412 416

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MIPS instructions• Signed numbers

– set less than (slt)– set less than

immediate (slti)– load byte (lb)

• Unsigned numbers– set less than unsigned (sltu)– set less than immediate

unsigned (sltiu)– load byte unsigned (lbu)

a0 = 11111111111111111111111111111111two

a1 = 00000000000000000000000000000000two

slt $t0, $a0, $a1 # signed comparisonsltu $t1, $a0, $a1 # unsigned comparison

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How do we add signed numbers?

Just like we always have!

signed unsigned -128 64 32 16 8 4 2 1

1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1

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What about overflow?

signed unsigned -128 64 32 16 8 4 2 1

0 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1

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Definition of overflow

• When a carry bit flows into the sign bit• This can only happen with signed

arithmetic• Machine raises an exception

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MIPS instructions (2)

• Signed numbers– set less than (slt)– set less than

immediate (slti)– load byte (lb)– add – add immediate (addi)

– sub– subtract immediate

(subi)

• Unsigned numbers– set less than unsigned (sltu)– set less than immediate

unsigned (sltiu)– load byte unsigned (lbu)– add unsigned (addu)– add immediate unsigned

(addiu)– subtract unsigned (subu)– subtract immediate

unsigned (subiu)

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Pseudo-instructions

• Definition: Instructions converted by the assembler into real machine instructions

• Examples:– Load immediate:

li $r1, 5 is replaced by:– Subtract immediate:

subi $r1, $r1, 5 is replaced by:

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Differences between signed and unsigned instructions

• Sign extension vs. Zero extension – lb/lbu

• Whether to signal overflow/underflow– add/addu, addi/addiu, sub/subu

• How to compare numbers– slt/sltu, slti/sltiu

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Remainder of today

• Logical operations– Bitwise operations– Shifting

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Bitwise operations• Perform the operation on each bit

position• Practice

– Initial values• $t0 = ..00000111• $t1 = ..00000001

– Problems• not $t2, $t0• and $t2, $t0, $t1• or $t2, $t0, $t1

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Shifting

• Assume $t0 = 0..00001100• Shift left logical (sll)

– sll $t1, $t0, 2– result:

• Shift right logical (srl)– srl $t1, $t0, 2– result:

• Note: Zeroes are shifted in