Arithmetic for Computers – Significance

Alexander Nelson

March 5, 2021

University of Arkansas - Department of Computer Science and Computer Engineering

Review of Computer Arithmetic

What have we learned so far:

We need circuits to perform binary arithmetic operations

• add, subtract, multiply, divide, and, or, etc.

We may need to perform operations on fewer bits

• Subword parallelism

We may need to perform this math on small or large numbers

• Floating point arithmetic

Specialized circuits perform these arithmetic operations

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Review of Computer Arithmetic

Why does this matter?

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Review of Computer Arithmetic

Why does this matter?

Keep the end goal in sight

A general purpose CPU (“central processing unit”) – Abstraction

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Review of Computer Arithmetic

Five classic components of

a computer:

• Input• Output• Memory• Datapath• Control 1Computer Arithmetic in the “Combinational Logic” block

1By Lambtron - Own work, CC BY-SA 4.0,

https://commons.wikimedia.org/w/index.php?curid=37716438

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Review of Computer Arithmetic

“Combinational Logic” receives a command to perform arithmetic

What kinds of arithmetic?

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Review of Computer Arithmetic

Too many types to draw into schematics/overviews

Need an abstraction!

Arithmetic-Logic Unit (ALU) – Abstraction for all computer

arithmetic operations

Allow CPU designers to focus on individual parts

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Arithmetic-Logic Unit

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2By Lambtron - Own work, CC BY-SA 4.0,

https://commons.wikimedia.org/w/index.php?curid=36975996

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Arithmetic-Logic Unit

Why does this work with MIPS?

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Arithmetic-Logic Unit

Why does this work with MIPS?

Two operands, one result!

MIPS instruction types:

R-type – Rs = A, Rt = B, Rd = Y

I-type – Rs = A, Rt = B or Y, Imm = B or branch3

J-type – Imm = memory offset (jump3)

3Separate adder for computing addresses

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Arithmetic-Logic Unit

What does the circuit of

the ALU look like?

Simple 4-bit ALU (74181

ALU)

75-logic gates, does 4-bit

add, sub, decrement,

AND, NAND, OR, NOR,

XOR, and shift

16-logical, 16-arithmetic

operations3

3CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=168473

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Arithmetic-Logic Unit

Note – The 74181 is a 4-bit ALU, with no floating point,

multiplication, or division support

32-bit ALU w/ these support can be quite large

Using abstraction is the only way to allow integration of all parts!

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Conclusion

What do I need to know about computer arithmetic?

Actual hardware implementation of arithmetic matters for

performance

Structure of the implementation (e.g. operands, result, status in,

status out) matter for design

In this class, (specifically in chapter 4) we focus on design

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Conclusion

For design, the following conclusions are important:

• Small number of generic computation formats (e.g. R-type,I-type)

• Simple operations can have status codes that are important(e.g. Overflow)

• Complex computations require extra care (e.g. floating point,multiply, divide)

• Specialized computations require extra care (e.g. Multimedia,SIMD instructions)

Final remark – For integer arithmetic, simple operations dominate

the instructions in SPEC programs

These simple instructions will be used to design the CPU in the

remainder of the course.

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Fallacies/Pitfalls

Pitfall 1 – Shifting right is not division for signed numbers!

srl – shift right logical – zero pads on the left, changing a negative

to a positive number

Even sra (shift right arithmetic) may not produce correct results

Pitfall 2 – Floating point addition not associative

Associative property – A + B + C = A + C + B

Floating point = approximation, meaning these may yield different

results

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Fallacies/Pitfalls

Fallacy 1 – Parallel execution strategies that work for integer data

types also work for floating point

Related to Pitfall 2, parallel execution is not associative, so

distributing work must be handled with care.

Pitfall 3 – The MIPS instruction add immediate unsigned (addiu)

sign-extends its 16-bit immediate field

There is no subtract immediate, so this must have the ability to

represent negative numbers

Only used for addition where we don’t care about overflow

(immediate is sign extended)

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