Unit 6: The Calculator

Do both, then pick one

A CPU does not have a separate chip for every sum. It has one chip that can do several things, with wires that say which.

The trick is the Mux from unit 3. Work out the And *and* the sum — both, always, at the same time — then let one control wire decide which answer leaves the box.

Here f chooses: 0 gives you And, 1 gives you plus.

HDL

CHIP Calc {
    IN x3, x2, x1, x0, y3, y2, y1, y0, f;
    OUT o3, o2, o1, o0;
    PARTS:
    And4(a3=x3, a2=x2, a1=x1, a0=x0, b3=y3, b2=y2, b1=y1, b0=y0,
         o3=and3, o2=and2, o1=and1, o0=and0);
    Add4(a3=x3, a2=x2, a1=x1, a0=x0, b3=y3, b2=y2, b1=y1, b0=y0,
         s3=sum3, s2=sum2, s1=sum1, s0=sum0);
    Mux4(a3=and3, a2=and2, a1=and1, a0=and0,
         b3=sum3, b2=sum2, b1=sum1, b0=sum0, sel=f,
         o3=o3, o2=o2, o1=o1, o0=o0);
}

USE Calc;
SET x3=0, x2=1, x1=1, x0=0, y3=0, y2=0, y1=1, y0=1, f=0;
SHOW;
SET f=1;
SHOW;

It prints

o3=0 o2=0 o1=1 o0=0
o3=1 o2=0 o1=0 o0=1

The real one has six controls

A real ALU adds four more switches, all of them tiny, in front of and behind the sum:

- zx — make x zero first
- nx — flip x first
- zy, ny — the same two, for y
- f — 0 for And, 1 for plus
- no — flip the answer on the way out

Six switches, 64 combinations. And that handful of Nots and Muxes is enough to compute almost everything a CPU ever needs.

Zero plus zero

Set zx and zy and both inputs become 0, whatever you fed in. Then f=1 adds them. The answer is 0 — and it is 0 no matter what x and y were.

That is how a CPU produces the number 0 without storing one anywhere.

HDL

CHIP Zero {
    IN x3, x2, x1, x0, y3, y2, y1, y0;
    OUT o3, o2, o1, o0;
    PARTS:
    ALU4(x3=x3, x2=x2, x1=x1, x0=x0, y3=y3, y2=y2, y1=y1, y0=y0,
         zx=1, nx=0, zy=1, ny=0, f=1, no=0,
         o3=o3, o2=o2, o1=o1, o0=o0);
}

USE Zero;
SET x3=1, x2=1, x1=1, x0=1, y3=1, y2=0, y1=1, y0=0;
SHOW;

It prints

o3=0 o2=0 o1=0 o0=0

Adding one, the ALU way

Here is a strange-looking setting: zx=0, nx=1, zy=1, ny=1, f=1, no=1. That works out as x + 1.

It is not obvious, and it does not need to be. What matters is that six switches in the right position turn one adder into "add one" — no extra chip.

HDL

CHIP Plus1 {
    IN x3, x2, x1, x0;
    OUT o3, o2, o1, o0;
    PARTS:
    ALU4(x3=x3, x2=x2, x1=x1, x0=x0, y3=0, y2=0, y1=0, y0=0,
         zx=0, nx=1, zy=1, ny=1, f=1, no=1,
         o3=o3, o2=o2, o1=o1, o0=o0);
}

USE Plus1;
SET x3=0, x2=1, x1=0, x0=1;
SHOW;

It prints

o3=0 o2=1 o1=1 o0=0

There is no subtractor

Nowhere in a CPU is there a chip that subtracts. There does not need to be one.

You learnt in unit 4 that flipping a number and adding one makes it negative. So x − y is just x + (−y) — the same adder, with the flip switches turned on.

zx=0, nx=1, zy=0, ny=0, f=1, no=1 is the setting for it.

HDL

CHIP Sub {
    IN x3, x2, x1, x0, y3, y2, y1, y0;
    OUT o3, o2, o1, o0;
    PARTS:
    ALU4(x3=x3, x2=x2, x1=x1, x0=x0, y3=y3, y2=y2, y1=y1, y0=y0,
         zx=0, nx=1, zy=0, ny=0, f=1, no=1,
         o3=o3, o2=o2, o1=o1, o0=o0);
}

USE Sub;
SET x3=0, x2=1, x1=1, x0=1, y3=0, y2=0, y1=1, y0=0;
SHOW;

It prints

o3=0 o2=1 o1=0 o0=1

Two extra wires that decide everything

The ALU has two spare outputs that are not part of the answer:

- zr — 1 when the answer came out as zero
- ng — 1 when the answer is negative (its top wire is 1)

These are what if is made of. A program that says "if a equals b" really subtracts them and looks at zr.

HDL

CHIP Compare {
    IN x3, x2, x1, x0, y3, y2, y1, y0;
    OUT equal;
    PARTS:
    ALU4(x3=x3, x2=x2, x1=x1, x0=x0, y3=y3, y2=y2, y1=y1, y0=y0,
         zx=0, nx=1, zy=0, ny=0, f=1, no=1,
         zr=equal);
}

USE Compare;
SET x3=0, x2=1, x1=0, x0=1, y3=0, y2=1, y1=0, y0=1;
SHOW equal;
SET y0=0;
SHOW equal;

It prints

equal=1
equal=0

Half a computer

You now have the part of a CPU that *does* things. Give it two numbers and six switches and it will add, subtract, compare, negate or zero them.

What it cannot do is remember anything. Turn the power off — or just change the inputs — and everything is gone. That is the next unit, and it needs a genuinely new idea: time.