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If you study modern microprocessors, you may notice the similarities between them and old mechanical computing devices such as mechanical calculators. More specifically, the arithmetic logic unit (ALU) in the microprocessor has many similarities with the old mechanical calculator. ALU can be thought of as the calculator of a microprocessor. It reads the numbers stored in the register (register), performs addition, subtraction or shift operations, and stores the results back to another register.

The arithmetic logic unit within the microprocessor performs arithmetic operations. The registers are R1, R2, and R3.

These basic operations are exactly the same as those performed by the calculator. It has three registers to hold the numbers used in the calculation:

Input register: a 5-bit register, you can pull the lever to define the input number.

Cumulative register: a 13-bit register used to hold the results of multiple calculations.

Count register: an 8-digit register that records how much addition you have made.

The Odner calculator, invented in Russia in 1873, I will not explain the microprocessor ALU or the old mechanical calculator in detail here. We will travel further back to the abacus. This different form of manual computing machine existed long before Roman times.

Why do you want to know about the abacus and its operation? Because understanding the abacus allows you to see something deeper, linking the way numbers were processed thousands of years ago to the methods of modern microprocessors. Modern microprocessors are so complex that it is easy to "go astray" when you try to understand them.

The abacus is like the ALU of a microprocessor. Modern computers are based on binary number systems, where each number is called a bit. The abacus shown in the following figure can handle up to four decimal digits. Each column on the abacus represents one of the numbers.

A diagram of a typical abacus. Each column represents a number in the decimal system. The first column from the right is a bit, the second column is ten, the third column is 100, and so on. This means that if I want to represent the number 4023 on the abacus, I will arrange the beads as shown above. With this arrangement, addition and subtraction can be easily implemented. If I want to add 12, all I have to do is Dora two beads in one position and Dora one bead in ten.

Addition and subtraction is very simple, so let's see how to multiply on the abacus. The way of multiplication is very similar to that of old mechanical calculators and early microprocessors, which have only ALU and no dedicated multiplication hardware.

With the abacus multiplication we will do a simple multiplication: 32 × 4, the result should be 128. To do this, we specify a specific meaning for each column. The first two columns are converted to counter registers, while the third column is defined as the input register. Finally, we use the last two columns as accumulation registers, which will hold the final result.

Multiplication can be thought of as performing multiple addition operations. In this case, we will add 4 32 times.

Multiplied by a bit, let's add the accumulator four times first. Four beads in a single column move down. To record the number of four beads we have added, we move one bead from the counter. In fact, we count down from 32 to 0.

We add another 4 to the accumulator and the counter is reduced to 30. At this point, the value of the accumulator is 8. We have to add another 30 and four.

Multiplying the ten-bit column and moving four beads on the individual column more than 30 times can be tedious. Fortunately, we don't have to do this. Instead, we perform a shift operation to speed up the addition. We count each moving bead as ten times four. How do you do that? Every time we move a bead on the ten bits of the counter, we move four beads on the ten bits of the accumulator. By doing this once, the counter is reduced from 30 to 20 and the accumulator is increased to 48 (8 + 4 × 10).

We repeat the same operation again, reducing the counter from 20 to 10 and adding another 40 to the accumulator, which now has a value of 88 (48 + 4 × 10).

Note that in the image above, we cannot move four more beads down, because there are only two beads left in the ten-bit column of the accumulator. We need to use the third column and use an accumulator to represent 100 digits. We traded ten beads out of ten for one bead in a hundred.

In addition, a total of 12 beads are needed for ten people, and we exchange it for one bead for a hundred and two beads for ten. At this point, we have spent all the beads on the counter, and we can see that the final result is 128.

Shift operation abacus, mechanical calculator or ALU can not do multiplication and division directly. However, we can easily multiply by multiplying 10. Multiplying 10,100,100 or 1000 by an input number is easy. All you have to do is move 1, 2 or 3 columns of beads. To do 2 × 4, you need to move 4 beads in each column twice. If you want to perform 20 x 4, move the four beads in the ten-bit column twice. If it is 200 x 4, this means moving 8 beads in the 100-bit column. What you are doing is actually a combination of addition and shift operations.

If you look at the mechanical calculator, you will see that the accumulation register is on a slider. Slide a notch to the left and the number added from the input register will be tenfold. Slide two grooves to the left and the number you enter will be 100 times that of the original. We can also slide to the right to make the added numbers smaller. An interesting effect is that the slider shifts, which also affects the count register. Suppose you specify an input register of 42. Next, slide the rod two grooves to the left. When you turn the handle to add input to the accumulator, you will add 4200. 100 is displayed on the count register. So, it looks like you twisted the handle a hundred times. In other words, shift allows us to significantly reduce the number of times we perform addition.

These shifts are no different from what we have implemented on the abacus. The key difference is that you need to update the accumulator and counter by moving the beads manually.

The use of shift operations in microprocessors helps us speed up the multiplication and division operations on the microprocessor ALU. Because modern microprocessors operate with binary digital systems, shifts operate on multiple inputs of 2. So instead of multiplying 10, 100, 1000 or 10000 by the input, we multiply the input by 2, 4, 8, 16, 32, and so on.

This knowledge is actually useful when you start programming. In the early 1990s, the multiplication of personal computers was very slow. We must use multiplication to calculate the position of pixels at specific x, y coordinates in the graphics memory. Because the screen resolution is 320 x 240, you have to multiply the y coordinates by 320 and then add the x coordinates to get the location in the screen memory.

The point is, I have to multiply any number y by 320 countless times. This is very slow. The solution is to change the calculation to a combination of shift and addition. The following is a brief introduction:

Replacing multiplication with two shift operations and an addition gives a key speed boost.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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