Tracing arithmetic expressions

0:00- [Presenter] How does the computer evaluate expressions with multiple operators, multiple function calls, or even nested function calls? That's a function call inside the parentheses of another function call. To examine this order of operations, let's trace a program with some complex arithmetic expressions.

0:16As always, the first thing the computer does when we run the program is load the first instruction into its working memory. When it sees an instruction like this where there's nested function calls, it's going to search for the innermost parentheses first. Then it starts evaluating from the inside out.

0:32Here in the innermost parentheses, we have the string. How many servings? This is already a single value, so it doesn't need simplifying. Okay, so the computer's going to peek outside of those parentheses and ask, "What did you want me to do with this value, How many servings?"

0:46And it sees the input function. The computer takes that value, displays it in the console, pops up a prompt, and then waits for the user to enter something. The computer's done with this part of the instruction, but it won't continue evaluating until it gets an answer back from the user.

1:02Let's say I type in five and hit enter. Where does that five go? Well, the input function returns out whatever value the user entered. Remember that the input function always returns that entered value as a string. So this isn't the integer five, it's the string five.

1:19Great. Now we have a much simpler expression. The computer goes to the parentheses first, sees that it's already a single value, so there's nothing to do. Peeks outside the parentheses and sees the int function. The int function takes that string five, casts it to an integer and returns out the value five.

1:36Now we just have a standard assignment statement. This statement tells the computer to go off to its memory, allocate a new chunk of it to remember a new value. Then it tags it with the name servings and sticks that value five in there. This instruction is now complete. Computer did everything.

1:52So it's gonna clear out that working memory and look for that next line of the program. Okay, we load the second instruction into working memory. Looks like we're just doing some math here. We follow PEMDAS or MDAS, that's parentheses, exponents, then multiplication and division, and then addition and subtraction.

2:09So we've got parentheses first, the computer evaluates six minus two. Then we're left with only multiplication and division operators, which are at the same level of precedence. So the computer just evaluates left to right. First thing it sees is this variable servings.

2:23It needs to go off and access that chunk of memory that was associated with the name servings. So it goes here, see servings, follows that line, and it finds the value five. So we substitute that in here. 5 times 4 is 20. Then we have 20 divided by 3. This is the float division operator.

2:40So we get 6.66666 repeating. Now it sees that equal sign and it says, "Ah, you want me to assign this value I finally have to the variable spices." It checks its memory and sees that it has no recollection of a spices. So it allocates a new trunk, tags it with a name and sticks that value in there for later.

2:59All done. Clear it out. This next line is blank so the computer skips that, and then the line after that is a comment because it starts with this hashtag character. So the computer ignores that too. Comments are for humans, not for computers. Load in that next line, and here we have two sets of parentheses, but those parentheses are at the same level of precedence.

3:21They're not one inside of the other. So that means we're just gonna evaluate them left to right. Starting with those left parentheses, the computer is going to grab that value of spices from its memory, substitute it in. Now, it has two values inside these parentheses separated by a comma, but they're both simplified all the way down.

3:39So we're good here. We pop out the parentheses and we see the min function. And the min function returns out the minimum of those two input values. Now it jumps to the next set of parentheses, substitute in the value of servings. That's five. Two single values.

3:55Pops outside the parentheses sees that it wants to take the maximum. Five is greater than four. And now we just have two numbers to add together. We have a single value now on the right hand side of the equal sign so we can go ahead and do that assignment. Checks in its memory, and sees, "Oh, I already know a spices, so I'm just gonna erase what's in there and replace it with this new value."

4:16Cool. Clear out working memory, move on. Next line now. That's a lot of parentheses. We're starting in and we're moving out. We've got an expression here that needs simplifying. We're substituting in spices. And then we're adding three to get that down to a single value.

4:33Now, everything inside here has been simplified, so we peek out and we see the round function. The round function tells the computer to take that first value and round it to the number of decimal places that the second value says. Here, we're rounding to two decimal places.

4:47Now we're in the next innermost parentheses. We already have a single value, so we pop outside those parentheses and we see the string function. The string function casts this to a string and returns out that value. Now we're in the last set of parentheses. We're concatenating two strings, so we just smoosh them together.

5:08And what should it do with this value? Oh, print it. There's no next line so it terminates the program execution. And when it terminates, that means the computer also forgets everything in its short term memory so all these values go away too. That's it for this program.

5:23Remember, if you ever have a complex expression, the computer always evaluates innermost parentheses to outermost.