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# 11.13 Using the FPU

Strangely enough, most of assembly language literature does not even mention the existence of the FPU, or floating point unit, let alone discuss programming it.

Yet, never does assembly language shine more than when we create highly optimized FPU code by doing things that can be done only in assembly language.

## 11.13.1 Organization of the FPU

The FPU consists of 8 80-bit floating-point registers. These are organized in a stack fashion--you can `push` a value on TOS (top of stack) and you can `pop` it.

That said, the assembly language op codes are not `push` and `pop` because those are already taken.

You can `push` a value on TOS by using `fld`, `fild`, and `fbld`. Several other op codes let you `push` many common constants--such as pi--on the TOS.

Similarly, you can `pop` a value by using `fst`, `fstp`, `fist`, `fistp`, and `fbstp`. Actually, only the op codes that end with a p will literally `pop` the value, the rest will `store` it somewhere else without removing it from the TOS.

We can transfer the data between the TOS and the computer memory either as a 32-bit, 64-bit, or 80-bit real, a 16-bit, 32-bit, or 64-bit integer, or an 80-bit packed decimal.

The 80-bit packed decimal is a special case of binary coded decimal which is very convenient when converting between the ASCII representation of data and the internal data of the FPU. It allows us to use 18 significant digits.

No matter how we represent data in the memory, the FPU always stores it in the 80-bit real format in its registers.

Its internal precision is at least 19 decimal digits, so even if we choose to display results as ASCII in the full 18-digit precision, we are still showing correct results.

We can perform mathematical operations on the TOS: We can calculate its sine, we can scale it (i.e., we can multiply or divide it by a power of 2), we can calculate its base-2 logarithm, and many other things.

We can also multiply or divide it by, add it to, or subtract it from, any of the FPU registers (including itself).

The official Intel op code for the TOS is `st`, and for the registers `st(0)`-`st(7)`. `st` and `st(0)`, then, refer to the same register.

For whatever reasons, the original author of nasm has decided to use different op codes, namely `st0`-`st7`. In other words, there are no parentheses, and the TOS is always `st0`, never just `st`.

### 11.13.1.1 The Packed Decimal Format

The packed decimal format uses 10 bytes (80 bits) of memory to represent 18 digits. The number represented there is always an integer.

> Tip: You can use it to get decimal places by multiplying the TOS by a power of 10 first.

The highest bit of the highest byte (byte 9) is the sign bit: If it is set, the number is negative, otherwise, it is positive. The rest of the bits of this byte are unused/ignored.

The remaining 9 bytes store the 18 digits of the number: 2 digits per byte.

The more significant digit is stored in the high nibble (4 bits), the less significant digit in the low nibble.

That said, you might think that `-1234567` would be stored in the memory like this (using hexadecimal notation):

    80 00 00 00 00 00 01 23 45 67

Alas it is not! As with everything else of Intel make, even the packed decimal is little-endian.

That means our `-1234567` is stored like this:

    67 45 23 01 00 00 00 00 00 80

Remember that, or you will be pulling your hair out in desperation!

> Note: The book to read--if you can find it--is Richard Startz' 8087/80287/80387 for the IBM PC & Compatibles. Though it does seem to take the fact about the little-endian storage of the packed decimal for granted. I kid you not about the desperation of trying to figure out what was wrong with the filter I show below before it occurred to me I should try the little-endian order even for this type of data.

## 11.13.2 Excursion to Pinhole Photography

To write meaningful software, we must not only understand our programming tools, but also the field we are creating software for.

Our next filter will help us whenever we want to build a pinhole camera, so, we need some background in pinhole photography before we can continue.

### 11.13.2.1 The Camera

The easiest way to describe any camera ever built is as some empty space enclosed in some lightproof material, with a small hole in the enclosure.

The enclosure is usually sturdy (e.g., a box), though sometimes it is flexible (the bellows). It is quite dark inside the camera. However, the hole lets light rays in through a single point (though in some cases there may be several). These light rays form an image, a representation of whatever is outside the camera, in front of the hole.

If some light sensitive material (such as film) is placed inside the camera, it can capture the image.

The hole often contains a lens, or a lens assembly, often called the objective.

### 11.13.2.2 The Pinhole

But, strictly speaking, the lens is not necessary: The original cameras did not use a lens but a pinhole. Even today, pinholes are used, both as a tool to study how cameras work, and to achieve a special kind of image.

The image produced by the pinhole is all equally sharp. Or blurred. There is an ideal size for a pinhole: If it is either larger or smaller, the image loses its sharpness.

### 11.13.2.3 Focal Length

This ideal pinhole diameter is a function of the square root of focal length, which is the distance of the pinhole from the film.

       D = PC * sqrt(FL)

In here, `D` is the ideal diameter of the pinhole, `FL` is the focal length, and `PC` is a pinhole constant. According to Jay Bender, its value is `0.04`, while Kenneth Connors has determined it to be `0.037`. Others have proposed other values. Plus, this value is for the daylight only: Other types of light will require a different constant, whose value can only be determined by experimentation.

### 11.13.2.4 The F-Number

The f-number is a very useful measure of how much light reaches the film. A light meter can determine that, for example, to expose a film of specific sensitivity with f5.6 may require the exposure to last 1/1000 sec.

It does not matter whether it is a 35-mm camera, or a 6x9cm camera, etc. As long as we know the f-number, we can determine the proper exposure.

The f-number is easy to calculate:

       F = FL / D

In other words, the f-number equals the focal length divided by the diameter of the pinhole. It also means a higher f-number either implies a smaller pinhole or a larger focal distance, or both. That, in turn, implies, the higher the f-number, the longer the exposure has to be.

Furthermore, while pinhole diameter and focal distance are one-dimensional measurements, both, the film and the pinhole, are two-dimensional. That means that if you have measured the exposure at f-number `A` as `t`, then the exposure at f-number `B` is:

       t * (B / A)²

### 11.13.2.5 Normalized F-Number

While many modern cameras can change the diameter of their pinhole, and thus their f-number, quite smoothly and gradually, such was not always the case.

To allow for different f-numbers, cameras typically contained a metal plate with several holes of different sizes drilled to them.

Their sizes were chosen according to the above formula in such a way that the resultant f-number was one of standard f-numbers used on all cameras everywhere. For example, a very old Kodak Duaflex IV camera in my possession has three such holes for f-numbers 8, 11, and 16.

A more recently made camera may offer f-numbers of 2.8, 4, 5.6, 8, 11, 16, 22, and 32 (as well as others). These numbers were not chosen arbitrarily: They all are powers of the square root of 2, though they may be rounded somewhat.

### 11.13.2.6 The F-Stop

A typical camera is designed in such a way that setting any of the normalized f-numbers changes the feel of the dial. It will naturally stop in that position. Because of that, these positions of the dial are called f-stops.

Since the f-numbers at each stop are powers of the square root of 2, moving the dial by 1 stop will double the amount of light required for proper exposure. Moving it by 2 stops will quadruple the required exposure. Moving the dial by 3 stops will require the increase in exposure 8 times, etc.

## 11.13.3 Designing the Pinhole Software

We are now ready to decide what exactly we want our pinhole software to do.

### 11.13.3.1 Processing Program Input

Since its main purpose is to help us design a working pinhole camera, we will use the focal length as the input to the program. This is something we can determine without software: Proper focal length is determined by the size of the film and by the need to shoot "regular" pictures, wide angle pictures, or telephoto pictures.

Most of the programs we have written so far worked with individual characters, or bytes, as their input: The hex program converted individual bytes into a hexadecimal number, the csv program either let a character through, or deleted it, or changed it to a different character, etc.

One program, ftuc used the state machine to consider at most two input bytes at a time.

But our pinhole program cannot just work with individual characters, it has to deal with larger syntactic units.

For example, if we want the program to calculate the pinhole diameter (and other values we will discuss later) at the focal lengths of `100 mm`, `150 mm`, and `210 mm`, we may want to enter something like this:

    100, 150, 210

Our program needs to consider more than a single byte of input at a time. When it sees the first `1`, it must understand it is seeing the first digit of a decimal number. When it sees the `0` and the other `0`, it must know it is seeing more digits of the same number.

When it encounters the first comma, it must know it is no longer receiving the digits of the first number. It must be able to convert the digits of the first number into the value of `100`. And the digits of the second number into the value of `150`. And, of course, the digits of the third number into the numeric value of `210`.

We need to decide what delimiters to accept: Do the input numbers have to be separated by a comma? If so, how do we treat two numbers separated by something else?

Personally, I like to keep it simple. Something either is a number, so I process it. Or it is not a number, so I discard it. I do not like the computer complaining about me typing in an extra character when it is obvious that it is an extra character. Duh!

Plus, it allows me to break up the monotony of computing and type in a query instead of just a number:

    What is the best pinhole diameter for the focal length of 150?

There is no reason for the computer to spit out a number of complaints:

    Syntax error: What
    Syntax error: is
    Syntax error: the
    Syntax error: best

Et cetera, et cetera, et cetera.

Secondly, I like the `#` character to denote the start of a comment which extends to the end of the line. This does not take too much effort to code, and lets me treat input files for my software as executable scripts.

In our case, we also need to decide what units the input should come in: We choose millimeters because that is how most photographers measure the focus length.

Finally, we need to decide whether to allow the use of the decimal point (in which case we must also consider the fact that much of the world uses a decimal comma).

In our case allowing for the decimal point/comma would offer a false sense of precision: There is little if any noticeable difference between the focus lengths of `50` and `51`, so allowing the user to input something like `50.5` is not a good idea. This is my opinion, mind you, but I am the one writing this program. You can make other choices in yours, of course.

### 11.13.3.2 Offering Options

The most important thing we need to know when building a pinhole camera is the diameter of the pinhole. Since we want to shoot sharp images, we will use the above formula to calculate the pinhole diameter from focal length. As experts are offering several different values for the `PC` constant, we will need to have the choice.

It is traditional in UNIX® programming to have two main ways of choosing program parameters, plus to have a default for the time the user does not make a choice.

Why have two ways of choosing?

One is to allow a (relatively) permanent choice that applies automatically each time the software is run without us having to tell it over and over what we want it to do.

The permanent choices may be stored in a configuration file, typically found in the user's home directory. The file usually has the same name as the application but is started with a dot. Often "rc" is added to the file name. So, ours could be `~/.pinhole` or `~/.pinholerc`. (The `~/` means current user's home directory.)

The configuration file is used mostly by programs that have many configurable parameters. Those that have only one (or a few) often use a different method: They expect to find the parameter in an environment variable. In our case, we might look at an environment variable named `PINHOLE`.

Usually, a program uses one or the other of the above methods. Otherwise, if a configuration file said one thing, but an environment variable another, the program might get confused (or just too complicated).

Because we only need to choose one such parameter, we will go with the second method and search the environment for a variable named `PINHOLE`.

The other way allows us to make ad hoc decisions: "Though I usually want you to use 0.039, this time I want 0.03872." In other words, it allows us to override the permanent choice.

This type of choice is usually done with command line parameters.

Finally, a program always needs a default. The user may not make any choices. Perhaps he does not know what to choose. Perhaps he is "just browsing." Preferably, the default will be the value most users would choose anyway. That way they do not need to choose. Or, rather, they can choose the default without an additional effort.

Given this system, the program may find conflicting options, and handle them this way:

1. If it finds an ad hoc choice (e.g., command line parameter), it should accept that choice. It must ignore any permanent choice and any default.

2. Otherwise, if it finds a permanent option (e.g., an environment variable), it should accept it, and ignore the default.

3. Otherwise, it should use the default.

We also need to decide what format our `PC` option should have.

At first site, it seems obvious to use the `PINHOLE=0.04` format for the environment variable, and `-p0.04` for the command line.

Allowing that is actually a security risk. The `PC` constant is a very small number. Naturally, we will test our software using various small values of `PC`. But what will happen if someone runs the program choosing a huge value?

It may crash the program because we have not designed it to handle huge numbers.

Or, we may spend more time on the program so it can handle huge numbers. We might do that if we were writing commercial software for computer illiterate audience.

Or, we might say, "Tough! The user should know better.""

Or, we just may make it impossible for the user to enter a huge number. This is the approach we will take: We will use an implied 0. prefix.

In other words, if the user wants `0.04`, we will expect him to type `-p04`, or set `PINHOLE=04` in his environment. So, if he says `-p9999999`, we will interpret it as `0.9999999`--still ridiculous but at least safer.

Secondly, many users will just want to go with either Bender's constant or Connors' constant. To make it easier on them, we will interpret `-b` as identical to `-p04`, and `-c` as identical to `-p037`.

### 11.13.3.3 The Output

We need to decide what we want our software to send to the output, and in what format.

Since our input allows for an unspecified number of focal length entries, it makes sense to use a traditional database-style output of showing the result of the calculation for each focal length on a separate line, while separating all values on one line by a `tab` character.

Optionally, we should also allow the user to specify the use of the CSV format we have studied earlier. In this case, we will print out a line of comma-separated names describing each field of every line, then show our results as before, but substituting a `comma` for the `tab`.

We need a command line option for the CSV format. We cannot use `-c` because that already means use Connors' constant. For some strange reason, many web sites refer to CSV files as "Excel spreadsheet" (though the CSV format predates Excel). We will, therefore, use the `-e` switch to inform our software we want the output in the CSV format.

We will start each line of the output with the focal length. This may sound repetitious at first, especially in the interactive mode: The user types in the focal length, and we are repeating it.

But the user can type several focal lengths on one line. The input can also come in from a file or from the output of another program. In that case the user does not see the input at all.

By the same token, the output can go to a file which we will want to examine later, or it could go to the printer, or become the input of another program.

So, it makes perfect sense to start each line with the focal length as entered by the user.

No, wait! Not as entered by the user. What if the user types in something like this:

    00000000150

Clearly, we need to strip those leading zeros.

So, we might consider reading the user input as is, converting it to binary inside the FPU, and printing it out from there.

But...

What if the user types something like this:

    17459765723452353453534535353530530534563507309676764423

Ha! The packed decimal FPU format lets us input 18-digit numbers. But the user has entered more than 18 digits. How do we handle that?

Well, we could modify our code to read the first 18 digits, enter it to the FPU, then read more, multiply what we already have on the TOS by 10 raised to the number of additional digits, then `add` to it.

Yes, we could do that. But in this program it would be ridiculous (in a different one it may be just the thing to do): Even the circumference of the Earth expressed in millimeters only takes 11 digits. Clearly, we cannot build a camera that large (not yet, anyway).

So, if the user enters such a huge number, he is either bored, or testing us, or trying to break into the system, or playing games--doing anything but designing a pinhole camera.

What will we do?

We will slap him in the face, in a manner of speaking:

    17459765723452353453534535353530530534563507309676764423   ??? ??? ??? ??? ???

To achieve that, we will simply ignore any leading zeros. Once we find a non-zero digit, we will initialize a counter to `0` and start taking three steps:

1. Send the digit to the output.

2. Append the digit to a buffer we will use later to produce the packed decimal we can send to the FPU.

3. Increase the counter.

Now, while we are taking these three steps, we also need to watch out for one of two conditions:

- If the counter grows above 18, we stop appending to the buffer. We continue reading the digits and sending them to the output.

- If, or rather when, the next input character is not a digit, we are done inputting for now.

  Incidentally, we can simply discard the non-digit, unless it is a `#`, which we must return to the input stream. It starts a comment, so we must see it after we are done producing output and start looking for more input.

That still leaves one possibility uncovered: If all the user enters is a zero (or several zeros), we will never find a non-zero to display.

We can determine this has happened whenever our counter stays at `0`. In that case we need to send `0` to the output, and perform another "slap in the face":

    0  ??? ??? ??? ??? ???

Once we have displayed the focal length and determined it is valid (greater than `0` but not exceeding 18 digits), we can calculate the pinhole diameter.

It is not by coincidence that pinhole contains the word pin. Indeed, many a pinhole literally is a pin hole, a hole carefully punched with the tip of a pin.

That is because a typical pinhole is very small. Our formula gets the result in millimeters. We will multiply it by `1000`, so we can output the result in microns.

At this point we have yet another trap to face: Too much precision.

Yes, the FPU was designed for high precision mathematics. But we are not dealing with high precision mathematics. We are dealing with physics (optics, specifically).

Suppose we want to convert a truck into a pinhole camera (we would not be the first ones to do that!). Suppose its box is `12` meters long, so we have the focal length of `12000`. Well, using Bender's constant, it gives us square root of `12000` multiplied by `0.04`, which is `4.381780460` millimeters, or `4381.780460` microns.

Put either way, the result is absurdly precise. Our truck is not exactly `12000` millimeters long. We did not measure its length with such a precision, so stating we need a pinhole with the diameter of `4.381780460` millimeters is, well, deceiving. `4.4` millimeters would do just fine.

> Note: I "only" used ten digits in the above example. Imagine the absurdity of going for all 18!

We need to limit the number of significant digits of our result. One way of doing it is by using an integer representing microns. So, our truck would need a pinhole with the diameter of `4382` microns. Looking at that number, we still decide that `4400` microns, or `4.4` millimeters is close enough.

Additionally, we can decide that no matter how big a result we get, we only want to display four significant digits (or any other number of them, of course). Alas, the FPU does not offer rounding to a specific number of digits (after all, it does not view the numbers as decimal but as binary).

We, therefore, must devise an algorithm to reduce the number of significant digits.

Here is mine (I think it is awkward--if you know a better one, please, let me know):

1. Initialize a counter to `0`.

2. While the number is greater than or equal to `10000`, divide it by `10` and increase the counter.

3. Output the result.

4. While the counter is greater than `0`, output `0` and decrease the counter.

> Note: The `10000` is only good if you want four significant digits. For any other number of significant digits, replace `10000` with `10` raised to the number of significant digits.

We will, then, output the pinhole diameter in microns, rounded off to four significant digits.

At this point, we know the focal length and the pinhole diameter. That means we have enough information to also calculate the f-number.

We will display the f-number, rounded to four significant digits. Chances are the f-number will tell us very little. To make it more meaningful, we can find the nearest normalized f-number, i.e., the nearest power of the square root of 2.

We do that by multiplying the actual f-number by itself, which, of course, will give us its `square`. We will then calculate its base-2 logarithm, which is much easier to do than calculating the base-square-root-of-2 logarithm! We will round the result to the nearest integer. Next, we will raise 2 to the result. Actually, the FPU gives us a good shortcut to do that: We can use the `fscale` op code to "scale" 1, which is analogous to `shift`ing an integer left. Finally, we calculate the square root of it all, and we have the nearest normalized f-number.

If all that sounds overwhelming--or too much work, perhaps--it may become much clearer if you see the code. It takes 9 op codes altogether:

       fmul    st0, st0
        fld1
        fld st1
        fyl2x
        frndint
        fld1
        fscale
        fsqrt
        fstp    st1

The first line, `fmul st0, st0`, squares the contents of the TOS (top of the stack, same as `st`, called `st0` by nasm). The `fld1` pushes `1` on the TOS.

The next line, `fld st1`, pushes the square back to the TOS. At this point the square is both in `st` and `st(2)` (it will become clear why we leave a second copy on the stack in a moment). `st(1)` contains `1`.

Next, `fyl2x` calculates base-2 logarithm of `st` multiplied by `st(1)`. That is why we placed `1` on `st(1)` before.

At this point, `st` contains the logarithm we have just calculated, `st(1)` contains the square of the actual f-number we saved for later.

`frndint` rounds the TOS to the nearest integer. `fld1` pushes a `1`. `fscale` shifts the `1` we have on the TOS by the value in `st(1)`, effectively raising 2 to `st(1)`.

Finally, `fsqrt` calculates the square root of the result, i.e., the nearest normalized f-number.

We now have the nearest normalized f-number on the TOS, the base-2 logarithm rounded to the nearest integer in `st(1)`, and the square of the actual f-number in `st(2)`. We are saving the value in `st(2)` for later.

But we do not need the contents of `st(1)` anymore. The last line, `fstp st1`, places the contents of `st` to `st(1)`, and pops. As a result, what was `st(1)` is now `st`, what was `st(2)` is now `st(1)`, etc. The new `st` contains the normalized f-number. The new `st(1)` contains the square of the actual f-number we have stored there for posterity.

At this point, we are ready to output the normalized f-number. Because it is normalized, we will not round it off to four significant digits, but will send it out in its full precision.

The normalized f-number is useful as long as it is reasonably small and can be found on our light meter. Otherwise we need a different method of determining proper exposure.

Earlier we have figured out the formula of calculating proper exposure at an arbitrary f-number from that measured at a different f-number.

Every light meter I have ever seen can determine proper exposure at f5.6. We will, therefore, calculate an "f5.6 multiplier," i.e., by how much we need to multiply the exposure measured at f5.6 to determine the proper exposure for our pinhole camera.

From the above formula we know this factor can be calculated by dividing our f-number (the actual one, not the normalized one) by `5.6`, and squaring the result.

Mathematically, dividing the square of our f-number by the square of `5.6` will give us the same result.

Computationally, we do not want to square two numbers when we can only square one. So, the first solution seems better at first.

But...

`5.6` is a constant. We do not have to have our FPU waste precious cycles. We can just tell it to divide the square of the f-number by whatever `5.6²` equals to. Or we can divide the f-number by `5.6`, and then square the result. The two ways now seem equal.

But, they are not!

Having studied the principles of photography above, we remember that the `5.6` is actually square root of 2 raised to the fifth power. An irrational number. The square of this number is exactly `32`.

Not only is `32` an integer, it is a power of 2. We do not need to divide the square of the f-number by `32`. We only need to use `fscale` to shift it right by five positions. In the FPU lingo it means we will `fscale` it with `st(1)` equal to `-5`. That is much faster than a division.

So, now it has become clear why we have saved the square of the f-number on the top of the FPU stack. The calculation of the f5.6 multiplier is the easiest calculation of this entire program! We will output it rounded to four significant digits.

There is one more useful number we can calculate: The number of stops our f-number is from f5.6. This may help us if our f-number is just outside the range of our light meter, but we have a shutter which lets us set various speeds, and this shutter uses stops.

Say, our f-number is 5 stops from f5.6, and the light meter says we should use 1/1000 sec. Then we can set our shutter speed to 1/1000 first, then move the dial by 5 stops.

This calculation is quite easy as well. All we have to do is to calculate the base-2 logarithm of the f5.6 multiplier we had just calculated (though we need its value from before we rounded it off). We then output the result rounded to the nearest integer. We do not need to worry about having more than four significant digits in this one: The result is most likely to have only one or two digits anyway.

## 11.13.4 FPU Optimizations

In assembly language we can optimize the FPU code in ways impossible in high languages, including C.

Whenever a C function needs to calculate a floating-point value, it loads all necessary variables and constants into FPU registers. It then does whatever calculation is required to get the correct result. Good C compilers can optimize that part of the code really well.

It "returns" the value by leaving the result on the TOS. However, before it returns, it cleans up. Any variables and constants it used in its calculation are now gone from the FPU.

It cannot do what we just did above: We calculated the square of the f-number and kept it on the stack for later use by another function.

We knew we would need that value later on. We also knew we had enough room on the stack (which only has room for 8 numbers) to store it there.

A C compiler has no way of knowing that a value it has on the stack will be required again in the very near future.

Of course, the C programmer may know it. But the only recourse he has is to store the value in a memory variable.

That means, for one, the value will be changed from the 80-bit precision used internally by the FPU to a C double (64 bits) or even single (32 bits).

That also means that the value must be moved from the TOS into the memory, and then back again. Alas, of all FPU operations, the ones that access the computer memory are the slowest.

So, whenever programming the FPU in assembly language, look for the ways of keeping intermediate results on the FPU stack.

We can take that idea even further! In our program we are using a constant (the one we named `PC`).

It does not matter how many pinhole diameters we are calculating: 1, 10, 20, 1000, we are always using the same constant. Therefore, we can optimize our program by keeping the constant on the stack all the time.

Early on in our program, we are calculating the value of the above constant. We need to divide our input by `10` for every digit in the constant.

It is much faster to multiply than to divide. So, at the start of our program, we divide `10` into `1` to obtain `0.1`, which we then keep on the stack: Instead of dividing the input by `10` for every digit, we multiply it by `0.1`.

By the way, we do not input `0.1` directly, even though we could. We have a reason for that: While `0.1` can be expressed with just one decimal place, we do not know how many binary places it takes. We, therefore, let the FPU calculate its binary value to its own high precision.

We are using other constants: We multiply the pinhole diameter by `1000` to convert it from millimeters to microns. We compare numbers to `10000` when we are rounding them off to four significant digits. So, we keep both, `1000` and `10000`, on the stack. And, of course, we reuse the `0.1` when rounding off numbers to four digits.

Last but not least, we keep `-5` on the stack. We need it to scale the square of the f-number, instead of dividing it by `32`. It is not by coincidence we load this constant last. That makes it the top of the stack when only the constants are on it. So, when the square of the f-number is being scaled, the `-5` is at `st(1)`, precisely where `fscale` expects it to be.

It is common to create certain constants from scratch instead of loading them from the memory. That is what we are doing with `-5`:

       fld1            ; TOS =  1
        fadd    st0, st0    ; TOS =  2
        fadd    st0, st0    ; TOS =  4
        fld1            ; TOS =  1
        faddp   st1, st0    ; TOS =  5
        fchs            ; TOS = -5

We can generalize all these optimizations into one rule: Keep repeat values on the stack!

> Tip: PostScript® is a stack-oriented programming language. There are many more books available about PostScript than about the FPU assembly language: Mastering PostScript will help you master the FPU.

## 11.13.5 pinhole--The Code

    ;;;;;;; pinhole.asm ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
    ;
    ; Find various parameters of a pinhole camera construction and use
    ;
    ; Started:   9-Jun-2001
    ; Updated:  10-Jun-2001
    ;
    ; Copyright (c) 2001 G. Adam Stanislav
    ; All rights reserved.
    ;
    ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

    %include    'system.inc'

    %define BUFSIZE 2048

    section .data
    align 4
    ten dd  10
    thousand    dd  1000
    tthou   dd  10000
    fd.in   dd  stdin
    fd.out  dd  stdout
    envar   db  'PINHOLE='  ; Exactly 8 bytes, or 2 dwords long
    pinhole db  '04,',      ; Bender's constant (0.04)
    connors db  '037', 0Ah  ; Connors' constant
    usg db  'Usage: pinhole [-b] [-c] [-e] [-p <value>] [-o <outfile>] [-i <infile>]', 0Ah
    usglen  equ $-usg
    iemsg   db  "pinhole: Can't open input file", 0Ah
    iemlen  equ $-iemsg
    oemsg   db  "pinhole: Can't create output file", 0Ah
    oemlen  equ $-oemsg
    pinmsg  db  "pinhole: The PINHOLE constant must not be 0", 0Ah
    pinlen  equ $-pinmsg
    toobig  db  "pinhole: The PINHOLE constant may not exceed 18 decimal places", 0Ah
    biglen  equ $-toobig
    huhmsg  db  9, '???'
    separ   db  9, '???'
    sep2    db  9, '???'
    sep3    db  9, '???'
    sep4    db  9, '???', 0Ah
    huhlen  equ $-huhmsg
    header  db  'focal length in millimeters,pinhole diameter in microns,'
        db  'F-number,normalized F-number,F-5.6 multiplier,stops '
        db  'from F-5.6', 0Ah
    headlen equ $-header

    section .bss
    ibuffer resb    BUFSIZE
    obuffer resb    BUFSIZE
    dbuffer resb    20      ; decimal input buffer
    bbuffer resb    10      ; BCD buffer

    section .text
    align 4
    huh:
        call    write
        push    dword huhlen
        push    dword huhmsg
        push    dword [fd.out]
        sys.write
        add esp, byte 12
        ret

    align 4
    perr:
        push    dword pinlen
        push    dword pinmsg
        push    dword stderr
        sys.write
        push    dword 4     ; return failure
        sys.exit

    align 4
    consttoobig:
        push    dword biglen
        push    dword toobig
        push    dword stderr
        sys.write
        push    dword 5     ; return failure
        sys.exit

    align 4
    ierr:
        push    dword iemlen
        push    dword iemsg
        push    dword stderr
        sys.write
        push    dword 1     ; return failure
        sys.exit

    align 4
    oerr:
        push    dword oemlen
        push    dword oemsg
        push    dword stderr
        sys.write
        push    dword 2
        sys.exit

    align 4
    usage:
        push    dword usglen
        push    dword usg
        push    dword stderr
        sys.write
        push    dword 3
        sys.exit

    align 4
    global  _start
    _start:
        add esp, byte 8 ; discard argc and argv[0]
        sub esi, esi

    .arg:
        pop ecx
        or  ecx, ecx
        je  near .getenv        ; no more arguments

        ; ECX contains the pointer to an argument
        cmp byte [ecx], '-'
        jne usage

        inc ecx
        mov ax, [ecx]
        inc ecx

    .o:
        cmp al, 'o'
        jne .i

        ; Make sure we are not asked for the output file twice
        cmp dword [fd.out], stdout
        jne usage

        ; Find the path to output file - it is either at [ECX+1],
        ; i.e., -ofile --
        ; or in the next argument,
        ; i.e., -o file

        or  ah, ah
        jne .openoutput
        pop ecx
        jecxz   usage

    .openoutput:
        push    dword 420   ; file mode (644 octal)
        push    dword 0200h | 0400h | 01h
        ; O_CREAT | O_TRUNC | O_WRONLY
        push    ecx
        sys.open
        jc  near oerr

        add esp, byte 12
        mov [fd.out], eax
        jmp short .arg

    .i:
        cmp al, 'i'
        jne .p

        ; Make sure we are not asked twice
        cmp dword [fd.in], stdin
        jne near usage

        ; Find the path to the input file
        or  ah, ah
        jne .openinput
        pop ecx
        or  ecx, ecx
        je near usage

    .openinput:
        push    dword 0     ; O_RDONLY
        push    ecx
        sys.open
        jc  near ierr       ; open failed

        add esp, byte 8
        mov [fd.in], eax
        jmp .arg

    .p:
        cmp al, 'p'
        jne .c
        or  ah, ah
        jne .pcheck

        pop ecx
        or  ecx, ecx
        je  near usage

        mov ah, [ecx]

    .pcheck:
        cmp ah, '0'
        jl  near usage
        cmp ah, '9'
        ja  near usage
        mov esi, ecx
        jmp .arg

    .c:
        cmp al, 'c'
        jne .b
        or  ah, ah
        jne near usage
        mov esi, connors
        jmp .arg

    .b:
        cmp al, 'b'
        jne .e
        or  ah, ah
        jne near usage
        mov esi, pinhole
        jmp .arg

    .e:
        cmp al, 'e'
        jne near usage
        or  ah, ah
        jne near usage
        mov al, ','
        mov [huhmsg], al
        mov [separ], al
        mov [sep2], al
        mov [sep3], al
        mov [sep4], al
        jmp .arg

    align 4
    .getenv:
        ; If ESI = 0, we did not have a -p argument,
        ; and need to check the environment for "PINHOLE="
        or  esi, esi
        jne .init

        sub ecx, ecx

    .nextenv:
        pop esi
        or  esi, esi
        je  .default    ; no PINHOLE envar found

        ; check if this envar starts with 'PINHOLE='
        mov edi, envar
        mov cl, 2       ; 'PINHOLE=' is 2 dwords long
    rep cmpsd
        jne .nextenv

        ; Check if it is followed by a digit
        mov al, [esi]
        cmp al, '0'
        jl  .default
        cmp al, '9'
        jbe .init
        ; fall through

    align 4
    .default:
        ; We got here because we had no -p argument,
        ; and did not find the PINHOLE envar.
        mov esi, pinhole
        ; fall through

    align 4
    .init:
        sub eax, eax
        sub ebx, ebx
        sub ecx, ecx
        sub edx, edx
        mov edi, dbuffer+1
        mov byte [dbuffer], '0'

        ; Convert the pinhole constant to real
    .constloop:
        lodsb
        cmp al, '9'
        ja  .setconst
        cmp al, '0'
        je  .processconst
        jb  .setconst

        inc dl

    .processconst:
        inc cl
        cmp cl, 18
        ja  near consttoobig
        stosb
        jmp short .constloop

    align 4
    .setconst:
        or  dl, dl
        je  near perr

        finit
        fild    dword [tthou]

        fld1
        fild    dword [ten]
        fdivp   st1, st0

        fild    dword [thousand]
        mov edi, obuffer

        mov ebp, ecx
        call    bcdload

    .constdiv:
        fmul    st0, st2
        loop    .constdiv

        fld1
        fadd    st0, st0
        fadd    st0, st0
        fld1
        faddp   st1, st0
        fchs

        ; If we are creating a CSV file,
        ; print header
        cmp byte [separ], ','
        jne .bigloop

        push    dword headlen
        push    dword header
        push    dword [fd.out]
        sys.write

    .bigloop:
        call    getchar
        jc  near done

        ; Skip to the end of the line if you got '#'
        cmp al, '#'
        jne .num
        call    skiptoeol
        jmp short .bigloop

    .num:
        ; See if you got a number
        cmp al, '0'
        jl  .bigloop
        cmp al, '9'
        ja  .bigloop

        ; Yes, we have a number
        sub ebp, ebp
        sub edx, edx

    .number:
        cmp al, '0'
        je  .number0
        mov dl, 1

    .number0:
        or  dl, dl      ; Skip leading 0's
        je  .nextnumber
        push    eax
        call    putchar
        pop eax
        inc ebp
        cmp ebp, 19
        jae .nextnumber
        mov [dbuffer+ebp], al

    .nextnumber:
        call    getchar
        jc  .work
        cmp al, '#'
        je  .ungetc
        cmp al, '0'
        jl  .work
        cmp al, '9'
        ja  .work
        jmp short .number

    .ungetc:
        dec esi
        inc ebx

    .work:
        ; Now, do all the work
        or  dl, dl
        je  near .work0

        cmp ebp, 19
        jae near .toobig

        call    bcdload

        ; Calculate pinhole diameter

        fld st0 ; save it
        fsqrt
        fmul    st0, st3
        fld st0
        fmul    st5
        sub ebp, ebp

        ; Round off to 4 significant digits
    .diameter:
        fcom    st0, st7
        fstsw   ax
        sahf
        jb  .printdiameter
        fmul    st0, st6
        inc ebp
        jmp short .diameter

    .printdiameter:
        call    printnumber ; pinhole diameter

        ; Calculate F-number

        fdivp   st1, st0
        fld st0

        sub ebp, ebp

    .fnumber:
        fcom    st0, st6
        fstsw   ax
        sahf
        jb  .printfnumber
        fmul    st0, st5
        inc ebp
        jmp short .fnumber

    .printfnumber:
        call    printnumber ; F number

        ; Calculate normalized F-number
        fmul    st0, st0
        fld1
        fld st1
        fyl2x
        frndint
        fld1
        fscale
        fsqrt
        fstp    st1

        sub ebp, ebp
        call    printnumber

        ; Calculate time multiplier from F-5.6

        fscale
        fld st0

        ; Round off to 4 significant digits
    .fmul:
        fcom    st0, st6
        fstsw   ax
        sahf

        jb  .printfmul
        inc ebp
        fmul    st0, st5
        jmp short .fmul

    .printfmul:
        call    printnumber ; F multiplier

        ; Calculate F-stops from 5.6

        fld1
        fxch    st1
        fyl2x

        sub ebp, ebp
        call    printnumber

        mov al, 0Ah
        call    putchar
        jmp .bigloop

    .work0:
        mov al, '0'
        call    putchar

    align 4
    .toobig:
        call    huh
        jmp .bigloop

    align 4
    done:
        call    write       ; flush output buffer

        ; close files
        push    dword [fd.in]
        sys.close

        push    dword [fd.out]
        sys.close

        finit

        ; return success
        push    dword 0
        sys.exit

    align 4
    skiptoeol:
        ; Keep reading until you come to cr, lf, or eof
        call    getchar
        jc  done
        cmp al, 0Ah
        jne .cr
        ret

    .cr:
        cmp al, 0Dh
        jne skiptoeol
        ret

    align 4
    getchar:
        or  ebx, ebx
        jne .fetch

        call    read

    .fetch:
        lodsb
        dec ebx
        clc
        ret

    read:
        jecxz   .read
        call    write

    .read:
        push    dword BUFSIZE
        mov esi, ibuffer
        push    esi
        push    dword [fd.in]
        sys.read
        add esp, byte 12
        mov ebx, eax
        or  eax, eax
        je  .empty
        sub eax, eax
        ret

    align 4
    .empty:
        add esp, byte 4
        stc
        ret

    align 4
    putchar:
        stosb
        inc ecx
        cmp ecx, BUFSIZE
        je  write
        ret

    align 4
    write:
        jecxz   .ret    ; nothing to write
        sub edi, ecx    ; start of buffer
        push    ecx
        push    edi
        push    dword [fd.out]
        sys.write
        add esp, byte 12
        sub eax, eax
        sub ecx, ecx    ; buffer is empty now
    .ret:
        ret

    align 4
    bcdload:
        ; EBP contains the number of chars in dbuffer
        push    ecx
        push    esi
        push    edi

        lea ecx, [ebp+1]
        lea esi, [dbuffer+ebp-1]
        shr ecx, 1

        std

        mov edi, bbuffer
        sub eax, eax
        mov [edi], eax
        mov [edi+4], eax
        mov [edi+2], ax

    .loop:
        lodsw
        sub ax, 3030h
        shl al, 4
        or  al, ah
        mov [edi], al
        inc edi
        loop    .loop

        fbld    [bbuffer]

        cld
        pop edi
        pop esi
        pop ecx
        sub eax, eax
        ret

    align 4
    printnumber:
        push    ebp
        mov al, [separ]
        call    putchar

        ; Print the integer at the TOS
        mov ebp, bbuffer+9
        fbstp   [bbuffer]

        ; Check the sign
        mov al, [ebp]
        dec ebp
        or  al, al
        jns .leading

        ; We got a negative number (should never happen)
        mov al, '-'
        call    putchar

    .leading:
        ; Skip leading zeros
        mov al, [ebp]
        dec ebp
        or  al, al
        jne .first
        cmp ebp, bbuffer
        jae .leading

        ; We are here because the result was 0.
        ; Print '0' and return
        mov al, '0'
        jmp putchar

    .first:
        ; We have found the first non-zero.
        ; But it is still packed
        test    al, 0F0h
        jz  .second
        push    eax
        shr al, 4
        add al, '0'
        call    putchar
        pop eax
        and al, 0Fh

    .second:
        add al, '0'
        call    putchar

    .next:
        cmp ebp, bbuffer
        jb  .done

        mov al, [ebp]
        push    eax
        shr al, 4
        add al, '0'
        call    putchar
        pop eax
        and al, 0Fh
        add al, '0'
        call    putchar

        dec ebp
        jmp short .next

    .done:
        pop ebp
        or  ebp, ebp
        je  .ret

    .zeros:
        mov al, '0'
        call    putchar
        dec ebp
        jne .zeros

    .ret:
        ret

The code follows the same format as all the other filters we have seen before, with one subtle exception:

> We are no longer assuming that the end of input implies the end of things to do, something we took for granted in the character-oriented filters.

> This filter does not process characters. It processes a language (albeit a very simple one, consisting only of numbers).

> When we have no more input, it can mean one of two things:

> - We are done and can quit. This is the same as before.

> - The last character we have read was a digit. We have stored it at the end of our ASCII-to-float conversion buffer. We now need to convert the contents of that buffer into a number and write the last line of our output.

> For that reason, we have modified our `getchar` and our `read` routines to return with the `carry flag` clear whenever we are fetching another character from the input, or the `carry flag` set whenever there is no more input.

> Of course, we are still using assembly language magic to do that! Take a good look at `getchar`. It always returns with the `carry flag` clear.

> Yet, our main code relies on the `carry flag` to tell it when to quit--and it works.

> The magic is in `read`. Whenever it receives more input from the system, it just returns to `getchar`, which fetches a character from the input buffer, clears the `carry flag` and returns.

> But when `read` receives no more input from the system, it does not return to `getchar` at all. Instead, the `add esp, byte 4` op code adds `4` to `ESP`, sets the `carry flag`, and returns.

> So, where does it return to? Whenever a program uses the `call` op code, the microprocessor `push`es the return address, i.e., it stores it on the top of the stack (not the FPU stack, the system stack, which is in the memory). When a program uses the `ret` op code, the microprocessor `pop`s the return value from the stack, and jumps to the address that was stored there.

> But since we added `4` to `ESP` (which is the stack pointer register), we have effectively given the microprocessor a minor case of amnesia: It no longer remembers it was `getchar` that `call`ed `read`.

> And since `getchar` never `push`ed anything before `call`ing `read`, the top of the stack now contains the return address to whatever or whoever `call`ed `getchar`. As far as that caller is concerned, he `call`ed `getchar`, which `ret`urned with the `carry flag` set!

Other than that, the `bcdload` routine is caught up in the middle of a Lilliputian conflict between the Big-Endians and the Little-Endians.

It is converting the text representation of a number into that number: The text is stored in the big-endian order, but the packed decimal is little-endian.

To solve the conflict, we use the `std` op code early on. We cancel it with `cld` later on: It is quite important we do not `call` anything that may depend on the default setting of the direction flag while `std` is active.

Everything else in this code should be quite clear, providing you have read the entire chapter that precedes it.

It is a classical example of the adage that programming requires a lot of thought and only a little coding. Once we have thought through every tiny detail, the code almost writes itself.

## 11.13.6 Using pinhole

Because we have decided to make the program ignore any input except for numbers (and even those inside a comment), we can actually perform textual queries. We do not have to, but we can.

In my humble opinion, forming a textual query, instead of having to follow a very strict syntax, makes software much more user friendly.

Suppose we want to build a pinhole camera to use the 4x5 inch film. The standard focal length for that film is about 150mm. We want to fine-tune our focal length so the pinhole diameter is as round a number as possible. Let us also suppose we are quite comfortable with cameras but somewhat intimidated by computers. Rather than just have to type in a bunch of numbers, we want to ask a couple of questions.

Our session might look like this:

    % pinhole

    Computer,

    What size pinhole do I need for the focal length of 150?
    150 490 306 362 2930    12
    Hmmm... How about 160?
    160 506 316 362 3125    12
    Let's make it 155, please.
    155 498 311 362 3027    12
    Ah, let's try 157...
    157 501 313 362 3066    12
    156?
    156 500 312 362 3047    12
    That's it! Perfect! Thank you very much!
    ^D

We have found that while for the focal length of 150, our pinhole diameter should be 490 microns, or 0.49 mm, if we go with the almost identical focal length of 156 mm, we can get away with a pinhole diameter of exactly one half of a millimeter.

## 11.13.7 Scripting

Because we have chosen the `#` character to denote the start of a comment, we can treat our pinhole software as a scripting language.

You have probably seen shell scripts that start with:

    #! /bin/sh

...or...

    #!/bin/sh

...because the blank space after the `#!` is optional.

Whenever UNIX is asked to run an executable file which starts with the `#!`, it assumes the file is a script. It adds the command to the rest of the first line of the script, and tries to execute that.

Suppose now that we have installed pinhole in /usr/local/bin/, we can now write a script to calculate various pinhole diameters suitable for various focal lengths commonly used with the 120 film.

The script might look something like this:

    #! /usr/local/bin/pinhole -b -i
    # Find the best pinhole diameter
    # for the 120 film

    ### Standard
    80

    ### Wide angle
    30, 40, 50, 60, 70

    ### Telephoto
    100, 120, 140

Because 120 is a medium size film, we may name this file medium.

We can set its permissions to execute, and run it as if it were a program:

    % chmod 755 medium
    % ./medium

UNIX will interpret that last command as:

    % /usr/local/bin/pinhole -b -i ./medium

It will run that command and display:

    80 358 224 256 1562    11
    30  219 137 128 586 9
    40  253 158 181 781 10
    50  283 177 181 977 10
    60  310 194 181 1172    10
    70  335 209 181 1367    10
    100 400 250 256 1953    11
    120 438 274 256 2344    11
    140 473 296 256 2734    11

Now, let us enter:

    % ./medium -c

UNIX will treat that as:

    % /usr/local/bin/pinhole -b -i ./medium -c

That gives it two conflicting options: `-b` and `-c` (Use Bender's constant and use Connors' constant). We have programmed it so later options override early ones--our program will calculate everything using Connors' constant:

    80 331 242 256 1826    11
    30  203 148 128 685 9
    40  234 171 181 913 10
    50  262 191 181 1141    10
    60  287 209 181 1370    10
    70  310 226 256 1598    11
    100 370 270 256 2283    11
    120 405 296 256 2739    11
    140 438 320 362 3196    12

We decide we want to go with Bender's constant after all. We want to save its values as a comma-separated file:

    % ./medium -b -e > bender
    % cat bender
    focal length in millimeters,pinhole diameter in microns,F-number,normalized F-number,F-5.6 multiplier,stops from F-5.6
    80,358,224,256,1562,11
    30,219,137,128,586,9
    40,253,158,181,781,10
    50,283,177,181,977,10
    60,310,194,181,1172,10
    70,335,209,181,1367,10
    100,400,250,256,1953,11
    120,438,274,256,2344,11
    140,473,296,256,2734,11
    %

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