Introduction to Fixed Point Math

9/14/2004

Back in the olden days of dragons and sorcerers, before floating point coprocessors, computers could only directly manipulate whole numbers (integers). Things like decimal points were new fangled technologies only used by crazy mathematicians with slide rules and pen and paper. To overcome this deficiency crufty old-school programmers, many of whom today are over 30 years old, had to resort to various hacks and tricks in order to represent wacky numerical concepts like "0.5". The most common technique was fixed point math, which is the topic of this article.

Preamble

I'm eventually, maybe, going to write a bunch of stuff here about real numbers and integers and number theory and domains and what not, but right now that doesn't interest me so I'm going to dodge the issue and do a quick summary for those late to class:

  • computers represent integers and real numbers differently
  • not all computers can perform math on real numbers quickly
  • computers that do perform math on real numbers typically use a floating point representation which dynamically adjusts range and precision depending on the inputs to a particular operation.
  • converting from a floating point value to an integer value is often very slow and always inext

Hope you're caught up. Next.

Fixed Point Representation

Given that there are processors out there that don't support floating point numbers, how do we represent something like "0.5" on such beasts? This is where fixed point math comes into play. As the name implies, a fixed point number places the "decimal" point between the whole and fractional parts of a number at a fixed location, providing f bits of fractional precision. For example, an 8.24 fixed point number has an eight bit integer portion and twenty four bits of fractional precision. Since the split between the whole and fractional portion is fixed, we know exactly what our range and precision will be.

So let's think about this for a second. Using a 16.16 fixed point format (which, handily enough, fits in a typical 32-bit integer in C), the high 16-bits represent the whole part, and the low 16-bits represent the fractional portion:

0xWWWWFFFF 

With 16-bits in the whole part, we can represent 216 (65536) discrete values (0..65535 unsigned or -32768 to +32767 signed). The fractional part gives us, again, 216 steps to represent the values from 0 to almost 1 -- specifically, 0 to 65535/65536 or approximately 0.99999. Our fractional resolution is 1/65536, or about 0.000015 -- that's pretty good resolution as well.

Let's look at a concrete example: the 16.16 fixed point value 0x00104ACF approximately equals the decimal value 16.29222. The high sixteen bits are 0x10 (16 decimal) and the low 16-bits are 0x4ACF, so 0x4ACF/65536.0 ~= 0.29222. The simple method to convert from a fixed point value of f fractional bits is to divide by 2f so:

http://www.bookofhook.com/Images/Fixed_Point_Images/fixrep_1.gif

However, we have to be careful since these integers are stored in 2's complement form. The value -16.29222 is 0xFFEFB531 -- notice that the fractional bits (0xB531) are not the same as for the positive value. So you can't just grab the fractional part by masking off the fractional bits -- sign matters.

2's Complement Summary

If you're reading this document there's a chance that you're not familiar with the concept of 2's complement arithmetic, which is fairly vital to understanding fixed point math. Fundamentally 2's complement solves the problem of representing negative numbers in binary form. Given a 32-bit integer, how can we represent negative numbers?

The obvious method is to incorporate a sign bit, which is precisely what floating point numbers in IEEE-754 format do. This signed magnitude form reserves one bit to represent positive/negative, and the rest of the bits represent the number's magnitude. Unfortunately there are a couple problems with this.

  1. Zero has two representations, positive (0x00000000) and negative (0x80000000), making comparisons cumbersome. It's also wasteful since two bit configurations represent the same value.
  2. Adding a positive integer to a negative integer requires logic. For example, -1 (0x80000001 in 32-bit sign magnitude form) summed with +1 (0x00000001 in 32-bit sign magnitude form) yields 0x8002, or -2 in 16-bit sign-bit form..

Some really smart guys came up with a much better system a long time ago called 2's complement arithmetic. In this system positive numbers are represented as you'd expect, however negative numbers are represented by their complement, plus one. This requires taking all bits in the positive representation, reversing them, then adding one.

So to encode -1 we take 0x00000001, invert (complement) the bits to get 0xFFFFFFFE, and then add 1, giving us 0xFFFFFFFF -- which you may recognize as the hex form for -1. This gives us a range of +/- 2(n-1) where n is the number of storage bits.

2's complement solves a host of problems.

  1. Zero has a single representation, simplifying comparisons and removing the wasted value.
  2. Addition and subtraction just work. A binary addition of 2's complement values will give the correct result regardless of sign. -1 +1 = 0 (0xFFFFFFFF + 0x00000001 = 0x00000000); -1 + -1 gives us -2 (0xFFFFFFFF + 0xFFFFFFFF = 0xFFFFFFFE); and 1 + 1 = 2 (0x00000001 + 0x00000001 = 0x00000002).

But...it introduces a couple unintuitive bits. Decoding a negative number takes a little bit of thought (you have to do the complement and add one), and the magnitude of a negative number is inversely proportional to the magnitude of its binary representation: -1 = 0xFFFFFFFF but -2147483648 = 0x80000000. This has at least one important ramification when dealing with fixed point -- the fractional bits of a fixed point number are not interpreted the same for positive and negative values.

Anyway, that's a quick refresher course. If you really need to know more about 2's complement, Google is your friend or, if you're feeling gung-ho, find a good book on discrete math or numerical analysis.

Range and Precision Trade Offs

The whole trick with fixed point is that you're sacrificing some range in exchange for precision. In this case, instead of a 32-bit integer you have a 16-bit integer space with 16-bits of fractional precision.

Different operations often require different amounts of range and precision. A format like 16.16 is good for many different things, but depending on the specific problem you may need highly precise formats such as 2.30, or longer range formats such as 28.4. For example, we know that the sine function only returns values in the range -1.0 to +1.0, so representing sine values with a 16.16 is wasteful since 14-bits of the integer component will always be zero. Not only that, but the difference between sine(90) and sine(89.9) is a very small number, ~.0000015230867, which our 16-bits of fractional precision cannot represent accurately. In this particular case we'd be much better off using a 2.30 format.

Conversely, some calculations will generate very large numbers, sometimes unexpectedly. The area of a screen sized triangle can easily exceed 216 (a triangle half the size of a 640x480 pixel screen covers 153600 pixels), so even though the individual coordinates of such a triangle might be well within what we'd intuitively believe to be acceptable range (e.g. (0,0),(639,0),(639,479)), the area formed can vastly outstrip our available integer bits.

This brings to light a serious problem with fixed point -- you can silently overflow or underflow without expecting it, especially when dealing with sums-of-products or power series. This is one of the most common sources of bugs when working on a fixed point code base: some set of values that you expect to stay within range rapidly overflow when you least expect it.

A very common case is vector normalization, which has a sum-of-products during the vector length calculation. Normalizing a vector consists of dividing each of the vector's elements by the length of the vector. The length of a vector is:

http://bookofhook.com/Images/Fixed_Point_Images/vector_length.gif

One might believe that the vector (125,125,125) can be easily normalized using a 16.16 signed fixed point format (-32K to +32K range)...right? Unfortunately, that's not the case. The problem is that the sum of those squares is 46875 -- greater than our maximum value of +215 (32767). After the square root it would drop down to a very manageable value (~216.51), but by then the damage is done. The length calculation will fail, and thus the normalization will fail -- but the scary thing is that the vector normalization will work correctly with a lot of values, but not with others. The vector (100,100,100) will normalize just fine, but the vector (40,40,180) will not, but you can't tell from a glance which is fine and which isn't. Fixed point math is highly input sensitive.

This is why choosing your range and precision is very important, and putting in a lot of range and overflow sanity checks is extremely important if you want robust software.

Conversion

Converting to and from fixed point is a simple affair if you're not too worried about rounding and details like that. A fixed point value is derived from a floating point value by multiplying by 2f. A fixed point value is generated from an integer value by shifting left f bits:

http://bookofhook.com/Images/Fixed_Point_Images/fixops_1.gif

F = fixed point value
i = integer value
r = floating point value
f = number of fractional bits for F

Converting back to floating point is trivial -- just perform the reverse of the float-to-fixed conversion:

http://bookofhook.com/Images/Fixed_Point_Images/fixops_2.gif

However, converting to an integer is a bit trickier due to rounding rules and the 2's complement encoding. In the world of floating point there are four common methods for converting a floating point value to an integer:

  1. round-towards-zero
  2. round-towards-positive-infinity
  3. round-towards-negative-infinity
  4. round-to-nearest

(Note: there are other rounding methods as well, such as round-to-even, round-to-odd, round-away-from-zero, etc. but those are rarer and I don't feel like devoting space to them). These are conceptually simpler in floating point than in fixed point -- for example, truncating a floating point value always rounds towards zero, but truncating the fractional bits of a fixed point number actually rounds towards negative infinity (due to the miracle of 2's complement, where the fractional bits move you away from 0 for positive numbers but towards 0 for negative numbers).

Round-Towards-Zero

Rounding towards zero "shrinks" a number so that it always gets closer to zero during the rounding operation. For a positive fixed point value a simple truncation works, however a negative number needs to have 1<<(f-1) added to it before the truncation. This is just "round towards negative-infinity if positive; round towards positive-infinity if negative".

Round-Towards-Negative-Infinity

Rounding towards negative-infinity is the simplest operation, it's the truncation we talked about earlier. A positive number that loses its fractional goes from A.B to A.0, and will always get closer to zero. However, negative numbers get larger as they approach 0 due to 2's complement encoding. This means that removing the fractional bits actually pushes the negative number away from zero (and thus towards negative infinity).

This can be somewhat confusing, but an example illustrate this. The fixed point value 0xFFFF9999 represents -0.40000 roughly. By truncating the fractional bits the new value is -1.0 -- we've rounded towards negative infinity. The fixed point value 0xFFFF0003 represents a value just above negative -1, so truncating the fractional bits will, again, give us a value of -1.

Note that we need to perform an arithmetic shift, also known as a sign-extended shift, to ensure that the sign bit is propagated correctly for negative numbers. Consider the 16.16 fixed point value for -1 (0xFFFF0000). If we do an unsigned shift to the right by 16 places we have 0x0000FFFF which would be 65535, which is incorrect. By sign extending the shift the result will be 0xFFFFFFFF, which is -1 in 2's complement integer representation.

http://bookofhook.com/Images/Fixed_Point_Images/fixops_3.gif

Round-Towards-Positive-Infinity

Rounding towards positive infinity requires adding the largest value under 1 ((1<<f)-1) to the fixed point number before the shift. This pushes the fixed point into the next highest integer value if the fixed point value is not already on an integer boundary.

http://bookofhook.com/Images/Fixed_Point_Images/fixops_4.gif

Rounding-to-Nearest

Rounding to the nearest value, which is what we often associate with rounding (this is what the C language does), is easy primarily because of the 2's complement encoding. By adding "one-half" (1<<(f-1)) before the shift we will always round to the nearest value. When a fixed point value is positive, this adds one- half away from 0. When a fixed point value is negative, this add one-half towards 0, so it adds +0.5 no matter what.

http://bookofhook.com/Images/Fixed_Point_Images/fixops_5.gif

Rounding Summary

The following table lists some example values and how they are rounded with the different rounding modes:

Input Value 	Round to 	    Output
-1.2 	        zero 	             -1
-1.2 	        nearest 	     -1
-1.2 	        positive infinity    -1
-1.2        	negative infinity    -2
-1.7    	zero 	             -1
-1.7    	nearest 	     -2
-1.7     	positive infinity    -1
-1.7     	negative infinity    -2
1.2     	zero 	              1
1.2      	nearest 	      1
1.2     	positive infinity     2	
1.2      	negative infinity     1
1.7     	zero 	              1
1.7     	nearest               2
1.7     	positive infinity     2
1.7     	negative infinity     1

Which rounding mode you use is entirely up to you and your application, just make sure you understand the ramifications of each mode.

Numerical Operations

Addition and Subtraction

Adding and subtracting two fixed point numbers of the same scale is simplicity itself -- just perform a standard integer addition or subtraction. It just works -- mathematically this is obvious since:

http://bookofhook.com/Images/Fixed_Point_Images/fixops_6.gif

If the two numbers are operating in different scales, then you will have to shift one or the other (or both) so that their "decimal points" line up. How you do this depends entirely on the final output desired. For example, adding a 16.16 to a 28.4 could be accomplished by shifting the 16.16 left by twelve; or by shifting the 28.4 right by 12; or converting both to 22.10; or whatever. It's up to you. The important thing is that during the addition, they have to be in the same format and that the intermediate calculations (and final output form) are of the appropriate range and precision for your needs.

Multiplication

Fixed point multiplication is a bit uglier, but not too bad -- multiply the two numbers, then shift down by some number of bits to get back into the output format you want. This can get confusing if you don't understand the basis for the multiplication operation, so I'm going to show that to you soon. But first, let's look at an intuitive example to get a feel for how fixed point multiplication works.

Let's assume we're multiplying two 16.16 values against each other, 0x00010000 and 0x00020000 (1.0 * 2.0). A straight integer multiplication will result in a 64-bit product:

http://bookofhook.com/Images/Fixed_Point_Images/fixops_7.gif

Doh! The product won't fit in a single 32-bit register, so we've already encountered our first problem: overflow. We can handle this by doing this piecewise (just like grade school long multiplication), or we can hope/pray for an architecture that supports 64-bit intermediate results from a multiplication. Thankfully, both the 32-bit x86 and newer ARM series of processors support 64-bit intermediates and most modern compilers have 64-bit types (long long in GCC and __int64 in many PC-based compilers such as Microsoft Visual C++).

So we're okay -- our initial 16.16 * 16.16 product is > 32-bits, but we'll assume that our compiler can handle that for us so we can safely get back down to 32-bit land for our final result.

Besides the overflow we can see another problem -- the output isn't shifted "correctly". We know that 1.0 * 2.0 = 2.0, and thus we'd expect our final result to be 0x00020000 (in 16.16 fixed point). Simple examination shows us that shifting down by 16-bits will give us the result we anticipate. So a 16.16 * 16.16 operation is performed as:

fixed16_16_t
mul_16_16( fixed_16_16_t a, fixed_16_16_t b )
{
    long long r = ( long long ) a * ( long long ) b;

    return ( fixed16_16_t ) ( r >> 16 );
}

But why do we shift to the right by 16-bits? And what if we're doing something like multiplying a 24.8 by a 10.22?

Now we get into a bit more of the nitty gritty -- it's just basic algebra, so this shouldn't be too frightening, even if you suck at math like I do.

Multiplying two fixed point numbers of m and n fractional bits, respectively, gives us:

http://bookofhook.com/Images/Fixed_Point_Images/multiply_1.gif

A = first number
m = number of fractional bits for A
B = second number, non-fixed point
n = number of fractional bits for B

Looking at the above we find that the product is scaled by the sum of m and n, so the product is "overscaled". We can fix this by multiplying through by some value that will bring it back down into the range we desire. Given that we eventually want, say, f bits of precision (i.e. the result is scaled by f bits) in our result, we need to convert the 2(m+n) term to 2f.

Multiplying through by:

http://bookofhook.com/Images/Fixed_Point_Images/multiply_4.gif

will scale things so that we have f fractional bits in our result since:

http://bookofhook.com/Images/Fixed_Point_Images/multiply_2.gif

So our final equation is:

http://bookofhook.com/Images/Fixed_Point_Images/multiply_3.gif

In plain English, that's just saying that if we want the product of two fixed point numbers, of m and n fractional bits respectively, to have f bits of precision, we multiply the two fixed point values then scale the result by 2(f-(m+n)).

Let's use our previous example of 0x10000 * 0x20000 (1.0 * 2.0 in 16.16 format) but this time with 24-bits of fractional precision in the result. We have m = 16, n = 16, and f = 24. The initial multiplication results in 0x200000000, which we then multiply by 2(24-(16+16)) = 2-8 or 1/256. Of course, we'd just shift right by 8 places, which in the end gives us 0x02000000 which is 2.0 in 8.24 format.

One thing to keep in mind is that along with loss of precision (any bits shifted out of the result are lost), we still have to contend with overflow in some situations. For example, if we had multipled 0xFF0000 and 0x20000 (255.0 x 2.0) the result should be 510.0 in 8.24 format, or 0x1FE000000...which is larger than 32-bits, so it'll be truncated to 0xFE000000 (254.0) on systems that don't support 64-bit intermediate products. This is a silent error, and overflow bugs like this can be excruciating to hunt down.

One way to get around this is to prescale the two multiplicands down, avoiding the final shift, but in the process we lose precision in exchange for avoiding overflow. This is not the ideal way of dealing with insufficient intermediate bits -- instead a piecewise multiplication (which I might explain one day when I'm real bored) would be used to preserve accuracy at the cost of some performance.

The moral being: never assume you have enough bits, always make sure that your inputs and outputs have the precision and range you need and expect!

Division

Division is similar to multiplication, with a couple caveats. Let's look at an example to get an intuitive feel for the process. Assuming we have 2.0 and 1.0 again in 16.16 fixed point form: 0x20000 and 0x10000. If we divide the former by the latter, we get the value 2. Not 0x20000, but just 2. Which is correct, granted, but it's not in a 16.16 output form. We could take the result and shift it back by the number of bits we need, but...that's just gross. And wrong, since shifting after the integer divide means we lose precision before the shift. In other words, if A=0x10000 and B=0x20000 our final result would be 0, instead of 0x8000 (0x8000 = 0.5 in 16.16 fixed point, since 0x8000/65536.0 = 0.5). But if we do the shift first, we manage to retain our precision and also have a properly scaled result.

Well, that all makes sense, but we're assuming that A and B are of the same fractional precision, and what if they're not? Now it's time to look at the "real" math of a fixed point division. Again, this is grade school stuff:

http://bookofhook.com/Images/Fixed_Point_Images/div_1.gif

What this says is that if you take two fixed point numbers of m and n fractional bits, and divide them, the quotient will be a value with m-n fractional bits. With our previous example of 16.16 divided by a 16.16, that means 0 fractional bits, which is precisely what we got. If we want to have f fractional bits, we must scale the numerator by 2f-(m-n). Since of order operations matters when performing integer operations, we want to scale the numerator before we do the divide, so our final equation looks like:

http://bookofhook.com/Images/Fixed_Point_Images/div_2.gif

Order of operations matters, so actual evaluation would use the middle form, where the numerator is fully expanded before division. The far right form is just to show what the final result will look like.

Overflow is still a concern, so we have to make sure that we do the adjustment before the divide that we don't run out of bits -- this is all too easy unfortunately. For example, a 16.16 divided by a 16.16 needs to be prescaled by 216, which is a guaranteed overflow. Again, modern CPUs can come to the rescue, this time by providing 64-bit / 32-bit division operations.

Summary

Fixed point is a wonderful way to get around the limitations of systems with slow (or non-existent) floating point units. It also allows you to write very fast inner loops that can generate integer values without a costly float-to-int conversion. However, fixed point math can be very tricky due to range and precision issues. This article attempts to cover the basics of fixed point math so that at least a theoretical grounding is established before tackling the tricky implementation problems (which I might write an article about later).

Thanks to Tom Hubina and Jonathan Blow for commenting on and proof reading this article.