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How to use Fixed Point (16.16) Math (Part 1 of 2) - by Night Stalker
????????????????[ Revised version 1.1, dated March 12th, 1995 ]?????????????????
[1] Introduction
Allright, a simple question: what exactly *IS* fixed point math?
Fixed point math is a very simple way to speed up any program that uses
floating point. The 16.16 describes how many bits are before and after the
decimal point. (In this case, 65535.999984741211 is the largest possible
number. If you want to know how to get this value, set the decimal portion
to all 1's (in binary), and you get 65535 with 16 bits. Take 65535 and divide
it by 65536 -- that should answer your question. Thanks goes to Jonathan K.
Butler for pointing this out.) There are
other variants such as 24.8, and 8.24. It depends on what your application
needs in regards to precision.
Even with a math coprocessor (as it is becoming a standard nowadays),
floating point speeds are slow. I will go into much detail about how fixed
point can be used in your programs, as well as some trigonometric functions.
I'll also warn you ahead of time. This document contains Intel 80386+
assembly code for performance. It was designed to compile with Watcom C/C++,
and I'm sure it could be ported to other 32-bit compilers. If you're going
to try to port it to a 16-bit compiler, I wish you best of luck. :)
But first, I must give credit to where credit is due. The code I am
referencing was created by David Boeren. If you wish to reach him, E-mail
me, and I will give you his address. (I'd rather not post it publicly,
although I don't think he'd mind...)
[2] Fixed point basics
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First, we'll start out on how we actually set our program up to handle
fixed point. You really only need one definition to use fixed point math:
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typedef long Fixed32; // 16.16 FixedPoint
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Then whenever you want a fixed point number, just use 'Fixed32' instead
of 'float'.
Allright, you've got fixed point now. How do you actually get a number
into this format?
This is a little trickier. If you're using an integer or a char, you
simply shift left over the decimal point. Longs are truncated because only
2 bytes fit in the upper portion of a Fixed32.
The following #defines will help you out from converting to and from
any format to Fixed32 and back:
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#define INT_TO_FIXED(x) ((x) << 16)
#define DOUBLE_TO_FIXED(x) ((long)(x * 65536.0 + 0.5))
#define FIXED_TO_INT(x) ((x) >> 16)
#define FIXED_TO_DOUBLE(x) (((double)x) / 65536.0)
#define ROUND_FIXED_TO_INT(x) (((x) + 0x8000) >> 16)
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I also use the following #defines as well to access some well-known
numbers:
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#define ONE INT_TO_FIXED(1)
#define FIXED_PI 205887L
#define FIXED_2PI 411775L
#define FIXED_E 178144L
#define FIXED_ROOT2 74804L
#define FIXED_ROOT3 113512L
#define FIXED_GOLDEN 106039L
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Notice that FIXED_2PI is not equal to 2 * PI. Why? Well, 2 * FIXED_PI
is technically close enough, but FIXED_2PI is closer. :)
[3] Fixed point math
????????????????????
Okay, you understand fixed point and now you want to do the basics of math.
How do we accomplish adding, subtracting, multipling, and dividing?
Adding and subtracting are simple. You simply add and subtract like you
would normal floats. (ie: c = a + b; d -= c; etc, etc.) Multiplication and
division require a little more. Here is the code to multiply and divide fixed
point numbers:
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Fixed32 FixedMul(Fixed32 num1, Fixed32 num2);
Fixed32 FixedDiv(Fixed32 numer, Fixed32 denom);
#pragma aux FixedMul = \
"imul edx" \
"add eax, 8000h" \
"adc edx, 0" \
"shrd eax, edx, 16" \
parm caller [eax] [edx] \
value [eax] \
modify [eax edx];
#pragma aux FixedDiv = \
"xor eax, eax" \
"shrd eax, edx, 16" \
"sar edx, 16" \
"idiv ebx" \
parm caller [edx] [ebx] \
value [eax] \
modify [eax ebx edx];
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If you're up on your assembly code, you'll notice that FixedDiv() doesn't
do any rounding. Just take that into note.
If some of you have never done assembly in Watcom, I'll try to help.
Watcom's way of adding assembly into your programs starts with a #pragma.
The standard format is:
????????????????????????????????????????????????????????????????????????????????
#pragma aux = \
"" \
"" \
"<...>" \
parm caller [] ... \
value [] \
modify [ ...];
????????????????????????????????????????????????????????????????????????????????
Watcom uses 'parm caller' to know where to put the variables that are
passed to an inline assembly function. For example in FixedDiv(), the
numerator is passed into the EDX register, and the denominator is passed
into the EBX register. The 'value' command is used to determine what
register contains the value to return to the program when the inline code
has completed. The 'modify' command tells Watcom what registers to push
onto the stack (since you will be modifying them). Watcom allows omitting
of 'parm caller' and 'value' if there is no need for them. If you have
a function which doesn't return a value, then omit 'value'. The same goes
for 'parm caller'.
Unfortunately, I've only had success at putting inline code into my .H
(or .HPP) files. If anyone knows the answer to this mystery, *PLEASE* tell
me. Thanks. :)
[4] More fixed point math
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Moving right along... now that you know the basic math, we can start on
how to do some of the harder math functions: squaring, one over (1/x), and
square root.
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Fixed32 FixedSquare(Fixed32 n);
Fixed32 OneOver(Fixed32 n);
Fixed32 FixedSqrtLP(Fixed32 n); // Low Precision (8.8)
Fixed32 FixedSqrtHP(Fixed32 n); // High Precision (8.16)
// This is faster than using FixedMul for squares.
#pragma aux FixedSquare = \
"imul eax" \
"add eax, 8000h" \
"adc edx, 0" \
"shrd eax, edx, 16" \
parm caller [eax] \
value [eax] \
modify [eax edx];
// This is faster than using FixedDiv.
#pragma aux OneOver = \
"xor eax, eax" \
"mov edx, 1" \
"idiv ebx" \
parm caller [ebx] \
value [eax] \
modify [eax ebx edx];
#pragma aux FixedSqrtLP = \
" xor eax, eax" \
" mov ebx, 40000000h" \
"sqrtLP1: mov edx, ecx" \
" sub edx, ebx" \
" jl sqrtLP2" \
" sub edx, eax" \
" jl sqrtLP2" \
" mov ecx,edx" \
" shr eax, 1" \
" or eax, ebx" \
" shr ebx, 2" \
" jnz sqrtLP1" \
" shl eax, 8" \
" jmp sqrtLP3" \
"sqrtLP2: shr eax, 1" \
" shr ebx, 2" \
" jnz sqrtLP1" \
" shl eax, 8" \
"sqrtLP3: nop" \
parm caller [ecx] \
value [eax] \
modify [eax ebx ecx edx];
#pragma aux FixedSqrtHP = \
" xor eax, eax" \
" mov ebx, 40000000h" \
"sqrtHP1: mov edx, ecx" \
" sub edx, ebx" \
" jb sqrtHP2" \
" sub edx, eax" \
" jb sqrtHP2" \
" mov ecx,edx" \
" shr eax, 1" \
" or eax, ebx" \
" shr ebx, 2" \
" jnz sqrtHP1" \
" jz sqrtHP5" \
"sqrtHP2: shr eax, 1" \
" shr ebx, 2" \
" jnz sqrtHP1" \
"sqrtHP5: mov ebx, 00004000h" \
" shl eax, 16" \
" shl ecx, 16" \
"sqrtHP3: mov edx, ecx" \
" sub edx, ebx" \
" jb sqrtHP4" \
" sub edx, eax" \
" jb sqrtHP4" \
" mov ecx, edx" \
" shr eax, 1" \
" or eax, ebx" \
" shr ebx, 2" \
" jnz sqrtHP3" \
" jmp sqrtHP6" \
"sqrtHP4: shr eax, 1" \
" shr ebx, 2" \
" jnz sqrtHP3" \
"sqrtHP6: nop" \
parm caller [ecx] \
value [eax] \
modify [eax ebx ecx edx];
????????????????????????????????????????????????????????????????????????????????
I hope you don't gawk at all that assembly code. Not to worry, there isn't
any more. :) Note that the high precision square root takes more time to
perform than it does for the lower precision.
One note here regarding the inline assembly. As you can see, jumps are
possible in the code. However, when you have a jump to the last command
(see 'sqrtHP6:' above..), you need an instruction or Watcom will yell at you.
NOP was the only logical choice I found, since it does just what it says, "No
Operation".
I also have a formula to do logarithm (log), but you need to be able to
calculate a natural logarithm (ln) before you can do it. The basic formula
is:
ln(n)
log n = -----
x ln(x)
However, I have no clue on how to calculate a natural logarithm. If anyone
can offer any ideas, I'll be glad to write one.
[5] Fixed point trigonometric functions
???????????????????????????????????????
Allright, this is the last section: How to do trigonometric functions
in fixed point.
Well, I found it easier to use lookup tables since the need for fixed
point in the first place was for speed. Why bother bogging down the processor
with floating point cosines and sines and converting them to fixed point?
First off, let's define what an 'Iangle' is. It technically stands for
'Integer angle'. Since it's faster to reference an integer than it is to
reference a floating point, we'll use 'Iangles'.
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typedef unsigned short Iangle; // Integer angle (0..1023)
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This means if you want to travel through a complete sine (or cosine) loop,
you must pass through 1024 'Iangles'. Don't get this confused with degrees or
radians. Neither one of them comes close to this definition. If you're still
a little confused, you might want to look at the table generation code below
to see what an 'Iangle' actually is.
Below is the code to do sine, cosine, and tangent formulas:
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#define MAX_TRIG 1024
void CosSin(Iangle theta, Fixed32 *Cos, Fixed32 *Sin)
{
theta &= (MAX_TRIG - 1);
*Sin = SinTab[theta];
*Cos = CosTab[theta];
}
Fixed32 Tan(Iangle theta)
{
// This shifting stuff is for better accuracy.
theta &= (MAX_TRIG - 1);
return (FixedDiv(SinTab[theta] << 16, CosTab[theta]) >> 16);
}
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The code to generate the lookup tables is below. It creates two .HPP
files, COSTAB.HPP and SINTAB.HPP which can simply be #included into your
source code.
The 'MAGIC' value I used was derived from being able to fit an entire
360? sine and cosine inside an array of 1024 fixed point numbers. These
tables eat up 8K of RAM, so I figured 1024 was plenty. Remember, fixed
point numbers are 4 bytes a piece:
(1024 entries * 4 bytes per entry) = 4096 bytes * 2 tables = 8192 bytes.
????????????????????????????????????????????????????????????????????????????????
#include
#include
#define MAGIC 162.974
#define MAX_TRIG 1024
#define DOUBLE_TO_FIXED(x) ((long)(x * 65536.0 + 0.5))
typedef long Fixed32;
void main()
{
FILE *f;
int i, pos = 0;
Fixed32 z;
f = fopen("sintab.hpp", "wt");
fputs("Fixed32 SinTab[MAX_TRIG] = {\n", f);
for (i = 0; i < 1023; i++)
{
if (pos == 0)
fputs(" ", f);
z = DOUBLE_TO_FIXED(sin((double)i / MAGIC));
fprintf(f, "%10lu, ", z);
pos += 12;
if (pos > 70)
{
fputs("\n", f);
pos = 0;
}
}
z = DOUBLE_TO_FIXED(sin(1023.0 / MAGIC));
fprintf(f, "%10lu };\n ", z);
fclose(f);
pos = 0;
f = fopen("costab.hpp", "wt");
fputs("Fixed32 CosTab[MAX_TRIG] = {\n", f);
for (i = 0; i < 1023; i++)
{
if (pos == 0)
fputs(" ", f);
z = DOUBLE_TO_FIXED(cos((double)i / MAGIC));
fprintf(f, "%10lu, ", z);
pos += 12;
if (pos > 70)
{
fputs("\n", f);
pos = 0;
}
}
z = DOUBLE_TO_FIXED(cos(1023.0 / MAGIC));
fprintf(f, "%10lu };\n ", z);
fclose(f);
}
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[6] Conclusion
??????????????
This is the first part of two files on fixed point math. The next file
covers doing Vector and Matrix calculations.
Enjoy the speed improvements in your program, and as always, if you ever
need any help, feel free to ask us. We may be working on a kick-butt demo,
but we can always spare a couple of minutes to help someone out. :)
- Night Stalker
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