Short: Very fast trigonom. funcs for 040/060.
Author: astegema@ix.urz.uni-heidelberg.de (Achim Stegemann)
Uploader: astegema ix urz uni-heidelberg de (Achim Stegemann)
Type: dev/asm
Architecture: m68k-amigaos
This is an updated version compared to last Aminet upload.
* The fastsin and fastcos functions speed as been improved.
* Time of table-initialization has been almost halved.
* The cycle timing has been corrected.
* I beg your pardon, but... the polynoms are of degree 2 and not 3 as stated in
the last version.
*** History ***
When I was programming my first version of Digital Almanac (Aminet:misc/sci),
I saw, that using trigonometric functions take a long time to calculate.
So I thought that there must be a way to be fast but also to be precise.
Using short tables would burst the speed, but wouldnt be precise enough.
Using Taylor series are only fast for values near zero, but would enlengthen
time for values that are closer to 1 or more, but would use no memory.
As today everybody has enough RAM in his/hers Amiga, memory consuption is no
more a problem. So I decided to write these functions and make them public.
*** Copyright ***
There is no copyright on my idea. Use them as you like it, even if you write
a commercial program. I dont care.
If you have any idea, how to improve speed, tell me !!
Any idea is welcome !!
*** Contents ***
This archive contains assembler source codes for PhxAss that shows, how to
program very fast trigonometric functions (sin,cos,asin,acos,atan2) on a
040/060 CPU.
The source contains interfaces for C, C++ and Assembler stubs.
I myself use PhxAss and StormC V3.
*** Background ***
As you know, the 040/060 CPU does not contain those functions. They have to be
emulated by the 680x0.library. This emulation is very time consuming and not
very multi-tasking friendly.
So StormC offers special algorithms to calculate values from those functions
just by using internal FPU commands. To be 100% compatible to FPU, these
functions need to calculate values with an accuracy of 16 digits.
In every-day-purpose a programmer usually does not need that amount of digits.
10 digits would suffice for their purpose.
This is where my functions join the game.
All fast functions offer an accuracy of about 10 to 13 digits (depending on the
initial value.
*** Speed comparison on a 68060/50 ***
This table shows a speed comparison of the corresponding FPU commands
on my 68060/50. The times are measured in cycles and might vary by
some cycles.
FF=Fast functions
Command 68060.library StormC FF (from C) FF (from Asm)
------------------------------------------------------------------------------
fsin.x fpx 333 284 83 61
fcos.x fpx 329 281 82 60
fsincos.x fpx,fpc:fps 416 --- 90 ---
fasin.x fpx 444 404 98 74
facos.x fpx 434 391 97 74
atan2(y,x) --- 320 211 211
As you can see, the fast functions are up to 5 times faster than the
original commands.
The tan command is not listed here, because tan=sin/cos !!
*** How is it done ? ***
Interpolation of sin, cos, asin and acos is realized by using square polynoms.
Starting your program, at first stage tables are filled with polynomial
coefficients.
When you want to receive the value of an input, the program simply evaluates
the corresponding array entry that belongs to your value.
E.g. sin(x) = (p[0]*x+p[1])*x+p[2]
where p is the double-pointer to the array of coefficients.
*** Memory consumption ***
The sin and cos use a table of 506 kB size.
The asin and acos use a table of each 234 kB size.
The atan2 is transformed into a acos, so its table will be used.
( atan2(y,x)=acos(x/sqrt(x*x+y+y)) )
The initialization of the tables will need about one to two seconds on a 060.
*** Register trashing ***
All functions follow the rules for trashing registers (D0/D1,A0/A1,FP0/FP1).
The assembler functions also restore the FP1 register, so only FP0 is filled with
the desired return value.
The C/C++ functions also restore the D0 and A0 registers (D1 and A1 are untouched
throughout the code.
*** C includes ***
The prototypes of the functions are easy.
double fastsin(double);
double fastcos(double);
void fastsincos(double,double &s,double &c); // (C++)
double fastasin(double);
double fastacos(double);
double fastatan2(double y,double x);
I myself have put them into the math.h include.