compressing Xilinx bitstreams

J

John Larkin

Guest
Forgive me if this has been asked before, but does anybody have
comments or links to simple methods of compressing/decompressing
Xilinx configuration bitstreams? I've been perusing a few of my .rbt
files, and they have long bunches of 1s and 0s (interestingly,
different designs seem to have more 1s, others mostly 0s.) I'd think
that something very simple might achieve pretty serious (as, maybe
2:1-ish) compression without a lot of runtime complexity. We generally
run a uP from EPROM, with the uP code and the packed Xilinx config
stuff in the same eprom, with the uP bit-banging the Xilinx FPGA at
powerup time. So a simple decompressor would be nice.

I did google for this... haven't found much.

Thanks,

John
 
John Larkin wrote:
Forgive me if this has been asked before, but does anybody have
comments or links to simple methods of compressing/decompressing
Xilinx configuration bitstreams? I've been perusing a few of my .rbt
files, and they have long bunches of 1s and 0s (interestingly,
different designs seem to have more 1s, others mostly 0s.) I'd think
that something very simple might achieve pretty serious (as, maybe
2:1-ish) compression without a lot of runtime complexity. We generally
run a uP from EPROM, with the uP code and the packed Xilinx config
stuff in the same eprom, with the uP bit-banging the Xilinx FPGA at
powerup time. So a simple decompressor would be nice.

I did google for this... haven't found much.
There was discussion on this some months ago.
Might show up here ? http://www.fpga-faq.com/

You can run std ZIP tools on the files, to get a quick
'practical limit' indication.
We did some work with Run length compression, which is
very simple (simple enough to code into CPLD), but has medium
compression gains. ISTR about half the gains of ZIP ?

It could be improved with a RLC-Compiler/optimiser that looked
for the best pattern/lengths for that chip, or even bitstream,
as you could store the RLC params as a 'header', but we did not
go that far.

Be a good project for Xilinx to do as an app note :)

It would make sense to target to the pattern-repeat sizes on
devices like FPGAs.

-jg
 
The bit generation tool has an option to compress the .bit file. I use this
when I'm loading over JTAG to save time. I assume Xilinx has info on in
system programming with a compressed .bit file.

However, I've observed the same phenomenon as you: when I zip a .bit file it
is usually less than 50% of the original size. My guess is even a trivial
run length encoding compression would be helpful. There are plenty of
resources for Lempel Ziv compression on the web:

see http://www.dogma.net/markn/articles/lzw/lzw.htm

If you get it working please post/send the result.


"John Larkin" <jjlarkin@highSNIPlandTHIStechPLEASEnology.com> wrote in
message news:hh34d0tud78se5vqejirkc2bvufbd8io3d@4ax.com...
Forgive me if this has been asked before, but does anybody have
comments or links to simple methods of compressing/decompressing
Xilinx configuration bitstreams? I've been perusing a few of my .rbt
files, and they have long bunches of 1s and 0s (interestingly,
different designs seem to have more 1s, others mostly 0s.) I'd think
that something very simple might achieve pretty serious (as, maybe
2:1-ish) compression without a lot of runtime complexity. We generally
run a uP from EPROM, with the uP code and the packed Xilinx config
stuff in the same eprom, with the uP bit-banging the Xilinx FPGA at
powerup time. So a simple decompressor would be nice.

I did google for this... haven't found much.

Thanks,

John
 
John,

I think that I had heard that zipping, and unzipping bit files led to
the most compression (2:1 or better). (classic unix or windows zip/unzip)

I think that a zip/unzip routine would be a great example of something a
uP could do without an unreasonable amount of memory (ROM+RAM) support.

Austin


John Larkin wrote:
Forgive me if this has been asked before, but does anybody have
comments or links to simple methods of compressing/decompressing
Xilinx configuration bitstreams? I've been perusing a few of my .rbt
files, and they have long bunches of 1s and 0s (interestingly,
different designs seem to have more 1s, others mostly 0s.) I'd think
that something very simple might achieve pretty serious (as, maybe
2:1-ish) compression without a lot of runtime complexity. We generally
run a uP from EPROM, with the uP code and the packed Xilinx config
stuff in the same eprom, with the uP bit-banging the Xilinx FPGA at
powerup time. So a simple decompressor would be nice.

I did google for this... haven't found much.

Thanks,

John
 
First, please be aware that the ACSII .rbt file is 8x the simple .bin file
size. Check the bitgen options and you'll find the ability to generate the
straight binary file - 1s and 0s at the bit level, not the ASCII character
level. Compression beyond that may be what you're looking for, but please -
start with the binary file.

"John Larkin" <jjlarkin@highSNIPlandTHIStechPLEASEnology.com> wrote in
message news:hh34d0tud78se5vqejirkc2bvufbd8io3d@4ax.com...
Forgive me if this has been asked before, but does anybody have
comments or links to simple methods of compressing/decompressing
Xilinx configuration bitstreams? I've been perusing a few of my .rbt
files, and they have long bunches of 1s and 0s (interestingly,
different designs seem to have more 1s, others mostly 0s.) I'd think
that something very simple might achieve pretty serious (as, maybe
2:1-ish) compression without a lot of runtime complexity. We generally
run a uP from EPROM, with the uP code and the packed Xilinx config
stuff in the same eprom, with the uP bit-banging the Xilinx FPGA at
powerup time. So a simple decompressor would be nice.

I did google for this... haven't found much.

Thanks,

John
 
Austin Lesea wrote:
John,

I think that I had heard that zipping, and unzipping bit files led to
the most compression (2:1 or better). (classic unix or windows zip/unzip)
Yes, but a compress targeted to FPGA content should be more efficent,
and use less resource than a generic compress.

I think that a zip/unzip routine would be a great example of something a
uP could do without an unreasonable amount of memory (ROM+RAM) support.
One engineer's reasonable is anothers excessive :)

There are two main classes of uC loader:
* Ones that store the compressed stream on-chip, and so can expect
to have good random access, for things like decompress tables.
Large-code uC also tend to have larger RAM
* Ones that store the compressed stream in low-cost serial flash.
In this class, table handling is not as easy.
uC used here could be as miniscule as the PIC10F in SOT23
PIC10F starts at 16 bytes RAM and 256 Words Code space...
CPLDs are also used for loaders, and they can do simple decompress.

=jg
 
"John Larkin" <jjlarkin@highSNIPlandTHIStechPLEASEnology.com> wrote in
message news:hh34d0tud78se5vqejirkc2bvufbd8io3d@4ax.com...
Forgive me if this has been asked before, but does anybody have
comments or links to simple methods of compressing/decompressing
Xilinx configuration bitstreams? I've been perusing a few of my .rbt
files, and they have long bunches of 1s and 0s (interestingly,
different designs seem to have more 1s, others mostly 0s.) I'd think
that something very simple might achieve pretty serious (as, maybe
2:1-ish) compression without a lot of runtime complexity. We generally
run a uP from EPROM, with the uP code and the packed Xilinx config
stuff in the same eprom, with the uP bit-banging the Xilinx FPGA at
powerup time. So a simple decompressor would be nice.

I did google for this... haven't found much.

Thanks,

John
VCC did a package called HOTMan that does compression. It takes the bit file
and turns it into a compressed file that looks like...

int testArray[2669]=\
{
0xddedda78,0xe55c8c5f,0xefe1c079.... }

We get at least 4 to 1 and small designs in big chip can get 50 to 1. The
above format allows you to compile the design into a C/C++ program.

Steve
 
John Larkin wrote:

Forgive me if this has been asked before, but does anybody have
comments or links to simple methods of compressing/decompressing
Xilinx configuration bitstreams? I've been perusing a few of my .rbt
files, and they have long bunches of 1s and 0s (interestingly,
different designs seem to have more 1s, others mostly 0s.) I'd think
that something very simple might achieve pretty serious (as, maybe
2:1-ish) compression without a lot of runtime complexity. We generally
run a uP from EPROM, with the uP code and the packed Xilinx config
stuff in the same eprom, with the uP bit-banging the Xilinx FPGA at
powerup time. So a simple decompressor would be nice.

I did google for this... haven't found much.

Thanks,

John
No links, but have you considered simple run-length limiting? I can
think of at least one scheme that would be guaranteed sub-optimal from a
compression standpoint but that wouldn't take much code -- just encode
any string of 0xff or 0x00 bytes as that byte followed by a count -- so
that 0x00 0x00 0x00 0x00 becomes 0x00 0x04, for instance. You have the
overhead that 0x00 becomes 0x00 0x01, and you also can't encode anything
that spans bytes -- but you may be happy with it none the less.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
 
On Thu, 17 Jun 2004 22:14:15 GMT, "John_H" <johnhandwork@mail.com>
wrote:

First, please be aware that the ACSII .rbt file is 8x the simple .bin file
size. Check the bitgen options and you'll find the ability to generate the
straight binary file - 1s and 0s at the bit level, not the ASCII character
level. Compression beyond that may be what you're looking for, but please -
start with the binary file.
Of course. We have a little utility, vaguely like a linker, that
gobbles up Motorola .s28 files and Xilinx .rbt files and builds a rom
image, all properly squashed into bits. It's cute... it even saves the
beginning of the rbt ASCII header in the rom image for FPGA version
verification. My observation was that the bits themselves include long
runs of 1s or 0s.

I'd like to design a board using a 28-pin eprom (space is at a premium
here) but plan hooks for using a bigger Xilinx chip some day, and then
I'd run out of rom space to store the config bits. So having a
compression scheme would give us the margin to use the small eprom.

Suppose the compressed data were an array of bytes. If the MS bit of a
byte were 0, the remaining 7 bits are to be loaded verbatum; if the MS
bit is a 1, the other 7 bits specify a run of up to 63 1's or 0's.

Something like that; the exact numbers may need tuning. Very easy to
unpack, not hard to encode. I'd have to test some actual config files
to see how good something like this could compress.

John
 
On Thu, 17 Jun 2004 14:41:01 -0700, John Larkin
<jjlarkin@highSNIPlandTHIStechPLEASEnology.com> wrote:

Forgive me if this has been asked before, but does anybody have
comments or links to simple methods of compressing/decompressing
Xilinx configuration bitstreams? I've been perusing a few of my .rbt
files, and they have long bunches of 1s and 0s (interestingly,
different designs seem to have more 1s, others mostly 0s.) I'd think
that something very simple might achieve pretty serious (as, maybe
2:1-ish) compression without a lot of runtime complexity. We generally
run a uP from EPROM, with the uP code and the packed Xilinx config
stuff in the same eprom, with the uP bit-banging the Xilinx FPGA at
powerup time. So a simple decompressor would be nice.

I did google for this... haven't found much.

Thanks,

John
See:

www.ee.washington.edu/people/faculty/hauck/publications/runlength.PDF
www.ee.washington.edu/people/faculty/hauck/publications/runlengthTR.PDF
www.ee.washington.edu/people/faculty/hauck/publications/runlengthJ.pdf

It should be straightforward to generate some RLL compression and
decompression code. You might want to test the algorithms on a PC to
make sure that the decompressed output ends up the same as the
uncompressed input. A garbled bitstream can have the same effect as
the MC6800 HCF opcode...



================================

Greg Neff
VP Engineering
*Microsym* Computers Inc.
greg@guesswhichwordgoeshere.com
 
Hi John,

Forgive me if this has been asked before, but does anybody have
comments or links to simple methods of compressing/decompressing
Xilinx configuration bitstreams?
Can't help you on the Xilinx front, but many of Altera's newest chips
(Cyclone, Stratix II) support on-the-fly decompression of the bitstream.
The Quartus software compresses the bitstream which is then programmed into
the device using pretty much any of the many methods of programming
available, and the chip's configuration controller will decompress the
bitstream that it sees. This typically achieves a 1.9-2.3:1 compression
ratio, depending on the device utilization, RAM contents and such.

Some of our programming devices also can decompress bitstreams on-the-fly,
allowing bitstream compression for other chip families that do not support
decompression internally.

See the Configuration Handbook Volume 2
(http://www.altera.com/literature/hb/cfg/cfg_volume2.pdf) for a detailed
description of device programming and compression options.

Regards,

Paul Leventis
Altera Corp.
 
Paul Leventis (at home) wrote:
Hi John,


Forgive me if this has been asked before, but does anybody have
comments or links to simple methods of compressing/decompressing
Xilinx configuration bitstreams?


Can't help you on the Xilinx front, but many of Altera's newest chips
(Cyclone, Stratix II) support on-the-fly decompression of the bitstream.
The Quartus software compresses the bitstream which is then programmed into
the device using pretty much any of the many methods of programming
available, and the chip's configuration controller will decompress the
bitstream that it sees. This typically achieves a 1.9-2.3:1 compression
ratio, depending on the device utilization, RAM contents and such.
Snip

Has anyone taken the simple step of run ZIP on some Altera compressed
files, to see how much more compression is possible ?

-jg
 
On Thu, 17 Jun 2004 22:09:31 GMT, "Clark Pope" <cepope@mindspring.com>
wrote:

The bit generation tool has an option to compress the .bit file. I use this
when I'm loading over JTAG to save time. I assume Xilinx has info on in
system programming with a compressed .bit file.
This 'compression' merely merges identical frames. The probability of
getting identical frames in a well utilised FPGA isn't very high, so
this doesn't result in much reduction in file size.

Some experiments I did a few years ago (on Virtex-E and Virtex-2
files) indicated that the this compression made subsequent compression
by tools such as gzip *worse*.
It is, however, the only way to speed up JTAG loading.

Regards,
Allan.
 
Jim Granville <no.spam@designtools.co.nz> wrote in message
We did some work with Run length compression, which is
very simple (simple enough to code into CPLD), but has medium
compression gains. ISTR about half the gains of ZIP ?
Interesting. A student of mine did the same as a semester project.
The goal was to find a compression method simple enough that the logic
to programm an FPGA from a NOR-Flash would fit into an XC9536.

For XC4K FPGAs he was very successfull. He achieved a compression in
the range of 50% to 65% size compared to the original with runlength
encoding of 1s only.
This is almost as good as zip.

The Virtex family seems to use it's configuration bits a lot more
efficiently. (encoded switch configurations ?). He could not find any
simple solution for those.

Kolja Sulimma
 
Kolja Sulimma wrote:
Jim Granville <no.spam@designtools.co.nz> wrote in message

We did some work with Run length compression, which is
very simple (simple enough to code into CPLD), but has medium
compression gains. ISTR about half the gains of ZIP ?


Interesting. A student of mine did the same as a semester project.
The goal was to find a compression method simple enough that the logic
to programm an FPGA from a NOR-Flash would fit into an XC9536.

For XC4K FPGAs he was very successfull. He achieved a compression in
the range of 50% to 65% size compared to the original with runlength
encoding of 1s only.
This is almost as good as zip.

The Virtex family seems to use it's configuration bits a lot more
efficiently. (encoded switch configurations ?). He could not find any
simple solution for those.
Maybe you could have allowed him to also use a 9572 ( or XC2C64) ?

If there was a big change between families, it sounds like Xilinx
followed the same path, and did a simple reduction in
bit-encode with some small RLC - after all, a 9536 level resource
will be miniscule in a FPGA.
 
While this doesn't exactly answer your question, the new Xilinx XCFP
serial PROMs support storage of compressed bitstream data. The data is
compressed when you translate to the PROM format and the PROM does the
decompression before delivery to the FPGA.

http://www.xilinx.com/bvdocs/publications/ds123.pdf



John Larkin wrote:

Forgive me if this has been asked before, but does anybody have
comments or links to simple methods of compressing/decompressing
Xilinx configuration bitstreams? I've been perusing a few of my .rbt
files, and they have long bunches of 1s and 0s (interestingly,
different designs seem to have more 1s, others mostly 0s.) I'd think
that something very simple might achieve pretty serious (as, maybe
2:1-ish) compression without a lot of runtime complexity. We generally
run a uP from EPROM, with the uP code and the packed Xilinx config
stuff in the same eprom, with the uP bit-banging the Xilinx FPGA at
powerup time. So a simple decompressor would be nice.

I did google for this... haven't found much.

Thanks,

John
 
Forgive me if this has been asked before, but does anybody have
comments or links to simple methods of compressing/decompressing
Xilinx configuration bitstreams? I've been perusing a few of my .rbt
My application uses a 128Kb flash micro (M16C) to program a SpartanIIE-100
in slave parallel mode. Back "then", external serial config. memory price
was outrageusly high. Besides, by keeping all inside the main micro, I now
can remote upgrade my FPGA code very very easily.

To free some more Kb for a new requirement, I compressed the bit-stream with
a simple LZW implementation, and decompressed
on the fly before sending to FPGA.

Here are some results from a real design.

Starting from uncompressed design BIN file:
CAMERA.BIN: 107980 bytes
LZW compression with different "dictionary table" bit length:

BIT SIZE
10 73853
11 63283
12 61442
13 58012
14 56803

Same design, but BIN file has been compressed by ISE:

CAMERA_COMP.BIN 97944 (what I was using before; leaved 30 Kb for my app.
code)
After LZW compression:

BIT SIZE
10 73475
11 65442
12 62865
13 58795
14 57687

As expected, you get slightly better results starting from uncompressed BIN
stream, when using simple compression algorithms.

Even after adding code decompression, I saved tens of flash Kb to implement
more features at no cost.
Initial configuration has slowed down, of course: before I was pumping out
data to FPGA as fast as possible with string move assembly instructions.
Now I have to decompress on the fly.
Times stays in the hundreds of ms range, tough.

RAM USAGE during decompression:

from bit width of string table, you define the table size this way (smaller
prime number larger than 2^bits):

BITS TABLE SIZE
14 18041
13 9029
12 5021
11 2053
<= 10 1031

During decompression, you will need 3*TABLE_SIZE bytes + a decode buffer;
decode buffer size can be in the 2000-4000 bytes range.
Yes, it's big. But...given that FPGA programming is the first thing I do at
startup, I can use up to all the M16C RAM available; after FPGA programming,
all that ram is available again for my code.
Even much less, with some checks during compression.


I used a slightly modified version of the code for LZW compression published
by Mark. Nelson on DDJ, Oct. 89.
Google found a copy of the article here:
http://www.dogma.net/markn/articles/lzw/lzw.htm


Please, post your results if you tested for have different implementations!
 
"John Larkin" <jjlarkin@highSNIPlandTHIStechPLEASEnology.com> escribió en el
mensaje news:hh34d0tud78se5vqejirkc2bvufbd8io3d@4ax.com...
Forgive me if this has been asked before, but does anybody have
comments or links to simple methods of compressing/decompressing
Xilinx configuration bitstreams? I've been perusing a few of my .rbt
files, and they have long bunches of 1s and 0s (interestingly,
different designs seem to have more 1s, others mostly 0s.) I'd think
that something very simple might achieve pretty serious (as, maybe
2:1-ish) compression without a lot of runtime complexity. We generally
run a uP from EPROM, with the uP code and the packed Xilinx config
stuff in the same eprom, with the uP bit-banging the Xilinx FPGA at
powerup time. So a simple decompressor would be nice.

I did google for this... haven't found much.
try searching for RLE (run length encoding) that's the encoding used for
..PCX graphic files

Thanks,

John
 
John Larkin <jjlarkin@highSNIPlandTHIStechPLEASEnology.com> wrote:

Forgive me if this has been asked before, but does anybody have
comments or links to simple methods of compressing/decompressing
Xilinx configuration bitstreams? I've been perusing a few of my .rbt
files, and they have long bunches of 1s and 0s (interestingly,
different designs seem to have more 1s, others mostly 0s.) I'd think
that something very simple might achieve pretty serious (as, maybe
2:1-ish) compression without a lot of runtime complexity. We generally
run a uP from EPROM, with the uP code and the packed Xilinx config
stuff in the same eprom, with the uP bit-banging the Xilinx FPGA at
powerup time. So a simple decompressor would be nice.

I did google for this... haven't found much.
Tried it but found the files aren't reduced in size much and more
important, the software required to decompress the file eats away all
the savings for a 400k device. In other words: Unless you have more
than around half a million gates of configuration data, it's not worth
it.

--
Reply to nico@nctdevpuntnl (punt=.)
Bedrijven en winkels vindt U op www.adresboekje.nl
 
On Sat, 19 Jun 2004 15:27:40 GMT, nico@puntnl.niks (Nico Coesel)
wrote:

John Larkin <jjlarkin@highSNIPlandTHIStechPLEASEnology.com> wrote:


Forgive me if this has been asked before, but does anybody have
comments or links to simple methods of compressing/decompressing
Xilinx configuration bitstreams? I've been perusing a few of my .rbt
files, and they have long bunches of 1s and 0s (interestingly,
different designs seem to have more 1s, others mostly 0s.) I'd think
that something very simple might achieve pretty serious (as, maybe
2:1-ish) compression without a lot of runtime complexity. We generally
run a uP from EPROM, with the uP code and the packed Xilinx config
stuff in the same eprom, with the uP bit-banging the Xilinx FPGA at
powerup time. So a simple decompressor would be nice.

I did google for this... haven't found much.

Tried it but found the files aren't reduced in size much and more
important, the software required to decompress the file eats away all
the savings for a 400k device. In other words: Unless you have more
than around half a million gates of configuration data, it's not worth
it.

OK, bear with me on this. Here's a piece of a .rbt for a Spartan XL...

01111111111111111111111111111111111111111111111111111111111011111111111111111111111111110111111110111111011111111110111111110101011101111110111111011111111111111111110011111111111111111111111111111111111111111111111111111110101
01111111111111111111111111111111111111111111111111111111111111111111111111111101111111111111111111111111110111111101111111111111111110111111111111110111111111111011111101111111111111111111111111111111111111111111111111111110011
01111111111111111111111111111111111111111111111111111100011111111111111111101111111100111111110011111111111111011101111111111100111011110011111011111111111111111111111110110111001111111111111111111111110111111011111111111111011
01111111111111111111111111111111111111111111111111111111011111111111111111101111111101011111111110011111111111111100111111111111011111111101111111111111111111110111101111111111110111111111111111111111111111111111111111111111110
01111111111111111111111111111111111111111111110111111111111111111111111111111111111111111111111011111111111111111011010111111110011111111011111111111011111011111011110101111111000111111111011111111111101111111111111110101101111
00111111111111111111111111111111111111111111000111111111111111111111111111111111111111111111111111111111110111111111110110111111011111111111111111111101111111111111111101111111110111111100011111111111111111111111111101101100000
01111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111110111111111101110111101011111111111111111111111111111111111111101111111101111111111111111111111111111111111111110111111100
00011111111111111111111111111111111111111111111111111111111100111111011111111111001111110110101111001111111101111111111111001111111100111111111001111101101011110110011111101010111101111111111111111111111010111100111111111111000
01101011111111111111111111111111111111111111111111111111111110101111111111111111101011111110011110111111110101001110111111101011011100111111111010010111001111110110101101111111111111111111111111111111110011111101111111010100111
01111011111111111111111111111111111111111111111111111111111111111111011111111111111111110111111111111111110110111111111111101011011111111111111111111101111111111111111101111111111101111111111111111111111111111111111111011111010
01101111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111110011111111111111111111111111111111111111111001111111111111111110111111111111111111111111111111111111111111111111100111110011000
01111111111111111111111111111111111111111111111111111111111011111111101111111110111111111011111111101111011110111111111110111101111001101111101111111110101011111011010111101111111110111111101111111111111111111110111111110100011
01101111111111111111111111111111111111111111111011111111111111111111111111111111111111111111111111111111111101011010111111111111110111101111111111111101011011111111111111011110111111111111111111111111111111111111111111111110101
00111100111111111111111111111111111111111111111011111111110100111111100011111101001111111000111111111111111110101011111101101011110010011111011011111111101011110110101111010001111110111111111111111111111111111101111100111110111


Where there are lots of 1's. Other hunks of this file are almost all
1's. So what we need is a not-very-general compression scheme, with
the only "dictionary" entry being "the following is a hunk of 1's". So
the decompressor could be very simple.

Interestingly, this is for a Spartan 2:

00000000000001001000000000000000
00000000000000000000000000000000
00000000000100100000000000000000
00000000000000000000000000000000
00000000000000000000000000000000
00000000000000000000000000000000
00000000000000000000000000000000
00000000000001001000000000000000
00000000000000000000000000000000
00000000000100100100000000000000
00000000000000000000000000000000
00000000000000000000000000000000
00000000000000000000000000000000
00000000000000000000000000000000
00000000000001001100000000000000
00000000000000000000000000000000
11111111000100110000000100000100
00000000010001000000000000010000
00000000000001110100100000000000
11010100000000000011010000000000
00000001000000000000000000001000
00111111110001000000000000000000

Which has long runs of zeroes!

Just eyeballing these files, it looks like something very simple could
get at least a 2:1 squash factor.

John
 

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