+---------------------------------------------------------------------------+
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| wm-FPU-emu an FPU emulator for 80386 and 80486SX microprocessors. |
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| |
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| Copyright (C) 1992,1993,1994,1995,1996,1997,1999 |
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| W. Metzenthen, 22 Parker St, Ormond, Vic 3163, |
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| Australia. E-mail billm@melbpc.org.au |
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| |
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| This program is free software; you can redistribute it and/or modify |
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| it under the terms of the GNU General Public License version 2 as |
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| published by the Free Software Foundation. |
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| |
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| This program is distributed in the hope that it will be useful, |
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| but WITHOUT ANY WARRANTY; without even the implied warranty of |
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| MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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| GNU General Public License for more details. |
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| |
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| You should have received a copy of the GNU General Public License |
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| along with this program; if not, write to the Free Software |
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| Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. |
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| |
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+---------------------------------------------------------------------------+
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|
|
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wm-FPU-emu is an FPU emulator for Linux. It is derived from wm-emu387
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which was my 80387 emulator for early versions of djgpp (gcc under
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msdos); wm-emu387 was in turn based upon emu387 which was written by
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DJ Delorie for djgpp. The interface to the Linux kernel is based upon
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the original Linux math emulator by Linus Torvalds.
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My target FPU for wm-FPU-emu is that described in the Intel486
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Programmer's Reference Manual (1992 edition). Unfortunately, numerous
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facets of the functioning of the FPU are not well covered in the
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Reference Manual. The information in the manual has been supplemented
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with measurements on real 80486's. Unfortunately, it is simply not
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possible to be sure that all of the peculiarities of the 80486 have
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been discovered, so there is always likely to be obscure differences
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in the detailed behaviour of the emulator and a real 80486.
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wm-FPU-emu does not implement all of the behaviour of the 80486 FPU,
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but is very close. See "Limitations" later in this file for a list of
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some differences.
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Please report bugs, etc to me at:
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billm@melbpc.org.au
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or b.metzenthen@medoto.unimelb.edu.au
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For more information on the emulator and on floating point topics, see
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my web pages, currently at http://www.suburbia.net/~billm/
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|
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--Bill Metzenthen
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December 1999
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----------------------- Internals of wm-FPU-emu -----------------------
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Numeric algorithms:
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(1) Add, subtract, and multiply. Nothing remarkable in these.
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(2) Divide has been tuned to get reasonable performance. The algorithm
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is not the obvious one which most people seem to use, but is designed
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to take advantage of the characteristics of the 80386. I expect that
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it has been invented many times before I discovered it, but I have not
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seen it. It is based upon one of those ideas which one carries around
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for years without ever bothering to check it out.
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(3) The sqrt function has been tuned to get good performance. It is based
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upon Newton's classic method. Performance was improved by capitalizing
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upon the properties of Newton's method, and the code is once again
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structured taking account of the 80386 characteristics.
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(4) The trig, log, and exp functions are based in each case upon quasi-
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"optimal" polynomial approximations. My definition of "optimal" was
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based upon getting good accuracy with reasonable speed.
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(5) The argument reducing code for the trig function effectively uses
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a value of pi which is accurate to more than 128 bits. As a consequence,
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the reduced argument is accurate to more than 64 bits for arguments up
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to a few pi, and accurate to more than 64 bits for most arguments,
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even for arguments approaching 2^63. This is far superior to an
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80486, which uses a value of pi which is accurate to 66 bits.
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The code of the emulator is complicated slightly by the need to
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account for a limited form of re-entrancy. Normally, the emulator will
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emulate each FPU instruction to completion without interruption.
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However, it may happen that when the emulator is accessing the user
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memory space, swapping may be needed. In this case the emulator may be
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temporarily suspended while disk i/o takes place. During this time
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another process may use the emulator, thereby perhaps changing static
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variables. The code which accesses user memory is confined to five
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files:
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fpu_entry.c
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reg_ld_str.c
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load_store.c
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get_address.c
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errors.c
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As from version 1.12 of the emulator, no static variables are used
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(apart from those in the kernel's per-process tables). The emulator is
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therefore now fully re-entrant, rather than having just the restricted
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form of re-entrancy which is required by the Linux kernel.
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----------------------- Limitations of wm-FPU-emu -----------------------
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There are a number of differences between the current wm-FPU-emu
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(version 2.01) and the 80486 FPU (apart from bugs). The differences
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are fewer than those which applied to the 1.xx series of the emulator.
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Some of the more important differences are listed below:
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The Roundup flag does not have much meaning for the transcendental
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functions and its 80486 value with these functions is likely to differ
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from its emulator value.
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In a few rare cases the Underflow flag obtained with the emulator will
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be different from that obtained with an 80486. This occurs when the
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following conditions apply simultaneously:
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(a) the operands have a higher precision than the current setting of the
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precision control (PC) flags.
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(b) the underflow exception is masked.
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(c) the magnitude of the exact result (before rounding) is less than 2^-16382.
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(d) the magnitude of the final result (after rounding) is exactly 2^-16382.
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(e) the magnitude of the exact result would be exactly 2^-16382 if the
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operands were rounded to the current precision before the arithmetic
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operation was performed.
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If all of these apply, the emulator will set the Underflow flag but a real
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80486 will not.
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NOTE: Certain formats of Extended Real are UNSUPPORTED. They are
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unsupported by the 80486. They are the Pseudo-NaNs, Pseudoinfinities,
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and Unnormals. None of these will be generated by an 80486 or by the
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emulator. Do not use them. The emulator treats them differently in
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detail from the way an 80486 does.
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Self modifying code can cause the emulator to fail. An example of such
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code is:
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movl %esp,[%ebx]
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fld1
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The FPU instruction may be (usually will be) loaded into the pre-fetch
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queue of the CPU before the mov instruction is executed. If the
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destination of the 'movl' overlaps the FPU instruction then the bytes
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in the prefetch queue and memory will be inconsistent when the FPU
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instruction is executed. The emulator will be invoked but will not be
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able to find the instruction which caused the device-not-present
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exception. For this case, the emulator cannot emulate the behaviour of
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an 80486DX.
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Handling of the address size override prefix byte (0x67) has not been
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extensively tested yet. A major problem exists because using it in
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vm86 mode can cause a general protection fault. Address offsets
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greater than 0xffff appear to be illegal in vm86 mode but are quite
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acceptable (and work) in real mode. A small test program developed to
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check the addressing, and which runs successfully in real mode,
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crashes dosemu under Linux and also brings Windows down with a general
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protection fault message when run under the MS-DOS prompt of Windows
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3.1. (The program simply reads data from a valid address).
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The emulator supports 16-bit protected mode, with one difference from
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an 80486DX. A 80486DX will allow some floating point instructions to
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write a few bytes below the lowest address of the stack. The emulator
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will not allow this in 16-bit protected mode: no instructions are
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allowed to write outside the bounds set by the protection.
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----------------------- Performance of wm-FPU-emu -----------------------
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Speed.
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-----
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The speed of floating point computation with the emulator will depend
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upon instruction mix. Relative performance is best for the instructions
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which require most computation. The simple instructions are adversely
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affected by the FPU instruction trap overhead.
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|
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Timing: Some simple timing tests have been made on the emulator functions.
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The times include load/store instructions. All times are in microseconds
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measured on a 33MHz 386 with 64k cache. The Turbo C tests were under
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ms-dos, the next two columns are for emulators running with the djgpp
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ms-dos extender. The final column is for wm-FPU-emu in Linux 0.97,
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using libm4.0 (hard).
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function Turbo C djgpp 1.06 WM-emu387 wm-FPU-emu
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+ 60.5 154.8 76.5 139.4
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- 61.1-65.5 157.3-160.8 76.2-79.5 142.9-144.7
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* 71.0 190.8 79.6 146.6
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/ 61.2-75.0 261.4-266.9 75.3-91.6 142.2-158.1
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sin() 310.8 4692.0 319.0 398.5
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cos() 284.4 4855.2 308.0 388.7
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tan() 495.0 8807.1 394.9 504.7
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atan() 328.9 4866.4 601.1 419.5-491.9
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sqrt() 128.7 crashed 145.2 227.0
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log() 413.1-419.1 5103.4-5354.21 254.7-282.2 409.4-437.1
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exp() 479.1 6619.2 469.1 850.8
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The performance under Linux is improved by the use of look-ahead code.
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The following results show the improvement which is obtained under
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Linux due to the look-ahead code. Also given are the times for the
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original Linux emulator with the 4.1 'soft' lib.
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[ Linus' note: I changed look-ahead to be the default under linux, as
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there was no reason not to use it after I had edited it to be
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disabled during tracing ]
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wm-FPU-emu w original w
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look-ahead 'soft' lib
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+ 106.4 190.2
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- 108.6-111.6 192.4-216.2
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* 113.4 193.1
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/ 108.8-124.4 700.1-706.2
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sin() 390.5 2642.0
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cos() 381.5 2767.4
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tan() 496.5 3153.3
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atan() 367.2-435.5 2439.4-3396.8
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sqrt() 195.1 4732.5
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log() 358.0-387.5 3359.2-3390.3
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exp() 619.3 4046.4
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These figures are now somewhat out-of-date. The emulator has become
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progressively slower for most functions as more of the 80486 features
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have been implemented.
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----------------------- Accuracy of wm-FPU-emu -----------------------
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The accuracy of the emulator is in almost all cases equal to or better
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than that of an Intel 80486 FPU.
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The results of the basic arithmetic functions (+,-,*,/), and fsqrt
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match those of an 80486 FPU. They are the best possible; the error for
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these never exceeds 1/2 an lsb. The fprem and fprem1 instructions
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return exact results; they have no error.
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The following table compares the emulator accuracy for the sqrt(),
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trig and log functions against the Turbo C "emulator". For this table,
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each function was tested at about 400 points. Ideal worst-case results
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would be 64 bits. The reduced Turbo C accuracy of cos() and tan() for
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arguments greater than pi/4 can be thought of as being related to the
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precision of the argument x; e.g. an argument of pi/2-(1e-10) which is
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accurate to 64 bits can result in a relative accuracy in cos() of
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about 64 + log2(cos(x)) = 31 bits.
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Function Tested x range Worst result Turbo C
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(relative bits)
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sqrt(x) 1 .. 2 64.1 63.2
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atan(x) 1e-10 .. 200 64.2 62.8
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cos(x) 0 .. pi/2-(1e-10) 64.4 (x <= pi/4) 62.4
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64.1 (x = pi/2-(1e-10)) 31.9
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sin(x) 1e-10 .. pi/2 64.0 62.8
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tan(x) 1e-10 .. pi/2-(1e-10) 64.0 (x <= pi/4) 62.1
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64.1 (x = pi/2-(1e-10)) 31.9
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exp(x) 0 .. 1 63.1 ** 62.9
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log(x) 1+1e-6 .. 2 63.8 ** 62.1
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** The accuracy for exp() and log() is low because the FPU (emulator)
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does not compute them directly; two operations are required.
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The emulator passes the "paranoia" tests (compiled with gcc 2.3.3 or
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later) for 'float' variables (24 bit precision numbers) when precision
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control is set to 24, 53 or 64 bits, and for 'double' variables (53
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bit precision numbers) when precision control is set to 53 bits (a
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properly performing FPU cannot pass the 'paranoia' tests for 'double'
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variables when precision control is set to 64 bits).
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The code for reducing the argument for the trig functions (fsin, fcos,
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fptan and fsincos) has been improved and now effectively uses a value
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for pi which is accurate to more than 128 bits precision. As a
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consequence, the accuracy of these functions for large arguments has
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been dramatically improved (and is now very much better than an 80486
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FPU). There is also now no degradation of accuracy for fcos and fptan
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for operands close to pi/2. Measured results are (note that the
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definition of accuracy has changed slightly from that used for the
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above table):
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Function Tested x range Worst result
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(absolute bits)
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cos(x) 0 .. 9.22e+18 62.0
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sin(x) 1e-16 .. 9.22e+18 62.1
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tan(x) 1e-16 .. 9.22e+18 61.8
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It is possible with some effort to find very large arguments which
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give much degraded precision. For example, the integer number
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8227740058411162616.0
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is within about 10e-7 of a multiple of pi. To find the tan (for
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example) of this number to 64 bits precision it would be necessary to
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have a value of pi which had about 150 bits precision. The FPU
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emulator computes the result to about 42.6 bits precision (the correct
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result is about -9.739715e-8). On the other hand, an 80486 FPU returns
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0.01059, which in relative terms is hopelessly inaccurate.
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For arguments close to critical angles (which occur at multiples of
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pi/2) the emulator is more accurate than an 80486 FPU. For very large
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arguments, the emulator is far more accurate.
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Prior to version 1.20 of the emulator, the accuracy of the results for
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the transcendental functions (in their principal range) was not as
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good as the results from an 80486 FPU. From version 1.20, the accuracy
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has been considerably improved and these functions now give measured
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worst-case results which are better than the worst-case results given
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by an 80486 FPU.
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The following table gives the measured results for the emulator. The
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number of randomly selected arguments in each case is about half a
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million. The group of three columns gives the frequency of the given
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accuracy in number of times per million, thus the second of these
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columns shows that an accuracy of between 63.80 and 63.89 bits was
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found at a rate of 133 times per one million measurements for fsin.
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The results show that the fsin, fcos and fptan instructions return
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results which are in error (i.e. less accurate than the best possible
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result (which is 64 bits)) for about one per cent of all arguments
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between -pi/2 and +pi/2. The other instructions have a lower
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frequency of results which are in error. The last two columns give
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the worst accuracy which was found (in bits) and the approximate value
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of the argument which produced it.
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frequency (per M)
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------------------- ---------------
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instr arg range # tests 63.7 63.8 63.9 worst at arg
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bits bits bits bits
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----- ------------ ------- ---- ---- ----- ----- --------
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fsin (0,pi/2) 547756 0 133 10673 63.89 0.451317
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fcos (0,pi/2) 547563 0 126 10532 63.85 0.700801
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fptan (0,pi/2) 536274 11 267 10059 63.74 0.784876
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fpatan 4 quadrants 517087 0 8 1855 63.88 0.435121 (4q)
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fyl2x (0,20) 541861 0 0 1323 63.94 1.40923 (x)
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fyl2xp1 (-.293,.414) 520256 0 0 5678 63.93 0.408542 (x)
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f2xm1 (-1,1) 538847 4 481 6488 63.79 0.167709
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Tests performed on an 80486 FPU showed results of lower accuracy. The
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following table gives the results which were obtained with an AMD
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486DX2/66 (other tests indicate that an Intel 486DX produces
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identical results). The tests were basically the same as those used
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to measure the emulator (the values, being random, were in general not
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the same). The total number of tests for each instruction are given
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at the end of the table, in case each about 100k tests were performed.
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Another line of figures at the end of the table shows that most of the
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instructions return results which are in error for more than 10
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percent of the arguments tested.
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The numbers in the body of the table give the approx number of times a
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result of the given accuracy in bits (given in the left-most column)
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was obtained per one million arguments. For three of the instructions,
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two columns of results are given: * The second column for f2xm1 gives
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the number cases where the results of the first column were for a
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positive argument, this shows that this instruction gives better
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results for positive arguments than it does for negative. * In the
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cases of fcos and fptan, the first column gives the results when all
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cases where arguments greater than 1.5 were removed from the results
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given in the second column. Unlike the emulator, an 80486 FPU returns
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results of relatively poor accuracy for these instructions when the
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argument approaches pi/2. The table does not show those cases when the
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accuracy of the results were less than 62 bits, which occurs quite
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often for fsin and fptan when the argument approaches pi/2. This poor
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accuracy is discussed above in relation to the Turbo C "emulator", and
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the accuracy of the value of pi.
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bits f2xm1 f2xm1 fpatan fcos fcos fyl2x fyl2xp1 fsin fptan fptan
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62.0 0 0 0 0 437 0 0 0 0 925
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62.1 0 0 10 0 894 0 0 0 0 1023
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62.2 14 0 0 0 1033 0 0 0 0 945
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62.3 57 0 0 0 1202 0 0 0 0 1023
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62.4 385 0 0 10 1292 0 23 0 0 1178
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62.5 1140 0 0 119 1649 0 39 0 0 1149
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62.6 2037 0 0 189 1620 0 16 0 0 1169
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62.7 5086 14 0 646 2315 10 101 35 39 1402
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62.8 8818 86 0 984 3050 59 287 131 224 2036
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62.9 11340 1355 0 2126 4153 79 605 357 321 1948
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63.0 15557 4750 0 3319 5376 246 1281 862 808 2688
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63.1 20016 8288 0 4620 6628 511 2569 1723 1510 3302
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63.2 24945 11127 10 6588 8098 1120 4470 2968 2990 4724
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63.3 25686 12382 69 8774 10682 1906 6775 4482 5474 7236
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63.4 29219 14722 79 11109 12311 3094 9414 7259 8912 10587
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63.5 30458 14936 393 13802 15014 5874 12666 9609 13762 15262
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63.6 32439 16448 1277 17945 19028 10226 15537 14657 19158 20346
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63.7 35031 16805 4067 23003 23947 18910 20116 21333 25001 26209
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63.8 33251 15820 7673 24781 25675 24617 25354 24440 29433 30329
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63.9 33293 16833 18529 28318 29233 31267 31470 27748 29676 30601
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Per cent with error:
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30.9 3.2 18.5 9.8 13.1 11.6 17.4
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Total arguments tested:
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70194 70099 101784 100641 100641 101799 128853 114893 102675 102675
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|
|
------------------------- Contributors -------------------------------
|
|
A number of people have contributed to the development of the
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emulator, often by just reporting bugs, sometimes with suggested
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fixes, and a few kind people have provided me with access in one way
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or another to an 80486 machine. Contributors include (to those people
|
who I may have forgotten, please forgive me):
|
|
Linus Torvalds
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Tommy.Thorn@daimi.aau.dk
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Andrew.Tridgell@anu.edu.au
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Nick Holloway, alfie@dcs.warwick.ac.uk
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Hermano Moura, moura@dcs.gla.ac.uk
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Jon Jagger, J.Jagger@scp.ac.uk
|
Lennart Benschop
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Brian Gallew, geek+@CMU.EDU
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Thomas Staniszewski, ts3v+@andrew.cmu.edu
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Martin Howell, mph@plasma.apana.org.au
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M Saggaf, alsaggaf@athena.mit.edu
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Peter Barker, PETER@socpsy.sci.fau.edu
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tom@vlsivie.tuwien.ac.at
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Dan Russel, russed@rpi.edu
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Daniel Carosone, danielce@ee.mu.oz.au
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cae@jpmorgan.com
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Hamish Coleman, t933093@minyos.xx.rmit.oz.au
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Bruce Evans, bde@kralizec.zeta.org.au
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Timo Korvola, Timo.Korvola@hut.fi
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Rick Lyons, rick@razorback.brisnet.org.au
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Rick, jrs@world.std.com
|
|
...and numerous others who responded to my request for help with
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a real 80486.
|
