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 +---------------------------------------------------------------------------+
 |  wm-FPU-emu   an FPU emulator for 80386 and 80486SX microprocessors.      |
 |                                                                           |
 | Copyright (C) 1992,1993,1994                                              |
 |                       W. Metzenthen, 22 Parker St, Ormond, Vic 3163,      |
 |                       Australia.  E-mail   [e-mail]    |
 |                                                                           |
 |    This program is free software; you can redistribute it and/or modify   |
 |    it under the terms of the GNU General Public License version 2 as      |
 |    published by the Free Software Foundation.                             |
 |                                                                           |
 |    This program is distributed in the hope that it will be useful,        |
 |    but WITHOUT ANY WARRANTY; without even the implied warranty of         |
 |    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the          |
 |    GNU General Public License for more details.                           |
 |                                                                           |
 |    You should have received a copy of the GNU General Public License      |
 |    along with this program; if not, write to the Free Software            |
 |    Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.              |
 |                                                                           |
 +---------------------------------------------------------------------------+



wm-FPU-emu is an FPU emulator for Linux. It is derived from wm-emu387
which is my 80387 emulator for djgpp (gcc under msdos); wm-emu387 was
in turn based upon emu387 which was written by DJ Delorie for djgpp.
The interface to the Linux kernel is based upon the original Linux
math emulator by Linus Torvalds.

My target FPU for wm-FPU-emu is that described in the Intel486
Programmer's Reference Manual (1992 edition). Unfortunately, numerous
facets of the functioning of the FPU are not well covered in the
Reference Manual. The information in the manual has been supplemented
with measurements on real 80486's. Unfortunately, it is simply not
possible to be sure that all of the peculiarities of the 80486 have
been discovered, so there is always likely to be obscure differences
in the detailed behaviour of the emulator and a real 80486.

wm-FPU-emu does not implement all of the behaviour of the 80486 FPU.
See "Limitations" later in this file for a list of some differences.

Please report bugs, etc to me at:
       [e-mail]
  or at:
       [e-mail]


--Bill Metzenthen
  Feb 1994


----------------------- Internals of wm-FPU-emu -----------------------

Numeric algorithms:
(1) Add, subtract, and multiply. Nothing remarkable in these.
(2) Divide has been tuned to get reasonable performance. The algorithm
    is not the obvious one which most people seem to use, but is designed
    to take advantage of the characteristics of the 80386. I expect that
    it has been invented many times before I discovered it, but I have not
    seen it. It is based upon one of those ideas which one carries around
    for years without ever bothering to check it out.
(3) The sqrt function has been tuned to get good performance. It is based
    upon Newton's classic method. Performance was improved by capitalizing
    upon the properties of Newton's method, and the code is once again
    structured taking account of the 80386 characteristics.
(4) The trig, log, and exp functions are based in each case upon quasi-
    "optimal" polynomial approximations. My definition of "optimal" was
    based upon getting good accuracy with reasonable speed.
(5) The argument reducing code for the trig function effectively uses
    a value of pi which is accurate to more than 128 bits. As a consequence,
    the reduced argument is accurate to more than 64 bits for arguments up
    to a few pi, and accurate to more than 64 bits for most arguments,
    even for arguments approaching 2^63. This is far superior to an
    80486, which uses a value of pi which is accurate to 66 bits.

The code of the emulator is complicated slightly by the need to
account for a limited form of re-entrancy. Normally, the emulator will
emulate each FPU instruction to completion without interruption.
However, it may happen that when the emulator is accessing the user
memory space, swapping may be needed. In this case the emulator may be
temporarily suspended while disk i/o takes place. During this time
another process may use the emulator, thereby changing some static
variables (eg FPU_st0_ptr, etc). The code which accesses user memory
is confined to five files:
    fpu_entry.c
    reg_ld_str.c
    load_store.c
    get_address.c
    errors.c

----------------------- Limitations of wm-FPU-emu -----------------------

There are a number of differences between the current wm-FPU-emu
(version beta 1.10) and the 80486 FPU (apart from bugs). Some of the
more important differences are listed below:

The Roundup flag does not have much meaning for the transcendental
functions and its 80486 value with these functions is likely to differ
from its emulator value.

In a few rare cases the Underflow flag obtained with the emulator will
be different from that obtained with an 80486. This occurs when the
following conditions apply simultaneously:
(a) the operands have a higher precision than the current setting of the
    precision control (PC) flags.
(b) the underflow exception is masked.
(c) the magnitude of the exact result (before rounding) is less than 2^-16382.
(d) the magnitude of the final result (after rounding) is exactly 2^-16382.
(e) the magnitude of the exact result would be exactly 2^-16382 if the
    operands were rounded to the current precision before the arithmetic
    operation was performed.
If all of these apply, the emulator will set the Underflow flag but a real
80486 will not.

NOTE: Certain formats of Extended Real are UNSUPPORTED. They are
unsupported by the 80486. They are the Pseudo-NaNs, Pseudoinfinities,
and Unnormals. None of these will be generated by an 80486 or by the
emulator. Do not use them. The emulator treats them differently in
detail from the way an 80486 does.

The emulator treats PseudoDenormals differently from an 80486. These
numbers are in fact properly normalised numbers with the exponent
offset by 1, and the emulator treats them as such. Unlike the 80486,
the emulator does not generate a Denormal Operand exception for these
numbers. The arithmetical results produced when using such a number as
an operand are the same for the emulator and a real 80486 (apart from
any slight precision difference for the transcendental functions).
Neither the emulator nor an 80486 produces one of these numbers as the
result of any arithmetic operation. An 80486 can keep one of these
numbers in an FPU register with its identity as a PseudoDenormal, but
the emulator will not; they are always converted to a valid number.

Self modifying code can cause the emulator to fail. An example of such
code is:
          movl %esp,[%ebx]
	  fld1
The FPU instruction may be (usually will be) loaded into the pre-fetch
queue of the cpu before the mov instruction is executed. If the
destination of the 'movl' overlaps the FPU instruction then the bytes
in the prefetch queue and memory will be inconsistent when the FPU
instruction is executed. The emulator will be invoked but will not be
able to find the instruction which caused the device-not-present
exception. For this case, the emulator cannot emulate the behaviour of
an 80486DX.

----------------------- Performance of wm-FPU-emu -----------------------

Speed.
-----

The speed of floating point computation with the emulator will depend
upon instruction mix. Relative performance is best for the instructions
which require most computation. The simple instructions are adversely
affected by the fpu instruction trap overhead.


Timing: Some simple timing tests have been made on the emulator functions.
The times include load/store instructions. All times are in microseconds