 +---------------------------------------------------------------------------+
 |  wm-FPU-emu   an FPU emulator for 80386 and 80486SX microprocessors.      |
 |                                                                           |
 | Copyright (C) 1992,1993                                                   |
 |                       W. Metzenthen, 22 Parker St, Ormond, Vic 3163,      |
 |                       Australia.  E-mail apm233m@vaxc.cc.monash.edu.au    |
 |                                                                           |
 |    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:
       apm233m@vaxc.cc.monash.edu.au


--Bill Metzenthen
  July 1993


----------------------- 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 1.5) and the 80486 FPU (apart from bugs). Some of the
more important differences are listed below:

Segment overrides don't do anything yet.

All internal computations are performed at 64 bit or higher precision
and the results rounded etc as required by the PC bits of the FPU
control word.  Under the crt0 version for Linux current at June 1993,
the FPU PC bits specify 64 bits precision.

The precision flag (PE of the FPU status word) and the Roundup flag
(C1 of the status word) are now implemented. Does anyone write code
which uses these features? 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.

----------------------- 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
measured on a 33MHz 386 with 64k cache. The Turbo C tests were under
ms-dos, the next two columns are for emulators running with the djgpp
ms-dos extender. The final column is for wm-FPU-emu in Linux 0.97,
using libm4.0 (hard).

function      Turbo C        djgpp 1.06        WM-emu387     wm-FPU-emu

   +          60.5           154.8              76.5          139.4
   -          61.1-65.5      157.3-160.8        76.2-79.5     142.9-144.7
   *          71.0           190.8              79.6          146.6
   /          61.2-75.0      261.4-266.9        75.3-91.6     142.2-158.1

 sin()        310.8          4692.0            319.0          398.5
 cos()        284.4          4855.2            308.0          388.7
 tan()        495.0          8807.1            394.9          504.7
 atan()       328.9          4866.4            601.1          419.5-491.9

 sqrt()       128.7          crashed           145.2          227.0
 log()        413.1-419.1    5103.4-5354.21    254.7-282.2    409.4-437.1
 exp()        479.1          6619.2            469.1          850.8


The performance under Linux is improved by the use of look-ahead code.
The following results show the improvement which is obtained under
Linux due to the look-ahead code. Also given are the times for the
original Linux emulator with the 4.1 'soft' lib.

 [ Linus' note: I changed look-ahead to be the default under linux, as
   there was no reason not to use it after I had edited it to be
   disabled during tracing ]

            wm-FPU-emu w     original w
            look-ahead       'soft' lib
   +         106.4             190.2
   -         108.6-111.6      192.4-216.2
   *         113.4             193.1
   /         108.8-124.4      700.1-706.2

 sin()       390.5            2642.0
 cos()       381.5            2767.4
 tan()       496.5            3153.3
 atan()      367.2-435.5     2439.4-3396.8

 sqrt()      195.1            4732.5
 log()       358.0-387.5     3359.2-3390.3
 exp()       619.3            4046.4


These figures are now somewhat out-of-date. The emulator has become
progressively slower for most functions as more of the 80486 features
have been implemented.


----------------------- Accuracy of wm-FPU-emu -----------------------


Accuracy: The following table gives the accuracy of the sqrt(), trig
and log functions. Each function was tested at about 400 points. Ideal
results would be 64 bits. The reduced accuracy of cos() and tan() for
arguments greater than pi/4 can be thought of as being due to the
precision of the argument x; e.g. an argument of pi/2-(1e-10) which is
accurate to 64 bits can result in a relative accuracy in cos() of about
64 + log2(cos(x)) = 31 bits. Results for the Turbo C emulator are given
in the last column.


Function      Tested x range            Worst result                Turbo C
                                        (relative bits)

sqrt(x)       1 .. 2                    64.1                         63.2
atan(x)       1e-10 .. 200              62.6                         62.8
cos(x)        0 .. pi/2-(1e-10)         63.2 (x <= pi/4)             62.4
                                        35.2 (x = pi/2-(1e-10))      31.9
sin(x)        1e-10 .. pi/2             63.0                         62.8
tan(x)        1e-10 .. pi/2-(1e-10)     62.4 (x <= pi/4)             62.1
                                        35.2 (x = pi/2-(1e-10))      31.9
exp(x)        0 .. 1                    63.1                         62.9
log(x)        1+1e-6 .. 2               62.4                         62.1


As of version 1.3 of the emulator, the accuracy of the basic
arithmetic has been improved (by a small fraction of a bit). Care has
been taken to ensure full accuracy of the rounding of the basic
arithmetic functions (+,-,*,/,and fsqrt), and they all now produce
results which are exact to the 64th bit (unless there are any bugs
left). To ensure this, it was necessary to effectively get information
of up to about 128 bits precision. The emulator now passes the
"paranoia" tests (compiled with gcc 2.3.3) for 'float' variables (24
bit precision numbers) when precision control is set to 24, 53 or 64
bits, and for 'double' variables (53 bit precision numbers) when
precision control is set to 53 bits (a properly performing FPU cannot
pass the 'paranoia' tests for 'double' variables when precision
control is set to 64 bits).

For version 1.5, the accuracy of fprem and fprem1 has been improved.
These functions now produce exact results. The code for reducing the
argument for the trig functions (fsin, fcos, fptan and fsincos) has
been improved and now effectively uses a value for pi which is
accurate to more than 128 bits precision. As a consquence, the
accuracy of these functions for large arguments has been dramatically
improved (and is now very much better than an 80486 FPU). There is
also now no degradation of accuracy for fcos and ftan for operands
close to pi/2. Measured results are (note that the definition of
accuracy has changed slightly from that used for the above table):

Function      Tested x range          Worst result
                                     (absolute bits)

cos(x)        0 .. 9.22e+18              62.0
sin(x)        1e-16 .. 9.22e+18          62.1
tan(x)        1e-16 .. 9.22e+18          61.8

It is possible with some effort to find very large arguments which
give much degraded precision. For example, the integer number
           8227740058411162616.0
is within about 10e-7 of a multiple of pi. To find the tan (for
example) of this number to 64 bits precision it would be necessary to
have a value of pi which had about 150 bits precision. The FPU
emulator computes the result to about 42.6 bits precision (the correct
result is about -9.739715e-8). On the other hand, an 80486 FPU returns
0.01059, which in relative terms is hopelessly inaccurate.

For arguments close to critical angles (which occur at multiples of
pi/2) the emulator is more accurate than an 80486 FPU. For very large
arguments, the emulator is far more accurate.

------------------------- Contributors -------------------------------

A number of people have contributed to the development of the
emulator, often by just reporting bugs, sometimes with suggested
fixes, and a few kind people have provided me with access in one way
or another to an 80486 machine. Contributors include (to those people
who I may have forgotten, please forgive me):

Linus Torvalds
Tommy.Thorn@daimi.aau.dk
Andrew.Tridgell@anu.edu.au
Nick Holloway, alfie@dcs.warwick.ac.uk
Hermano Moura, moura@dcs.gla.ac.uk
Jon Jagger, J.Jagger@scp.ac.uk
Lennart Benschop
Brian Gallew, geek+@CMU.EDU
Thomas Staniszewski, ts3v+@andrew.cmu.edu
Martin Howell, mph@plasma.apana.org.au
M Saggaf, alsaggaf@athena.mit.edu
Peter Barker, PETER@socpsy.sci.fau.edu
tom@vlsivie.tuwien.ac.at
Dan Russel, russed@rpi.edu
Daniel Carosone, danielce@ee.mu.oz.au
cae@jpmorgan.com
Hamish Coleman, t933093@minyos.xx.rmit.oz.au
Bruce Evans, bde@kralizec.zeta.org.au

...and numerous others who responded to my request for help with
a real 80486.

