 +---------------------------------------------------------------------------+
 |  wm-FPU-emu   an FPU emulator for 80386 and 80486SX microprocessors.      |
 |                                                                           |
 | Copyright (C) 1992    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.              |
 |                                                                           |
 +---------------------------------------------------------------------------+


***NOTE***       THIS SHOULD BE REGARDED AS AN ALPHA TEST VERSION
                 (although the beta version may be identical)


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). Numerous facets of the
functioning of the FPU are not well covered in the Reference Manual;
in the absence of clear details I have made guesses about the most
reasonable behaviour.

wm-FPU-emu does not implement all of the behaviour of the 80486 FPU. 
See "Limitations" later in this file for a partial list of some
differences.  I believe that the missing features are never used by
normal C or FORTRAN programs. 

Please report bugs, etc to me at:
       apm233m@vaxc.cc.monash.edu.au


--Bill Metzenthen
  Oct 1992

----------------------- 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.

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 ALPHA 0.7) and the 80486 FPU (apart from bugs). Some of the
more important differences are listed below:

Internal computations do not use de-normal numbers (but External
de-normals ARE recognised and generated). The design of wm-FPU-emu
allows a larger exponent range than the 80486 FPU for internal
computations.

All computations are performed at full 64 bit precision (the PC bits
of the FPU control word are ignored). Under Linux, the FPU normally
runs at 64 bits precision.

The precision flag (PE of the FPU status word) is not implemented.
Does anyone write code which uses this feature?

The Roundup flag (C1) is not implemented.

The functions which load/store the FPU state are partially implemented,
but the implementation should be sufficient for handling FPU errors etc
in 32 bit protected mode.

----------------------- 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


----------------------- 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 (bits)         Turbo C

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

