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+ +---------------------------------------------------------------------------+
+ |  wm-FPU-emu   an FPU emulator for 80386 and 80486SX microprocessors.      |
+ |                                                                           |
+ | Copyright (C) 1992,1993,1994,1995,1996,1997,1999                          |
+ |                       W. Metzenthen, 22 Parker St, Ormond, Vic 3163,      |
+ |                       Australia.  E-mail billm@melbpc.org.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 was my 80387 emulator for early versions of 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,
+but is very close.  See "Limitations" later in this file for a list of
+some differences.
+
+Please report bugs, etc to me at:
+       billm@melbpc.org.au
+or     b.metzenthen@medoto.unimelb.edu.au
+
+For more information on the emulator and on floating point topics, see
+my web pages, currently at  http://www.suburbia.net/~billm/
+
+
+--Bill Metzenthen
+  December 1999
+
+
+----------------------- 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 perhaps changing static
+variables. 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
+As from version 1.12 of the emulator, no static variables are used
+(apart from those in the kernel's per-process tables). The emulator is
+therefore now fully re-entrant, rather than having just the restricted
+form of re-entrancy which is required by the Linux kernel.
+
+----------------------- Limitations of wm-FPU-emu -----------------------
+
+There are a number of differences between the current wm-FPU-emu
+(version 2.01) and the 80486 FPU (apart from bugs).  The differences
+are fewer than those which applied to the 1.xx series of the emulator.
+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.
+
+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.
+
+Handling of the address size override prefix byte (0x67) has not been
+extensively tested yet. A major problem exists because using it in
+vm86 mode can cause a general protection fault. Address offsets
+greater than 0xffff appear to be illegal in vm86 mode but are quite
+acceptable (and work) in real mode. A small test program developed to
+check the addressing, and which runs successfully in real mode,
+crashes dosemu under Linux and also brings Windows down with a general
+protection fault message when run under the MS-DOS prompt of Windows
+3.1. (The program simply reads data from a valid address).
+
+The emulator supports 16-bit protected mode, with one difference from
+an 80486DX.  A 80486DX will allow some floating point instructions to
+write a few bytes below the lowest address of the stack.  The emulator
+will not allow this in 16-bit protected mode: no instructions are
+allowed to write outside the bounds set by the protection.
+
+----------------------- 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 -----------------------
+
+
+The accuracy of the emulator is in almost all cases equal to or better
+than that of an Intel 80486 FPU.
+
+The results of the basic arithmetic functions (+,-,*,/), and fsqrt
+match those of an 80486 FPU. They are the best possible; the error for
+these never exceeds 1/2 an lsb. The fprem and fprem1 instructions
+return exact results; they have no error.
+
+
+The following table compares the emulator accuracy for the sqrt(),
+trig and log functions against the Turbo C "emulator". For this table,
+each function was tested at about 400 points. Ideal worst-case results
+would be 64 bits. The reduced Turbo C accuracy of cos() and tan() for
+arguments greater than pi/4 can be thought of as being related 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.
+
+
+Function      Tested x range            Worst result                Turbo C
+                                        (relative bits)
+
+sqrt(x)       1 .. 2                    64.1                         63.2
+atan(x)       1e-10 .. 200              64.2                         62.8
+cos(x)        0 .. pi/2-(1e-10)         64.4 (x <= pi/4)             62.4
+                                        64.1 (x = pi/2-(1e-10))      31.9
+sin(x)        1e-10 .. pi/2             64.0                         62.8
+tan(x)        1e-10 .. pi/2-(1e-10)     64.0 (x <= pi/4)             62.1
+                                        64.1 (x = pi/2-(1e-10))      31.9
+exp(x)        0 .. 1                    63.1 **                      62.9
+log(x)        1+1e-6 .. 2               63.8 **                      62.1
+
+** The accuracy for exp() and log() is low because the FPU (emulator)
+does not compute them directly; two operations are required.
+
+
+The emulator passes the "paranoia" tests (compiled with gcc 2.3.3 or
+later) 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).
+
+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
+consequence, 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 fptan
+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.
+
+
+Prior to version 1.20 of the emulator, the accuracy of the results for
+the transcendental functions (in their principal range) was not as
+good as the results from an 80486 FPU. From version 1.20, the accuracy
+has been considerably improved and these functions now give measured
+worst-case results which are better than the worst-case results given
+by an 80486 FPU.
+
+The following table gives the measured results for the emulator. The
+number of randomly selected arguments in each case is about half a
+million.  The group of three columns gives the frequency of the given
+accuracy in number of times per million, thus the second of these
+columns shows that an accuracy of between 63.80 and 63.89 bits was
+found at a rate of 133 times per one million measurements for fsin.
+The results show that the fsin, fcos and fptan instructions return
+results which are in error (i.e. less accurate than the best possible
+result (which is 64 bits)) for about one per cent of all arguments
+between -pi/2 and +pi/2.  The other instructions have a lower
+frequency of results which are in error.  The last two columns give
+the worst accuracy which was found (in bits) and the approximate value
+of the argument which produced it.
+
+                                frequency (per M)
+                               -------------------   ---------------
+instr   arg range    # tests   63.7   63.8    63.9   worst   at arg
+                               bits   bits    bits    bits
+-----  ------------  -------   ----   ----   -----   -----  --------
+fsin     (0,pi/2)     547756      0    133   10673   63.89  0.451317
+fcos     (0,pi/2)     547563      0    126   10532   63.85  0.700801
+fptan    (0,pi/2)     536274     11    267   10059   63.74  0.784876
+fpatan  4 quadrants   517087      0      8    1855   63.88  0.435121 (4q)
+fyl2x     (0,20)      541861      0      0    1323   63.94  1.40923  (x)
+fyl2xp1 (-.293,.414)  520256      0      0    5678   63.93  0.408542 (x)
+f2xm1     (-1,1)      538847      4    481    6488   63.79  0.167709
+
+
+Tests performed on an 80486 FPU showed results of lower accuracy. The
+following table gives the results which were obtained with an AMD
+486DX2/66 (other tests indicate that an Intel 486DX produces
+identical results).  The tests were basically the same as those used
+to measure the emulator (the values, being random, were in general not
+the same).  The total number of tests for each instruction are given
+at the end of the table, in case each about 100k tests were performed.
+Another line of figures at the end of the table shows that most of the
+instructions return results which are in error for more than 10
+percent of the arguments tested.
+
+The numbers in the body of the table give the approx number of times a
+result of the given accuracy in bits (given in the left-most column)
+was obtained per one million arguments. For three of the instructions,
+two columns of results are given: * The second column for f2xm1 gives
+the number cases where the results of the first column were for a
+positive argument, this shows that this instruction gives better
+results for positive arguments than it does for negative.  * In the
+cases of fcos and fptan, the first column gives the results when all
+cases where arguments greater than 1.5 were removed from the results
+given in the second column. Unlike the emulator, an 80486 FPU returns
+results of relatively poor accuracy for these instructions when the
+argument approaches pi/2. The table does not show those cases when the
+accuracy of the results were less than 62 bits, which occurs quite
+often for fsin and fptan when the argument approaches pi/2. This poor
+accuracy is discussed above in relation to the Turbo C "emulator", and
+the accuracy of the value of pi.
+
+
+bits   f2xm1  f2xm1 fpatan   fcos   fcos  fyl2x fyl2xp1  fsin  fptan  fptan
+62.0       0      0      0      0    437      0      0      0      0    925
+62.1       0      0     10      0    894      0      0      0      0   1023
+62.2      14      0      0      0   1033      0      0      0      0    945
+62.3      57      0      0      0   1202      0      0      0      0   1023
+62.4     385      0      0     10   1292      0     23      0      0   1178
+62.5    1140      0      0    119   1649      0     39      0      0   1149
+62.6    2037      0      0    189   1620      0     16      0      0   1169
+62.7    5086     14      0    646   2315     10    101     35     39   1402
+62.8    8818     86      0    984   3050     59    287    131    224   2036
+62.9   11340   1355      0   2126   4153     79    605    357    321   1948
+63.0   15557   4750      0   3319   5376    246   1281    862    808   2688
+63.1   20016   8288      0   4620   6628    511   2569   1723   1510   3302
+63.2   24945  11127     10   6588   8098   1120   4470   2968   2990   4724
+63.3   25686  12382     69   8774  10682   1906   6775   4482   5474   7236
+63.4   29219  14722     79  11109  12311   3094   9414   7259   8912  10587
+63.5   30458  14936    393  13802  15014   5874  12666   9609  13762  15262
+63.6   32439  16448   1277  17945  19028  10226  15537  14657  19158  20346
+63.7   35031  16805   4067  23003  23947  18910  20116  21333  25001  26209
+63.8   33251  15820   7673  24781  25675  24617  25354  24440  29433  30329
+63.9   33293  16833  18529  28318  29233  31267  31470  27748  29676  30601
+
+Per cent with error:
+        30.9           3.2          18.5    9.8   13.1   11.6          17.4
+Total arguments tested:
+       70194  70099 101784 100641 100641 101799 128853 114893 102675 102675
+
+
+------------------------- 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
+Timo Korvola, Timo.Korvola@hut.fi
+Rick Lyons, rick@razorback.brisnet.org.au
+Rick, jrs@world.std.com
+ 
+...and numerous others who responded to my request for help with
+a real 80486.
+