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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.
+