+++ /dev/null
- +---------------------------------------------------------------------------+
- | 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.
-
git://ftp.safe.ca / safe / jmp / linux-2.6 / blobdiff