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diff --git a/arch/i386/math-emu/README b/arch/i386/math-emu/README deleted file mode 100644 index e6235491d6e..00000000000 --- a/arch/i386/math-emu/README +++ /dev/null @@ -1,427 +0,0 @@ - +---------------------------------------------------------------------------+ - | 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. - |