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diff --git a/arch/x86/math-emu/README b/arch/x86/math-emu/README new file mode 100644 index 0000000..e623549 --- /dev/null +++ b/arch/x86/math-emu/README @@ -0,0 +1,427 @@ + +---------------------------------------------------------------------------+ + | 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. + |
