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+---------------------------------------------------------------------------+ | wm-FPU-emu an FPU emulator for 80386 and 80486SX microprocessors. | | | | Copyright (C) 1992,1993,1994 | | W. Metzenthen, 22 Parker St, Ormond, Vic 3163, | | Australia. E-mail billm@vaxc.cc.monash.edu.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 is my 80387 emulator for 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. See "Limitations" later in this file for a list of some differences. Please report bugs, etc to me at: billm@vaxc.cc.monash.edu.au or at: billm@jacobi.maths.monash.edu.au --Bill Metzenthen March 1994 ----------------------- 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 changing some static variables (eg FPU_st0_ptr, etc). 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 ----------------------- Limitations of wm-FPU-emu ----------------------- There are a number of differences between the current wm-FPU-emu (version beta 1.11) and the 80486 FPU (apart from bugs). 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. The emulator treats PseudoDenormals differently from an 80486. These numbers are in fact properly normalised numbers with the exponent offset by 1, and the emulator treats them as such. Unlike the 80486, the emulator does not generate a Denormal Operand exception for these numbers. The arithmetical results produced when using such a number as an operand are the same for the emulator and a real 80486 (apart from any slight precision difference for the transcendental functions). Neither the emulator nor an 80486 produces one of these numbers as the result of any arithmetic operation. An 80486 can keep one of these numbers in an FPU register with its identity as a PseudoDenormal, but the emulator will not; they are always converted to a valid number. 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). ----------------------- 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 ----------------------- Accuracy: The following table gives the accuracy of the sqrt(), trig and log functions. Each function was tested at about 400 points. Ideal results would be 64 bits. The reduced accuracy of cos() and tan() for arguments greater than pi/4 can be thought of as being due 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. Results for the Turbo C emulator are given in the last column. Function Tested x range Worst result Turbo C (relative bits) sqrt(x) 1 .. 2 64.1 63.2 atan(x) 1e-10 .. 200 62.6 62.8 cos(x) 0 .. pi/2-(1e-10) 63.2 (x <= pi/4) 62.4 35.2 (x = pi/2-(1e-10)) 31.9 sin(x) 1e-10 .. pi/2 63.0 62.8 tan(x) 1e-10 .. pi/2-(1e-10) 62.4 (x <= pi/4) 62.1 35.2 (x = pi/2-(1e-10)) 31.9 exp(x) 0 .. 1 63.1 62.9 log(x) 1+1e-6 .. 2 62.4 62.1 As of version 1.3 of the emulator, the accuracy of the basic arithmetic has been improved (by a small fraction of a bit). Care has been taken to ensure full accuracy of the rounding of the basic arithmetic functions (+,-,*,/,and fsqrt), and they all now produce results which are exact to the 64th bit (unless there are any bugs left). To ensure this, it was necessary to effectively get information of up to about 128 bits precision. The emulator now passes the "paranoia" tests (compiled with gcc 2.3.3) 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). For version 1.5, the accuracy of fprem and fprem1 has been improved. These functions now produce exact results. 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 consquence, 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 ftan 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. ------------------------- 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 ...and numerous others who responded to my request for help with a real 80486.

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