| +---------------------------------------------------------------------------+ |
| | wm-FPU-emu an FPU emulator for 80386 and 80486SX microprocessors. | |
| | | |
| | Copyright (C) 1992,1993 | |
| | W. Metzenthen, 22 Parker St, Ormond, Vic 3163, | |
| | Australia. E-mail apm233m@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: |
| apm233m@vaxc.cc.monash.edu.au |
| |
| |
| --Bill Metzenthen |
| July 1993 |
| |
| |
| ----------------------- 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 1.5) and the 80486 FPU (apart from bugs). Some of the |
| more important differences are listed below: |
| |
| Segment overrides don't do anything yet. |
| |
| All internal computations are performed at 64 bit or higher precision |
| and the results rounded etc as required by the PC bits of the FPU |
| control word. Under the crt0 version for Linux current at June 1993, |
| the FPU PC bits specify 64 bits precision. |
| |
| The precision flag (PE of the FPU status word) and the Roundup flag |
| (C1 of the status word) are now implemented. Does anyone write code |
| which uses these features? 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. |
| |
| ----------------------- 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 |
| |
| ...and numerous others who responded to my request for help with |
| a real 80486. |
| |
