linux / linux / kernel / git / bpf / bpf / b63b50ab0613a3e8b53441668219e96a61a17b6b / . / arch / x86 / math-emu / README

+---------------------------------------------------------------------------+ | |

| 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. | |