1 +---------------------------------------------------------------------------+
2 | wm-FPU-emu an FPU emulator for 80386 and 80486SX microprocessors. |
4 | Copyright (C) 1992,1993,1994,1995,1996,1997,1999 |
5 | W. Metzenthen, 22 Parker St, Ormond, Vic 3163, |
6 | Australia. E-mail billm@melbpc.org.au |
8 | This program is free software; you can redistribute it and/or modify |
9 | it under the terms of the GNU General Public License version 2 as |
10 | published by the Free Software Foundation. |
12 | This program is distributed in the hope that it will be useful, |
13 | but WITHOUT ANY WARRANTY; without even the implied warranty of |
14 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
15 | GNU General Public License for more details. |
17 | You should have received a copy of the GNU General Public License |
18 | along with this program; if not, write to the Free Software |
19 | Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. |
21 +---------------------------------------------------------------------------+
25 wm-FPU-emu is an FPU emulator for Linux. It is derived from wm-emu387
26 which was my 80387 emulator for early versions of djgpp (gcc under
27 msdos); wm-emu387 was in turn based upon emu387 which was written by
28 DJ Delorie for djgpp. The interface to the Linux kernel is based upon
29 the original Linux math emulator by Linus Torvalds.
31 My target FPU for wm-FPU-emu is that described in the Intel486
32 Programmer's Reference Manual (1992 edition). Unfortunately, numerous
33 facets of the functioning of the FPU are not well covered in the
34 Reference Manual. The information in the manual has been supplemented
35 with measurements on real 80486's. Unfortunately, it is simply not
36 possible to be sure that all of the peculiarities of the 80486 have
37 been discovered, so there is always likely to be obscure differences
38 in the detailed behaviour of the emulator and a real 80486.
40 wm-FPU-emu does not implement all of the behaviour of the 80486 FPU,
41 but is very close. See "Limitations" later in this file for a list of
44 Please report bugs, etc to me at:
46 or b.metzenthen@medoto.unimelb.edu.au
48 For more information on the emulator and on floating point topics, see
49 my web pages, currently at http://www.suburbia.net/~billm/
56 ----------------------- Internals of wm-FPU-emu -----------------------
59 (1) Add, subtract, and multiply. Nothing remarkable in these.
60 (2) Divide has been tuned to get reasonable performance. The algorithm
61 is not the obvious one which most people seem to use, but is designed
62 to take advantage of the characteristics of the 80386. I expect that
63 it has been invented many times before I discovered it, but I have not
64 seen it. It is based upon one of those ideas which one carries around
65 for years without ever bothering to check it out.
66 (3) The sqrt function has been tuned to get good performance. It is based
67 upon Newton's classic method. Performance was improved by capitalizing
68 upon the properties of Newton's method, and the code is once again
69 structured taking account of the 80386 characteristics.
70 (4) The trig, log, and exp functions are based in each case upon quasi-
71 "optimal" polynomial approximations. My definition of "optimal" was
72 based upon getting good accuracy with reasonable speed.
73 (5) The argument reducing code for the trig function effectively uses
74 a value of pi which is accurate to more than 128 bits. As a consequence,
75 the reduced argument is accurate to more than 64 bits for arguments up
76 to a few pi, and accurate to more than 64 bits for most arguments,
77 even for arguments approaching 2^63. This is far superior to an
78 80486, which uses a value of pi which is accurate to 66 bits.
80 The code of the emulator is complicated slightly by the need to
81 account for a limited form of re-entrancy. Normally, the emulator will
82 emulate each FPU instruction to completion without interruption.
83 However, it may happen that when the emulator is accessing the user
84 memory space, swapping may be needed. In this case the emulator may be
85 temporarily suspended while disk i/o takes place. During this time
86 another process may use the emulator, thereby perhaps changing static
87 variables. The code which accesses user memory is confined to five
94 As from version 1.12 of the emulator, no static variables are used
95 (apart from those in the kernel's per-process tables). The emulator is
96 therefore now fully re-entrant, rather than having just the restricted
97 form of re-entrancy which is required by the Linux kernel.
99 ----------------------- Limitations of wm-FPU-emu -----------------------
101 There are a number of differences between the current wm-FPU-emu
102 (version 2.01) and the 80486 FPU (apart from bugs). The differences
103 are fewer than those which applied to the 1.xx series of the emulator.
104 Some of the more important differences are listed below:
106 The Roundup flag does not have much meaning for the transcendental
107 functions and its 80486 value with these functions is likely to differ
108 from its emulator value.
110 In a few rare cases the Underflow flag obtained with the emulator will
111 be different from that obtained with an 80486. This occurs when the
112 following conditions apply simultaneously:
113 (a) the operands have a higher precision than the current setting of the
114 precision control (PC) flags.
115 (b) the underflow exception is masked.
116 (c) the magnitude of the exact result (before rounding) is less than 2^-16382.
117 (d) the magnitude of the final result (after rounding) is exactly 2^-16382.
118 (e) the magnitude of the exact result would be exactly 2^-16382 if the
119 operands were rounded to the current precision before the arithmetic
120 operation was performed.
121 If all of these apply, the emulator will set the Underflow flag but a real
124 NOTE: Certain formats of Extended Real are UNSUPPORTED. They are
125 unsupported by the 80486. They are the Pseudo-NaNs, Pseudoinfinities,
126 and Unnormals. None of these will be generated by an 80486 or by the
127 emulator. Do not use them. The emulator treats them differently in
128 detail from the way an 80486 does.
130 Self modifying code can cause the emulator to fail. An example of such
134 The FPU instruction may be (usually will be) loaded into the pre-fetch
135 queue of the CPU before the mov instruction is executed. If the
136 destination of the 'movl' overlaps the FPU instruction then the bytes
137 in the prefetch queue and memory will be inconsistent when the FPU
138 instruction is executed. The emulator will be invoked but will not be
139 able to find the instruction which caused the device-not-present
140 exception. For this case, the emulator cannot emulate the behaviour of
143 Handling of the address size override prefix byte (0x67) has not been
144 extensively tested yet. A major problem exists because using it in
145 vm86 mode can cause a general protection fault. Address offsets
146 greater than 0xffff appear to be illegal in vm86 mode but are quite
147 acceptable (and work) in real mode. A small test program developed to
148 check the addressing, and which runs successfully in real mode,
149 crashes dosemu under Linux and also brings Windows down with a general
150 protection fault message when run under the MS-DOS prompt of Windows
151 3.1. (The program simply reads data from a valid address).
153 The emulator supports 16-bit protected mode, with one difference from
154 an 80486DX. A 80486DX will allow some floating point instructions to
155 write a few bytes below the lowest address of the stack. The emulator
156 will not allow this in 16-bit protected mode: no instructions are
157 allowed to write outside the bounds set by the protection.
159 ----------------------- Performance of wm-FPU-emu -----------------------
164 The speed of floating point computation with the emulator will depend
165 upon instruction mix. Relative performance is best for the instructions
166 which require most computation. The simple instructions are adversely
167 affected by the FPU instruction trap overhead.
170 Timing: Some simple timing tests have been made on the emulator functions.
171 The times include load/store instructions. All times are in microseconds
172 measured on a 33MHz 386 with 64k cache. The Turbo C tests were under
173 ms-dos, the next two columns are for emulators running with the djgpp
174 ms-dos extender. The final column is for wm-FPU-emu in Linux 0.97,
175 using libm4.0 (hard).
177 function Turbo C djgpp 1.06 WM-emu387 wm-FPU-emu
179 + 60.5 154.8 76.5 139.4
180 - 61.1-65.5 157.3-160.8 76.2-79.5 142.9-144.7
181 * 71.0 190.8 79.6 146.6
182 / 61.2-75.0 261.4-266.9 75.3-91.6 142.2-158.1
184 sin() 310.8 4692.0 319.0 398.5
185 cos() 284.4 4855.2 308.0 388.7
186 tan() 495.0 8807.1 394.9 504.7
187 atan() 328.9 4866.4 601.1 419.5-491.9
189 sqrt() 128.7 crashed 145.2 227.0
190 log() 413.1-419.1 5103.4-5354.21 254.7-282.2 409.4-437.1
191 exp() 479.1 6619.2 469.1 850.8
194 The performance under Linux is improved by the use of look-ahead code.
195 The following results show the improvement which is obtained under
196 Linux due to the look-ahead code. Also given are the times for the
197 original Linux emulator with the 4.1 'soft' lib.
199 [ Linus' note: I changed look-ahead to be the default under linux, as
200 there was no reason not to use it after I had edited it to be
201 disabled during tracing ]
203 wm-FPU-emu w original w
204 look-ahead 'soft' lib
206 - 108.6-111.6 192.4-216.2
208 / 108.8-124.4 700.1-706.2
213 atan() 367.2-435.5 2439.4-3396.8
216 log() 358.0-387.5 3359.2-3390.3
220 These figures are now somewhat out-of-date. The emulator has become
221 progressively slower for most functions as more of the 80486 features
222 have been implemented.
225 ----------------------- Accuracy of wm-FPU-emu -----------------------
228 The accuracy of the emulator is in almost all cases equal to or better
229 than that of an Intel 80486 FPU.
231 The results of the basic arithmetic functions (+,-,*,/), and fsqrt
232 match those of an 80486 FPU. They are the best possible; the error for
233 these never exceeds 1/2 an lsb. The fprem and fprem1 instructions
234 return exact results; they have no error.
237 The following table compares the emulator accuracy for the sqrt(),
238 trig and log functions against the Turbo C "emulator". For this table,
239 each function was tested at about 400 points. Ideal worst-case results
240 would be 64 bits. The reduced Turbo C accuracy of cos() and tan() for
241 arguments greater than pi/4 can be thought of as being related to the
242 precision of the argument x; e.g. an argument of pi/2-(1e-10) which is
243 accurate to 64 bits can result in a relative accuracy in cos() of
244 about 64 + log2(cos(x)) = 31 bits.
247 Function Tested x range Worst result Turbo C
250 sqrt(x) 1 .. 2 64.1 63.2
251 atan(x) 1e-10 .. 200 64.2 62.8
252 cos(x) 0 .. pi/2-(1e-10) 64.4 (x <= pi/4) 62.4
253 64.1 (x = pi/2-(1e-10)) 31.9
254 sin(x) 1e-10 .. pi/2 64.0 62.8
255 tan(x) 1e-10 .. pi/2-(1e-10) 64.0 (x <= pi/4) 62.1
256 64.1 (x = pi/2-(1e-10)) 31.9
257 exp(x) 0 .. 1 63.1 ** 62.9
258 log(x) 1+1e-6 .. 2 63.8 ** 62.1
260 ** The accuracy for exp() and log() is low because the FPU (emulator)
261 does not compute them directly; two operations are required.
264 The emulator passes the "paranoia" tests (compiled with gcc 2.3.3 or
265 later) for 'float' variables (24 bit precision numbers) when precision
266 control is set to 24, 53 or 64 bits, and for 'double' variables (53
267 bit precision numbers) when precision control is set to 53 bits (a
268 properly performing FPU cannot pass the 'paranoia' tests for 'double'
269 variables when precision control is set to 64 bits).
271 The code for reducing the argument for the trig functions (fsin, fcos,
272 fptan and fsincos) has been improved and now effectively uses a value
273 for pi which is accurate to more than 128 bits precision. As a
274 consequence, the accuracy of these functions for large arguments has
275 been dramatically improved (and is now very much better than an 80486
276 FPU). There is also now no degradation of accuracy for fcos and fptan
277 for operands close to pi/2. Measured results are (note that the
278 definition of accuracy has changed slightly from that used for the
281 Function Tested x range Worst result
284 cos(x) 0 .. 9.22e+18 62.0
285 sin(x) 1e-16 .. 9.22e+18 62.1
286 tan(x) 1e-16 .. 9.22e+18 61.8
288 It is possible with some effort to find very large arguments which
289 give much degraded precision. For example, the integer number
290 8227740058411162616.0
291 is within about 10e-7 of a multiple of pi. To find the tan (for
292 example) of this number to 64 bits precision it would be necessary to
293 have a value of pi which had about 150 bits precision. The FPU
294 emulator computes the result to about 42.6 bits precision (the correct
295 result is about -9.739715e-8). On the other hand, an 80486 FPU returns
296 0.01059, which in relative terms is hopelessly inaccurate.
298 For arguments close to critical angles (which occur at multiples of
299 pi/2) the emulator is more accurate than an 80486 FPU. For very large
300 arguments, the emulator is far more accurate.
303 Prior to version 1.20 of the emulator, the accuracy of the results for
304 the transcendental functions (in their principal range) was not as
305 good as the results from an 80486 FPU. From version 1.20, the accuracy
306 has been considerably improved and these functions now give measured
307 worst-case results which are better than the worst-case results given
310 The following table gives the measured results for the emulator. The
311 number of randomly selected arguments in each case is about half a
312 million. The group of three columns gives the frequency of the given
313 accuracy in number of times per million, thus the second of these
314 columns shows that an accuracy of between 63.80 and 63.89 bits was
315 found at a rate of 133 times per one million measurements for fsin.
316 The results show that the fsin, fcos and fptan instructions return
317 results which are in error (i.e. less accurate than the best possible
318 result (which is 64 bits)) for about one per cent of all arguments
319 between -pi/2 and +pi/2. The other instructions have a lower
320 frequency of results which are in error. The last two columns give
321 the worst accuracy which was found (in bits) and the approximate value
322 of the argument which produced it.
325 ------------------- ---------------
326 instr arg range # tests 63.7 63.8 63.9 worst at arg
328 ----- ------------ ------- ---- ---- ----- ----- --------
329 fsin (0,pi/2) 547756 0 133 10673 63.89 0.451317
330 fcos (0,pi/2) 547563 0 126 10532 63.85 0.700801
331 fptan (0,pi/2) 536274 11 267 10059 63.74 0.784876
332 fpatan 4 quadrants 517087 0 8 1855 63.88 0.435121 (4q)
333 fyl2x (0,20) 541861 0 0 1323 63.94 1.40923 (x)
334 fyl2xp1 (-.293,.414) 520256 0 0 5678 63.93 0.408542 (x)
335 f2xm1 (-1,1) 538847 4 481 6488 63.79 0.167709
338 Tests performed on an 80486 FPU showed results of lower accuracy. The
339 following table gives the results which were obtained with an AMD
340 486DX2/66 (other tests indicate that an Intel 486DX produces
341 identical results). The tests were basically the same as those used
342 to measure the emulator (the values, being random, were in general not
343 the same). The total number of tests for each instruction are given
344 at the end of the table, in case each about 100k tests were performed.
345 Another line of figures at the end of the table shows that most of the
346 instructions return results which are in error for more than 10
347 percent of the arguments tested.
349 The numbers in the body of the table give the approx number of times a
350 result of the given accuracy in bits (given in the left-most column)
351 was obtained per one million arguments. For three of the instructions,
352 two columns of results are given: * The second column for f2xm1 gives
353 the number cases where the results of the first column were for a
354 positive argument, this shows that this instruction gives better
355 results for positive arguments than it does for negative. * In the
356 cases of fcos and fptan, the first column gives the results when all
357 cases where arguments greater than 1.5 were removed from the results
358 given in the second column. Unlike the emulator, an 80486 FPU returns
359 results of relatively poor accuracy for these instructions when the
360 argument approaches pi/2. The table does not show those cases when the
361 accuracy of the results were less than 62 bits, which occurs quite
362 often for fsin and fptan when the argument approaches pi/2. This poor
363 accuracy is discussed above in relation to the Turbo C "emulator", and
364 the accuracy of the value of pi.
367 bits f2xm1 f2xm1 fpatan fcos fcos fyl2x fyl2xp1 fsin fptan fptan
368 62.0 0 0 0 0 437 0 0 0 0 925
369 62.1 0 0 10 0 894 0 0 0 0 1023
370 62.2 14 0 0 0 1033 0 0 0 0 945
371 62.3 57 0 0 0 1202 0 0 0 0 1023
372 62.4 385 0 0 10 1292 0 23 0 0 1178
373 62.5 1140 0 0 119 1649 0 39 0 0 1149
374 62.6 2037 0 0 189 1620 0 16 0 0 1169
375 62.7 5086 14 0 646 2315 10 101 35 39 1402
376 62.8 8818 86 0 984 3050 59 287 131 224 2036
377 62.9 11340 1355 0 2126 4153 79 605 357 321 1948
378 63.0 15557 4750 0 3319 5376 246 1281 862 808 2688
379 63.1 20016 8288 0 4620 6628 511 2569 1723 1510 3302
380 63.2 24945 11127 10 6588 8098 1120 4470 2968 2990 4724
381 63.3 25686 12382 69 8774 10682 1906 6775 4482 5474 7236
382 63.4 29219 14722 79 11109 12311 3094 9414 7259 8912 10587
383 63.5 30458 14936 393 13802 15014 5874 12666 9609 13762 15262
384 63.6 32439 16448 1277 17945 19028 10226 15537 14657 19158 20346
385 63.7 35031 16805 4067 23003 23947 18910 20116 21333 25001 26209
386 63.8 33251 15820 7673 24781 25675 24617 25354 24440 29433 30329
387 63.9 33293 16833 18529 28318 29233 31267 31470 27748 29676 30601
390 30.9 3.2 18.5 9.8 13.1 11.6 17.4
391 Total arguments tested:
392 70194 70099 101784 100641 100641 101799 128853 114893 102675 102675
395 ------------------------- Contributors -------------------------------
397 A number of people have contributed to the development of the
398 emulator, often by just reporting bugs, sometimes with suggested
399 fixes, and a few kind people have provided me with access in one way
400 or another to an 80486 machine. Contributors include (to those people
401 who I may have forgotten, please forgive me):
404 Tommy.Thorn@daimi.aau.dk
405 Andrew.Tridgell@anu.edu.au
406 Nick Holloway, alfie@dcs.warwick.ac.uk
407 Hermano Moura, moura@dcs.gla.ac.uk
408 Jon Jagger, J.Jagger@scp.ac.uk
410 Brian Gallew, geek+@CMU.EDU
411 Thomas Staniszewski, ts3v+@andrew.cmu.edu
412 Martin Howell, mph@plasma.apana.org.au
413 M Saggaf, alsaggaf@athena.mit.edu
414 Peter Barker, PETER@socpsy.sci.fau.edu
415 tom@vlsivie.tuwien.ac.at
416 Dan Russel, russed@rpi.edu
417 Daniel Carosone, danielce@ee.mu.oz.au
419 Hamish Coleman, t933093@minyos.xx.rmit.oz.au
420 Bruce Evans, bde@kralizec.zeta.org.au
421 Timo Korvola, Timo.Korvola@hut.fi
422 Rick Lyons, rick@razorback.brisnet.org.au
423 Rick, jrs@world.std.com
425 ...and numerous others who responded to my request for help with
projects / linux-2.6 / blob