1 +--------------------------------------------
2 | wm-FPU-emu an FPU emulator for 80386 and
3 |
4 | Copyright (C) 1992,1993,1994,1995,1996,1997
5 | W. Metzenthen, 22 Par
6 | Australia. E-mail bi
7 |
8 | This program is free software; you can r
9 | it under the terms of the GNU General Pu
10 | published by the Free Software Foundatio
11 |
12 | This program is distributed in the hope
13 | but WITHOUT ANY WARRANTY; without even t
14 | MERCHANTABILITY or FITNESS FOR A PARTICU
15 | GNU General Public License for more deta
16 |
17 | You should have received a copy of the G
18 | along with this program; if not, write t
19 | Foundation, Inc., 675 Mass Ave, Cambridg
20 |
21 +--------------------------------------------
22
23
24
25 wm-FPU-emu is an FPU emulator for Linux. It is
26 which was my 80387 emulator for early versions
27 msdos); wm-emu387 was in turn based upon emu38
28 DJ Delorie for djgpp. The interface to the Li
29 the original Linux math emulator by Linus Torv
30
31 My target FPU for wm-FPU-emu is that described
32 Programmer's Reference Manual (1992 edition).
33 facets of the functioning of the FPU are not w
34 Reference Manual. The information in the manua
35 with measurements on real 80486's. Unfortunate
36 possible to be sure that all of the peculiarit
37 been discovered, so there is always likely to
38 in the detailed behaviour of the emulator and
39
40 wm-FPU-emu does not implement all of the behav
41 but is very close. See "Limitations" later in
42 some differences.
43
44 Please report bugs, etc to me at:
45 billm@melbpc.org.au
46 or b.metzenthen@medoto.unimelb.edu.au
47
48 For more information on the emulator and on fl
49 my web pages, currently at http://www.suburbi
50
51
52 --Bill Metzenthen
53 December 1999
54
55
56 ----------------------- Internals of wm-FPU-em
57
58 Numeric algorithms:
59 (1) Add, subtract, and multiply. Nothing remar
60 (2) Divide has been tuned to get reasonable pe
61 is not the obvious one which most people s
62 to take advantage of the characteristics o
63 it has been invented many times before I d
64 seen it. It is based upon one of those ide
65 for years without ever bothering to check
66 (3) The sqrt function has been tuned to get go
67 upon Newton's classic method. Performance
68 upon the properties of Newton's method, an
69 structured taking account of the 80386 cha
70 (4) The trig, log, and exp functions are based
71 "optimal" polynomial approximations. My de
72 based upon getting good accuracy with reas
73 (5) The argument reducing code for the trig fu
74 a value of pi which is accurate to more th
75 the reduced argument is accurate to more t
76 to a few pi, and accurate to more than 64
77 even for arguments approaching 2^63. This
78 80486, which uses a value of pi which is a
79
80 The code of the emulator is complicated slight
81 account for a limited form of re-entrancy. Nor
82 emulate each FPU instruction to completion wit
83 However, it may happen that when the emulator
84 memory space, swapping may be needed. In this
85 temporarily suspended while disk i/o takes pla
86 another process may use the emulator, thereby
87 variables. The code which accesses user memory
88 files:
89 fpu_entry.c
90 reg_ld_str.c
91 load_store.c
92 get_address.c
93 errors.c
94 As from version 1.12 of the emulator, no stati
95 (apart from those in the kernel's per-process
96 therefore now fully re-entrant, rather than ha
97 form of re-entrancy which is required by the L
98
99 ----------------------- Limitations of wm-FPU-
100
101 There are a number of differences between the
102 (version 2.01) and the 80486 FPU (apart from b
103 are fewer than those which applied to the 1.xx
104 Some of the more important differences are lis
105
106 The Roundup flag does not have much meaning fo
107 functions and its 80486 value with these funct
108 from its emulator value.
109
110 In a few rare cases the Underflow flag obtaine
111 be different from that obtained with an 80486.
112 following conditions apply simultaneously:
113 (a) the operands have a higher precision than
114 precision control (PC) flags.
115 (b) the underflow exception is masked.
116 (c) the magnitude of the exact result (before
117 (d) the magnitude of the final result (after r
118 (e) the magnitude of the exact result would be
119 operands were rounded to the current preci
120 operation was performed.
121 If all of these apply, the emulator will set t
122 80486 will not.
123
124 NOTE: Certain formats of Extended Real are UNS
125 unsupported by the 80486. They are the Pseudo-
126 and Unnormals. None of these will be generated
127 emulator. Do not use them. The emulator treats
128 detail from the way an 80486 does.
129
130 Self modifying code can cause the emulator to
131 code is:
132 movl %esp,[%ebx]
133 fld1
134 The FPU instruction may be (usually will be) l
135 queue of the CPU before the mov instruction is
136 destination of the 'movl' overlaps the FPU ins
137 in the prefetch queue and memory will be incon
138 instruction is executed. The emulator will be
139 able to find the instruction which caused the
140 exception. For this case, the emulator cannot
141 an 80486DX.
142
143 Handling of the address size override prefix b
144 extensively tested yet. A major problem exists
145 vm86 mode can cause a general protection fault
146 greater than 0xffff appear to be illegal in vm
147 acceptable (and work) in real mode. A small te
148 check the addressing, and which runs successfu
149 crashes dosemu under Linux and also brings Win
150 protection fault message when run under the MS
151 3.1. (The program simply reads data from a val
152
153 The emulator supports 16-bit protected mode, w
154 an 80486DX. A 80486DX will allow some floatin
155 write a few bytes below the lowest address of
156 will not allow this in 16-bit protected mode:
157 allowed to write outside the bounds set by the
158
159 ----------------------- Performance of wm-FPU-
160
161 Speed.
162 -----
163
164 The speed of floating point computation with t
165 upon instruction mix. Relative performance is
166 which require most computation. The simple ins
167 affected by the FPU instruction trap overhead.
168
169
170 Timing: Some simple timing tests have been mad
171 The times include load/store instructions. All
172 measured on a 33MHz 386 with 64k cache. The Tu
173 ms-dos, the next two columns are for emulators
174 ms-dos extender. The final column is for wm-FP
175 using libm4.0 (hard).
176
177 function Turbo C djgpp 1.06
178
179 + 60.5 154.8
180 - 61.1-65.5 157.3-160.8
181 * 71.0 190.8
182 / 61.2-75.0 261.4-266.9
183
184 sin() 310.8 4692.0
185 cos() 284.4 4855.2
186 tan() 495.0 8807.1
187 atan() 328.9 4866.4
188
189 sqrt() 128.7 crashed
190 log() 413.1-419.1 5103.4-5354.21
191 exp() 479.1 6619.2
192
193
194 The performance under Linux is improved by the
195 The following results show the improvement whi
196 Linux due to the look-ahead code. Also given a
197 original Linux emulator with the 4.1 'soft' li
198
199 [ Linus' note: I changed look-ahead to be the
200 there was no reason not to use it after I h
201 disabled during tracing ]
202
203 wm-FPU-emu w original w
204 look-ahead 'soft' lib
205 + 106.4 190.2
206 - 108.6-111.6 192.4-216.2
207 * 113.4 193.1
208 / 108.8-124.4 700.1-706.2
209
210 sin() 390.5 2642.0
211 cos() 381.5 2767.4
212 tan() 496.5 3153.3
213 atan() 367.2-435.5 2439.4-3396.8
214
215 sqrt() 195.1 4732.5
216 log() 358.0-387.5 3359.2-3390.3
217 exp() 619.3 4046.4
218
219
220 These figures are now somewhat out-of-date. Th
221 progressively slower for most functions as mor
222 have been implemented.
223
224
225 ----------------------- Accuracy of wm-FPU-emu
226
227
228 The accuracy of the emulator is in almost all
229 than that of an Intel 80486 FPU.
230
231 The results of the basic arithmetic functions
232 match those of an 80486 FPU. They are the best
233 these never exceeds 1/2 an lsb. The fprem and
234 return exact results; they have no error.
235
236
237 The following table compares the emulator accu
238 trig and log functions against the Turbo C "em
239 each function was tested at about 400 points.
240 would be 64 bits. The reduced Turbo C accuracy
241 arguments greater than pi/4 can be thought of
242 precision of the argument x; e.g. an argument
243 accurate to 64 bits can result in a relative a
244 about 64 + log2(cos(x)) = 31 bits.
245
246
247 Function Tested x range Worst
248 (relat
249
250 sqrt(x) 1 .. 2 64.1
251 atan(x) 1e-10 .. 200 64.2
252 cos(x) 0 .. pi/2-(1e-10) 64.4 (
253 64.1 (
254 sin(x) 1e-10 .. pi/2 64.0
255 tan(x) 1e-10 .. pi/2-(1e-10) 64.0 (
256 64.1 (
257 exp(x) 0 .. 1 63.1 *
258 log(x) 1+1e-6 .. 2 63.8 *
259
260 ** The accuracy for exp() and log() is low bec
261 does not compute them directly; two operations
262
263
264 The emulator passes the "paranoia" tests (comp
265 later) for 'float' variables (24 bit precision
266 control is set to 24, 53 or 64 bits, and for '
267 bit precision numbers) when precision control
268 properly performing FPU cannot pass the 'paran
269 variables when precision control is set to 64
270
271 The code for reducing the argument for the tri
272 fptan and fsincos) has been improved and now e
273 for pi which is accurate to more than 128 bits
274 consequence, the accuracy of these functions f
275 been dramatically improved (and is now very mu
276 FPU). There is also now no degradation of accu
277 for operands close to pi/2. Measured results a
278 definition of accuracy has changed slightly fr
279 above table):
280
281 Function Tested x range Worst re
282 (absolute
283
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
287
288 It is possible with some effort to find very l
289 give much degraded precision. For example, the
290 8227740058411162616.0
291 is within about 10e-7 of a multiple of pi. To
292 example) of this number to 64 bits precision i
293 have a value of pi which had about 150 bits pr
294 emulator computes the result to about 42.6 bit
295 result is about -9.739715e-8). On the other ha
296 0.01059, which in relative terms is hopelessly
297
298 For arguments close to critical angles (which
299 pi/2) the emulator is more accurate than an 80
300 arguments, the emulator is far more accurate.
301
302
303 Prior to version 1.20 of the emulator, the acc
304 the transcendental functions (in their princip
305 good as the results from an 80486 FPU. From ve
306 has been considerably improved and these funct
307 worst-case results which are better than the w
308 by an 80486 FPU.
309
310 The following table gives the measured results
311 number of randomly selected arguments in each
312 million. The group of three columns gives the
313 accuracy in number of times per million, thus
314 columns shows that an accuracy of between 63.8
315 found at a rate of 133 times per one million m
316 The results show that the fsin, fcos and fptan
317 results which are in error (i.e. less accurate
318 result (which is 64 bits)) for about one per c
319 between -pi/2 and +pi/2. The other instructio
320 frequency of results which are in error. The
321 the worst accuracy which was found (in bits) a
322 of the argument which produced it.
323
324 frequency (per
325 ---------------
326 instr arg range # tests 63.7 63.8
327 bits bits
328 ----- ------------ ------- ---- ---- -
329 fsin (0,pi/2) 547756 0 133 1
330 fcos (0,pi/2) 547563 0 126 1
331 fptan (0,pi/2) 536274 11 267 1
332 fpatan 4 quadrants 517087 0 8
333 fyl2x (0,20) 541861 0 0
334 fyl2xp1 (-.293,.414) 520256 0 0
335 f2xm1 (-1,1) 538847 4 481
336
337
338 Tests performed on an 80486 FPU showed results
339 following table gives the results which were o
340 486DX2/66 (other tests indicate that an Intel
341 identical results). The tests were basically
342 to measure the emulator (the values, being ran
343 the same). The total number of tests for each
344 at the end of the table, in case each about 10
345 Another line of figures at the end of the tabl
346 instructions return results which are in error
347 percent of the arguments tested.
348
349 The numbers in the body of the table give the
350 result of the given accuracy in bits (given in
351 was obtained per one million arguments. For th
352 two columns of results are given: * The second
353 the number cases where the results of the firs
354 positive argument, this shows that this instru
355 results for positive arguments than it does fo
356 cases of fcos and fptan, the first column give
357 cases where arguments greater than 1.5 were re
358 given in the second column. Unlike the emulato
359 results of relatively poor accuracy for these
360 argument approaches pi/2. The table does not s
361 accuracy of the results were less than 62 bits
362 often for fsin and fptan when the argument app
363 accuracy is discussed above in relation to the
364 the accuracy of the value of pi.
365
366
367 bits f2xm1 f2xm1 fpatan fcos fcos fyl2
368 62.0 0 0 0 0 437
369 62.1 0 0 10 0 894
370 62.2 14 0 0 0 1033
371 62.3 57 0 0 0 1202
372 62.4 385 0 0 10 1292
373 62.5 1140 0 0 119 1649
374 62.6 2037 0 0 189 1620
375 62.7 5086 14 0 646 2315 1
376 62.8 8818 86 0 984 3050 5
377 62.9 11340 1355 0 2126 4153 7
378 63.0 15557 4750 0 3319 5376 24
379 63.1 20016 8288 0 4620 6628 51
380 63.2 24945 11127 10 6588 8098 112
381 63.3 25686 12382 69 8774 10682 190
382 63.4 29219 14722 79 11109 12311 309
383 63.5 30458 14936 393 13802 15014 587
384 63.6 32439 16448 1277 17945 19028 1022
385 63.7 35031 16805 4067 23003 23947 1891
386 63.8 33251 15820 7673 24781 25675 2461
387 63.9 33293 16833 18529 28318 29233 3126
388
389 Per cent with error:
390 30.9 3.2 18.5 9.
391 Total arguments tested:
392 70194 70099 101784 100641 100641 10179
393
394
395 ------------------------- Contributors -------
396
397 A number of people have contributed to the dev
398 emulator, often by just reporting bugs, someti
399 fixes, and a few kind people have provided me
400 or another to an 80486 machine. Contributors i
401 who I may have forgotten, please forgive me):
402
403 Linus Torvalds
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
409 Lennart Benschop
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
418 cae@jpmorgan.com
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
424
425 ...and numerous others who responded to my req
426 a real 80486.
427
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