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 +---------------------------------------------------------------------------+

 |  wm-FPU-emu   an FPU emulator for 80386 and 80486SX microprocessors.      |

 |                                                                           |

 | Copyright (C) 1992,1993,1994,1995,1996                                    |

 |                       W. Metzenthen, 22 Parker St, Ormond, Vic 3163,      |

 |                       Australia.  E-mail billm@suburbia.net               |

 |                                                                           |

 |    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@suburbia.net

--Bill Metzenthen

  October 1996

----------------------- 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 1.20) 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).

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.