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LINUX 1.0 内核c源代码

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These figures are now somewhat out-of-date.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
[e-mail]
[e-mail]
Nick Holloway, [e-mail]
Hermano Moura, [e-mail]
Jon Jagger, [e-mail]
Lennart Benschop
Brian Gallew, [e-mail]
Thomas Staniszewski, [e-mail]
Martin Howell, [e-mail]
M Saggaf, [e-mail]
Peter Barker, [e-mail]
[e-mail]
Dan Russel, [e-mail]
Daniel Carosone, [e-mail]
[e-mail]
Hamish Coleman, [e-mail]
Bruce Evans, [e-mail]
Timo Korvola, [e-mail]
 
...and numerous others who responded to my request for help with
a real 80486.