^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 1) +---------------------------------------------------------------------------+
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 2) | wm-FPU-emu an FPU emulator for 80386 and 80486SX microprocessors. |
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 3) | |
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 4) | Copyright (C) 1992,1993,1994,1995,1996,1997,1999 |
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 5) | W. Metzenthen, 22 Parker St, Ormond, Vic 3163, |
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 6) | Australia. E-mail billm@melbpc.org.au |
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 7) | |
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 8) | This program is free software; you can redistribute it and/or modify |
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 9) | it under the terms of the GNU General Public License version 2 as |
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 10) | published by the Free Software Foundation. |
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 11) | |
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 12) | This program is distributed in the hope that it will be useful, |
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 13) | but WITHOUT ANY WARRANTY; without even the implied warranty of |
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 14) | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 15) | GNU General Public License for more details. |
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 16) | |
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 17) | You should have received a copy of the GNU General Public License |
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 18) | along with this program; if not, write to the Free Software |
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 19) | Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA. |
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 20) | |
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 21) +---------------------------------------------------------------------------+
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 22)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 23)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 24)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 25) wm-FPU-emu is an FPU emulator for Linux. It is derived from wm-emu387
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 26) which was my 80387 emulator for early versions of djgpp (gcc under
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 27) msdos); wm-emu387 was in turn based upon emu387 which was written by
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 28) DJ Delorie for djgpp. The interface to the Linux kernel is based upon
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 29) the original Linux math emulator by Linus Torvalds.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 30)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 31) My target FPU for wm-FPU-emu is that described in the Intel486
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 32) Programmer's Reference Manual (1992 edition). Unfortunately, numerous
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 33) facets of the functioning of the FPU are not well covered in the
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 34) Reference Manual. The information in the manual has been supplemented
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 35) with measurements on real 80486's. Unfortunately, it is simply not
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 36) possible to be sure that all of the peculiarities of the 80486 have
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 37) been discovered, so there is always likely to be obscure differences
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 38) in the detailed behaviour of the emulator and a real 80486.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 39)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 40) wm-FPU-emu does not implement all of the behaviour of the 80486 FPU,
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 41) but is very close. See "Limitations" later in this file for a list of
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 42) some differences.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 43)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 44) Please report bugs, etc to me at:
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 45) billm@melbpc.org.au
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 46) or b.metzenthen@medoto.unimelb.edu.au
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 47)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 48) For more information on the emulator and on floating point topics, see
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 49) my web pages, currently at http://www.suburbia.net/~billm/
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 50)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 51)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 52) --Bill Metzenthen
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 53) December 1999
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 54)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 55)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 56) ----------------------- Internals of wm-FPU-emu -----------------------
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 57)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 58) Numeric algorithms:
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 59) (1) Add, subtract, and multiply. Nothing remarkable in these.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 60) (2) Divide has been tuned to get reasonable performance. The algorithm
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 61) is not the obvious one which most people seem to use, but is designed
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 62) to take advantage of the characteristics of the 80386. I expect that
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 63) it has been invented many times before I discovered it, but I have not
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 64) seen it. It is based upon one of those ideas which one carries around
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 65) for years without ever bothering to check it out.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 66) (3) The sqrt function has been tuned to get good performance. It is based
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 67) upon Newton's classic method. Performance was improved by capitalizing
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 68) upon the properties of Newton's method, and the code is once again
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 69) structured taking account of the 80386 characteristics.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 70) (4) The trig, log, and exp functions are based in each case upon quasi-
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 71) "optimal" polynomial approximations. My definition of "optimal" was
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 72) based upon getting good accuracy with reasonable speed.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 73) (5) The argument reducing code for the trig function effectively uses
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 74) a value of pi which is accurate to more than 128 bits. As a consequence,
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 75) the reduced argument is accurate to more than 64 bits for arguments up
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 76) to a few pi, and accurate to more than 64 bits for most arguments,
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 77) even for arguments approaching 2^63. This is far superior to an
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 78) 80486, which uses a value of pi which is accurate to 66 bits.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 79)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 80) The code of the emulator is complicated slightly by the need to
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 81) account for a limited form of re-entrancy. Normally, the emulator will
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 82) emulate each FPU instruction to completion without interruption.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 83) However, it may happen that when the emulator is accessing the user
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 84) memory space, swapping may be needed. In this case the emulator may be
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 85) temporarily suspended while disk i/o takes place. During this time
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 86) another process may use the emulator, thereby perhaps changing static
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 87) variables. The code which accesses user memory is confined to five
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 88) files:
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 89) fpu_entry.c
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 90) reg_ld_str.c
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 91) load_store.c
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 92) get_address.c
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 93) errors.c
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 94) As from version 1.12 of the emulator, no static variables are used
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 95) (apart from those in the kernel's per-process tables). The emulator is
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 96) therefore now fully re-entrant, rather than having just the restricted
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 97) form of re-entrancy which is required by the Linux kernel.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 98)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 99) ----------------------- Limitations of wm-FPU-emu -----------------------
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 100)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 101) There are a number of differences between the current wm-FPU-emu
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 102) (version 2.01) and the 80486 FPU (apart from bugs). The differences
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 103) are fewer than those which applied to the 1.xx series of the emulator.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 104) Some of the more important differences are listed below:
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 105)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 106) The Roundup flag does not have much meaning for the transcendental
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 107) functions and its 80486 value with these functions is likely to differ
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 108) from its emulator value.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 109)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 110) In a few rare cases the Underflow flag obtained with the emulator will
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 111) be different from that obtained with an 80486. This occurs when the
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 112) following conditions apply simultaneously:
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 113) (a) the operands have a higher precision than the current setting of the
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 114) precision control (PC) flags.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 115) (b) the underflow exception is masked.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 116) (c) the magnitude of the exact result (before rounding) is less than 2^-16382.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 117) (d) the magnitude of the final result (after rounding) is exactly 2^-16382.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 118) (e) the magnitude of the exact result would be exactly 2^-16382 if the
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 119) operands were rounded to the current precision before the arithmetic
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 120) operation was performed.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 121) If all of these apply, the emulator will set the Underflow flag but a real
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 122) 80486 will not.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 123)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 124) NOTE: Certain formats of Extended Real are UNSUPPORTED. They are
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 125) unsupported by the 80486. They are the Pseudo-NaNs, Pseudoinfinities,
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 126) and Unnormals. None of these will be generated by an 80486 or by the
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 127) emulator. Do not use them. The emulator treats them differently in
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 128) detail from the way an 80486 does.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 129)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 130) Self modifying code can cause the emulator to fail. An example of such
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 131) code is:
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 132) movl %esp,[%ebx]
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 133) fld1
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 134) The FPU instruction may be (usually will be) loaded into the pre-fetch
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 135) queue of the CPU before the mov instruction is executed. If the
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 136) destination of the 'movl' overlaps the FPU instruction then the bytes
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 137) in the prefetch queue and memory will be inconsistent when the FPU
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 138) instruction is executed. The emulator will be invoked but will not be
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 139) able to find the instruction which caused the device-not-present
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 140) exception. For this case, the emulator cannot emulate the behaviour of
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 141) an 80486DX.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 142)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 143) Handling of the address size override prefix byte (0x67) has not been
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 144) extensively tested yet. A major problem exists because using it in
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 145) vm86 mode can cause a general protection fault. Address offsets
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 146) greater than 0xffff appear to be illegal in vm86 mode but are quite
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 147) acceptable (and work) in real mode. A small test program developed to
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 148) check the addressing, and which runs successfully in real mode,
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 149) crashes dosemu under Linux and also brings Windows down with a general
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 150) protection fault message when run under the MS-DOS prompt of Windows
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 151) 3.1. (The program simply reads data from a valid address).
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 152)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 153) The emulator supports 16-bit protected mode, with one difference from
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 154) an 80486DX. A 80486DX will allow some floating point instructions to
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 155) write a few bytes below the lowest address of the stack. The emulator
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 156) will not allow this in 16-bit protected mode: no instructions are
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 157) allowed to write outside the bounds set by the protection.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 158)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 159) ----------------------- Performance of wm-FPU-emu -----------------------
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 160)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 161) Speed.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 162) -----
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 163)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 164) The speed of floating point computation with the emulator will depend
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 165) upon instruction mix. Relative performance is best for the instructions
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 166) which require most computation. The simple instructions are adversely
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 167) affected by the FPU instruction trap overhead.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 168)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 169)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 170) Timing: Some simple timing tests have been made on the emulator functions.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 171) The times include load/store instructions. All times are in microseconds
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 172) measured on a 33MHz 386 with 64k cache. The Turbo C tests were under
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 173) ms-dos, the next two columns are for emulators running with the djgpp
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 174) ms-dos extender. The final column is for wm-FPU-emu in Linux 0.97,
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 175) using libm4.0 (hard).
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 176)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 177) function Turbo C djgpp 1.06 WM-emu387 wm-FPU-emu
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 178)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 179) + 60.5 154.8 76.5 139.4
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 180) - 61.1-65.5 157.3-160.8 76.2-79.5 142.9-144.7
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 181) * 71.0 190.8 79.6 146.6
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 182) / 61.2-75.0 261.4-266.9 75.3-91.6 142.2-158.1
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 183)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 184) sin() 310.8 4692.0 319.0 398.5
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 185) cos() 284.4 4855.2 308.0 388.7
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 186) tan() 495.0 8807.1 394.9 504.7
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 187) atan() 328.9 4866.4 601.1 419.5-491.9
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 188)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 189) sqrt() 128.7 crashed 145.2 227.0
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 190) log() 413.1-419.1 5103.4-5354.21 254.7-282.2 409.4-437.1
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 191) exp() 479.1 6619.2 469.1 850.8
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 192)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 193)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 194) The performance under Linux is improved by the use of look-ahead code.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 195) The following results show the improvement which is obtained under
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 196) Linux due to the look-ahead code. Also given are the times for the
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 197) original Linux emulator with the 4.1 'soft' lib.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 198)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 199) [ Linus' note: I changed look-ahead to be the default under linux, as
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 200) there was no reason not to use it after I had edited it to be
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 201) disabled during tracing ]
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 202)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 203) wm-FPU-emu w original w
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 204) look-ahead 'soft' lib
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 205) + 106.4 190.2
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 206) - 108.6-111.6 192.4-216.2
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 207) * 113.4 193.1
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 208) / 108.8-124.4 700.1-706.2
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 209)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 210) sin() 390.5 2642.0
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 211) cos() 381.5 2767.4
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 212) tan() 496.5 3153.3
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 213) atan() 367.2-435.5 2439.4-3396.8
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 214)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 215) sqrt() 195.1 4732.5
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 216) log() 358.0-387.5 3359.2-3390.3
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 217) exp() 619.3 4046.4
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 218)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 219)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 220) These figures are now somewhat out-of-date. The emulator has become
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 221) progressively slower for most functions as more of the 80486 features
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 222) have been implemented.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 223)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 224)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 225) ----------------------- Accuracy of wm-FPU-emu -----------------------
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 226)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 227)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 228) The accuracy of the emulator is in almost all cases equal to or better
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 229) than that of an Intel 80486 FPU.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 230)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 231) The results of the basic arithmetic functions (+,-,*,/), and fsqrt
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 232) match those of an 80486 FPU. They are the best possible; the error for
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 233) these never exceeds 1/2 an lsb. The fprem and fprem1 instructions
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 234) return exact results; they have no error.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 235)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 236)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 237) The following table compares the emulator accuracy for the sqrt(),
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 238) trig and log functions against the Turbo C "emulator". For this table,
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 239) each function was tested at about 400 points. Ideal worst-case results
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 240) would be 64 bits. The reduced Turbo C accuracy of cos() and tan() for
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 241) arguments greater than pi/4 can be thought of as being related to the
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 242) precision of the argument x; e.g. an argument of pi/2-(1e-10) which is
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 243) accurate to 64 bits can result in a relative accuracy in cos() of
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 244) about 64 + log2(cos(x)) = 31 bits.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 245)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 246)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 247) Function Tested x range Worst result Turbo C
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 248) (relative bits)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 249)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 250) sqrt(x) 1 .. 2 64.1 63.2
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 251) atan(x) 1e-10 .. 200 64.2 62.8
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 252) cos(x) 0 .. pi/2-(1e-10) 64.4 (x <= pi/4) 62.4
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 253) 64.1 (x = pi/2-(1e-10)) 31.9
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 254) sin(x) 1e-10 .. pi/2 64.0 62.8
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 255) tan(x) 1e-10 .. pi/2-(1e-10) 64.0 (x <= pi/4) 62.1
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 256) 64.1 (x = pi/2-(1e-10)) 31.9
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 257) exp(x) 0 .. 1 63.1 ** 62.9
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 258) log(x) 1+1e-6 .. 2 63.8 ** 62.1
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 259)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 260) ** The accuracy for exp() and log() is low because the FPU (emulator)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 261) does not compute them directly; two operations are required.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 262)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 263)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 264) The emulator passes the "paranoia" tests (compiled with gcc 2.3.3 or
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 265) later) for 'float' variables (24 bit precision numbers) when precision
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 266) control is set to 24, 53 or 64 bits, and for 'double' variables (53
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 267) bit precision numbers) when precision control is set to 53 bits (a
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 268) properly performing FPU cannot pass the 'paranoia' tests for 'double'
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 269) variables when precision control is set to 64 bits).
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 270)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 271) The code for reducing the argument for the trig functions (fsin, fcos,
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 272) fptan and fsincos) has been improved and now effectively uses a value
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 273) for pi which is accurate to more than 128 bits precision. As a
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 274) consequence, the accuracy of these functions for large arguments has
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 275) been dramatically improved (and is now very much better than an 80486
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 276) FPU). There is also now no degradation of accuracy for fcos and fptan
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 277) for operands close to pi/2. Measured results are (note that the
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 278) definition of accuracy has changed slightly from that used for the
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 279) above table):
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 280)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 281) Function Tested x range Worst result
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 282) (absolute bits)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 283)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 284) cos(x) 0 .. 9.22e+18 62.0
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 285) sin(x) 1e-16 .. 9.22e+18 62.1
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 286) tan(x) 1e-16 .. 9.22e+18 61.8
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 287)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 288) It is possible with some effort to find very large arguments which
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 289) give much degraded precision. For example, the integer number
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 290) 8227740058411162616.0
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 291) is within about 10e-7 of a multiple of pi. To find the tan (for
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 292) example) of this number to 64 bits precision it would be necessary to
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 293) have a value of pi which had about 150 bits precision. The FPU
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 294) emulator computes the result to about 42.6 bits precision (the correct
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 295) result is about -9.739715e-8). On the other hand, an 80486 FPU returns
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 296) 0.01059, which in relative terms is hopelessly inaccurate.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 297)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 298) For arguments close to critical angles (which occur at multiples of
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 299) pi/2) the emulator is more accurate than an 80486 FPU. For very large
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 300) arguments, the emulator is far more accurate.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 301)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 302)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 303) Prior to version 1.20 of the emulator, the accuracy of the results for
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 304) the transcendental functions (in their principal range) was not as
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 305) good as the results from an 80486 FPU. From version 1.20, the accuracy
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 306) has been considerably improved and these functions now give measured
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 307) worst-case results which are better than the worst-case results given
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 308) by an 80486 FPU.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 309)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 310) The following table gives the measured results for the emulator. The
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 311) number of randomly selected arguments in each case is about half a
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 312) million. The group of three columns gives the frequency of the given
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 313) accuracy in number of times per million, thus the second of these
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 314) columns shows that an accuracy of between 63.80 and 63.89 bits was
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 315) found at a rate of 133 times per one million measurements for fsin.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 316) The results show that the fsin, fcos and fptan instructions return
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 317) results which are in error (i.e. less accurate than the best possible
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 318) result (which is 64 bits)) for about one per cent of all arguments
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 319) between -pi/2 and +pi/2. The other instructions have a lower
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 320) frequency of results which are in error. The last two columns give
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 321) the worst accuracy which was found (in bits) and the approximate value
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 322) of the argument which produced it.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 323)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 324) frequency (per M)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 325) ------------------- ---------------
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 326) instr arg range # tests 63.7 63.8 63.9 worst at arg
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 327) bits bits bits bits
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 328) ----- ------------ ------- ---- ---- ----- ----- --------
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 329) fsin (0,pi/2) 547756 0 133 10673 63.89 0.451317
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 330) fcos (0,pi/2) 547563 0 126 10532 63.85 0.700801
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 331) fptan (0,pi/2) 536274 11 267 10059 63.74 0.784876
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 332) fpatan 4 quadrants 517087 0 8 1855 63.88 0.435121 (4q)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 333) fyl2x (0,20) 541861 0 0 1323 63.94 1.40923 (x)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 334) fyl2xp1 (-.293,.414) 520256 0 0 5678 63.93 0.408542 (x)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 335) f2xm1 (-1,1) 538847 4 481 6488 63.79 0.167709
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 336)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 337)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 338) Tests performed on an 80486 FPU showed results of lower accuracy. The
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 339) following table gives the results which were obtained with an AMD
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 340) 486DX2/66 (other tests indicate that an Intel 486DX produces
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 341) identical results). The tests were basically the same as those used
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 342) to measure the emulator (the values, being random, were in general not
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 343) the same). The total number of tests for each instruction are given
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 344) at the end of the table, in case each about 100k tests were performed.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 345) Another line of figures at the end of the table shows that most of the
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 346) instructions return results which are in error for more than 10
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 347) percent of the arguments tested.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 348)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 349) The numbers in the body of the table give the approx number of times a
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 350) result of the given accuracy in bits (given in the left-most column)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 351) was obtained per one million arguments. For three of the instructions,
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 352) two columns of results are given: * The second column for f2xm1 gives
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 353) the number cases where the results of the first column were for a
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 354) positive argument, this shows that this instruction gives better
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 355) results for positive arguments than it does for negative. * In the
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 356) cases of fcos and fptan, the first column gives the results when all
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 357) cases where arguments greater than 1.5 were removed from the results
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 358) given in the second column. Unlike the emulator, an 80486 FPU returns
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 359) results of relatively poor accuracy for these instructions when the
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 360) argument approaches pi/2. The table does not show those cases when the
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 361) accuracy of the results were less than 62 bits, which occurs quite
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 362) often for fsin and fptan when the argument approaches pi/2. This poor
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 363) accuracy is discussed above in relation to the Turbo C "emulator", and
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 364) the accuracy of the value of pi.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 365)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 366)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 367) bits f2xm1 f2xm1 fpatan fcos fcos fyl2x fyl2xp1 fsin fptan fptan
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 368) 62.0 0 0 0 0 437 0 0 0 0 925
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 369) 62.1 0 0 10 0 894 0 0 0 0 1023
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 370) 62.2 14 0 0 0 1033 0 0 0 0 945
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 371) 62.3 57 0 0 0 1202 0 0 0 0 1023
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 372) 62.4 385 0 0 10 1292 0 23 0 0 1178
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 373) 62.5 1140 0 0 119 1649 0 39 0 0 1149
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 374) 62.6 2037 0 0 189 1620 0 16 0 0 1169
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 375) 62.7 5086 14 0 646 2315 10 101 35 39 1402
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 376) 62.8 8818 86 0 984 3050 59 287 131 224 2036
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 377) 62.9 11340 1355 0 2126 4153 79 605 357 321 1948
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 378) 63.0 15557 4750 0 3319 5376 246 1281 862 808 2688
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 379) 63.1 20016 8288 0 4620 6628 511 2569 1723 1510 3302
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 380) 63.2 24945 11127 10 6588 8098 1120 4470 2968 2990 4724
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 381) 63.3 25686 12382 69 8774 10682 1906 6775 4482 5474 7236
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 382) 63.4 29219 14722 79 11109 12311 3094 9414 7259 8912 10587
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 383) 63.5 30458 14936 393 13802 15014 5874 12666 9609 13762 15262
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 384) 63.6 32439 16448 1277 17945 19028 10226 15537 14657 19158 20346
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 385) 63.7 35031 16805 4067 23003 23947 18910 20116 21333 25001 26209
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 386) 63.8 33251 15820 7673 24781 25675 24617 25354 24440 29433 30329
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 387) 63.9 33293 16833 18529 28318 29233 31267 31470 27748 29676 30601
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 388)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 389) Per cent with error:
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 390) 30.9 3.2 18.5 9.8 13.1 11.6 17.4
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 391) Total arguments tested:
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 392) 70194 70099 101784 100641 100641 101799 128853 114893 102675 102675
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 393)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 394)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 395) ------------------------- Contributors -------------------------------
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 396)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 397) A number of people have contributed to the development of the
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 398) emulator, often by just reporting bugs, sometimes with suggested
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 399) fixes, and a few kind people have provided me with access in one way
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 400) or another to an 80486 machine. Contributors include (to those people
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 401) who I may have forgotten, please forgive me):
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 402)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 403) Linus Torvalds
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 404) Tommy.Thorn@daimi.aau.dk
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 405) Andrew.Tridgell@anu.edu.au
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 406) Nick Holloway, alfie@dcs.warwick.ac.uk
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 407) Hermano Moura, moura@dcs.gla.ac.uk
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 408) Jon Jagger, J.Jagger@scp.ac.uk
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 409) Lennart Benschop
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 410) Brian Gallew, geek+@CMU.EDU
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 411) Thomas Staniszewski, ts3v+@andrew.cmu.edu
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 412) Martin Howell, mph@plasma.apana.org.au
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 413) M Saggaf, alsaggaf@athena.mit.edu
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 414) Peter Barker, PETER@socpsy.sci.fau.edu
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 415) tom@vlsivie.tuwien.ac.at
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 416) Dan Russel, russed@rpi.edu
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 417) Daniel Carosone, danielce@ee.mu.oz.au
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 418) cae@jpmorgan.com
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 419) Hamish Coleman, t933093@minyos.xx.rmit.oz.au
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 420) Bruce Evans, bde@kralizec.zeta.org.au
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 421) Timo Korvola, Timo.Korvola@hut.fi
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 422) Rick Lyons, rick@razorback.brisnet.org.au
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 423) Rick, jrs@world.std.com
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 424)
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 425) ...and numerous others who responded to my request for help with
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 426) a real 80486.
^8f3ce5b39 (kx 2023-10-28 12:00:06 +0300 427)
Orange Pi5 kernel
Deprecated Linux kernel 5.10.110 for OrangePi 5/5B/5+ boards
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