| 1 |
3 |
xianfeng |
|
|
| 2 |
|
|
| slogn.sa 3.1 12/10/90
|
| 3 |
|
|
|
|
| 4 |
|
|
| slogn computes the natural logarithm of an
|
| 5 |
|
|
| input value. slognd does the same except the input value is a
|
| 6 |
|
|
| denormalized number. slognp1 computes log(1+X), and slognp1d
|
| 7 |
|
|
| computes log(1+X) for denormalized X.
|
| 8 |
|
|
|
|
| 9 |
|
|
| Input: Double-extended value in memory location pointed to by address
|
| 10 |
|
|
| register a0.
|
| 11 |
|
|
|
|
| 12 |
|
|
| Output: log(X) or log(1+X) returned in floating-point register Fp0.
|
| 13 |
|
|
|
|
| 14 |
|
|
| Accuracy and Monotonicity: The returned result is within 2 ulps in
|
| 15 |
|
|
| 64 significant bit, i.e. within 0.5001 ulp to 53 bits if the
|
| 16 |
|
|
| result is subsequently rounded to double precision. The
|
| 17 |
|
|
| result is provably monotonic in double precision.
|
| 18 |
|
|
|
|
| 19 |
|
|
| Speed: The program slogn takes approximately 190 cycles for input
|
| 20 |
|
|
| argument X such that |X-1| >= 1/16, which is the usual
|
| 21 |
|
|
| situation. For those arguments, slognp1 takes approximately
|
| 22 |
|
|
| 210 cycles. For the less common arguments, the program will
|
| 23 |
|
|
| run no worse than 10% slower.
|
| 24 |
|
|
|
|
| 25 |
|
|
| Algorithm:
|
| 26 |
|
|
| LOGN:
|
| 27 |
|
|
| Step 1. If |X-1| < 1/16, approximate log(X) by an odd polynomial in
|
| 28 |
|
|
| u, where u = 2(X-1)/(X+1). Otherwise, move on to Step 2.
|
| 29 |
|
|
|
|
| 30 |
|
|
| Step 2. X = 2**k * Y where 1 <= Y < 2. Define F to be the first seven
|
| 31 |
|
|
| significant bits of Y plus 2**(-7), i.e. F = 1.xxxxxx1 in base
|
| 32 |
|
|
| 2 where the six "x" match those of Y. Note that |Y-F| <= 2**(-7).
|
| 33 |
|
|
|
|
| 34 |
|
|
| Step 3. Define u = (Y-F)/F. Approximate log(1+u) by a polynomial in u,
|
| 35 |
|
|
| log(1+u) = poly.
|
| 36 |
|
|
|
|
| 37 |
|
|
| Step 4. Reconstruct log(X) = log( 2**k * Y ) = k*log(2) + log(F) + log(1+u)
|
| 38 |
|
|
| by k*log(2) + (log(F) + poly). The values of log(F) are calculated
|
| 39 |
|
|
| beforehand and stored in the program.
|
| 40 |
|
|
|
|
| 41 |
|
|
| lognp1:
|
| 42 |
|
|
| Step 1: If |X| < 1/16, approximate log(1+X) by an odd polynomial in
|
| 43 |
|
|
| u where u = 2X/(2+X). Otherwise, move on to Step 2.
|
| 44 |
|
|
|
|
| 45 |
|
|
| Step 2: Let 1+X = 2**k * Y, where 1 <= Y < 2. Define F as done in Step 2
|
| 46 |
|
|
| of the algorithm for LOGN and compute log(1+X) as
|
| 47 |
|
|
| k*log(2) + log(F) + poly where poly approximates log(1+u),
|
| 48 |
|
|
| u = (Y-F)/F.
|
| 49 |
|
|
|
|
| 50 |
|
|
| Implementation Notes:
|
| 51 |
|
|
| Note 1. There are 64 different possible values for F, thus 64 log(F)'s
|
| 52 |
|
|
| need to be tabulated. Moreover, the values of 1/F are also
|
| 53 |
|
|
| tabulated so that the division in (Y-F)/F can be performed by a
|
| 54 |
|
|
| multiplication.
|
| 55 |
|
|
|
|
| 56 |
|
|
| Note 2. In Step 2 of lognp1, in order to preserved accuracy, the value
|
| 57 |
|
|
| Y-F has to be calculated carefully when 1/2 <= X < 3/2.
|
| 58 |
|
|
|
|
| 59 |
|
|
| Note 3. To fully exploit the pipeline, polynomials are usually separated
|
| 60 |
|
|
| into two parts evaluated independently before being added up.
|
| 61 |
|
|
|
|
| 62 |
|
|
|
| 63 |
|
|
| Copyright (C) Motorola, Inc. 1990
|
| 64 |
|
|
| All Rights Reserved
|
| 65 |
|
|
|
|
| 66 |
|
|
| For details on the license for this file, please see the
|
| 67 |
|
|
| file, README, in this same directory.
|
| 68 |
|
|
|
| 69 |
|
|
|slogn idnt 2,1 | Motorola 040 Floating Point Software Package
|
| 70 |
|
|
|
| 71 |
|
|
|section 8
|
| 72 |
|
|
|
| 73 |
|
|
#include "fpsp.h"
|
| 74 |
|
|
|
| 75 |
|
|
BOUNDS1: .long 0x3FFEF07D,0x3FFF8841
|
| 76 |
|
|
BOUNDS2: .long 0x3FFE8000,0x3FFFC000
|
| 77 |
|
|
|
| 78 |
|
|
LOGOF2: .long 0x3FFE0000,0xB17217F7,0xD1CF79AC,0x00000000
|
| 79 |
|
|
|
| 80 |
|
|
one: .long 0x3F800000
|
| 81 |
|
|
zero: .long 0x00000000
|
| 82 |
|
|
infty: .long 0x7F800000
|
| 83 |
|
|
negone: .long 0xBF800000
|
| 84 |
|
|
|
| 85 |
|
|
LOGA6: .long 0x3FC2499A,0xB5E4040B
|
| 86 |
|
|
LOGA5: .long 0xBFC555B5,0x848CB7DB
|
| 87 |
|
|
|
| 88 |
|
|
LOGA4: .long 0x3FC99999,0x987D8730
|
| 89 |
|
|
LOGA3: .long 0xBFCFFFFF,0xFF6F7E97
|
| 90 |
|
|
|
| 91 |
|
|
LOGA2: .long 0x3FD55555,0x555555a4
|
| 92 |
|
|
LOGA1: .long 0xBFE00000,0x00000008
|
| 93 |
|
|
|
| 94 |
|
|
LOGB5: .long 0x3F175496,0xADD7DAD6
|
| 95 |
|
|
LOGB4: .long 0x3F3C71C2,0xFE80C7E0
|
| 96 |
|
|
|
| 97 |
|
|
LOGB3: .long 0x3F624924,0x928BCCFF
|
| 98 |
|
|
LOGB2: .long 0x3F899999,0x999995EC
|
| 99 |
|
|
|
| 100 |
|
|
LOGB1: .long 0x3FB55555,0x55555555
|
| 101 |
|
|
TWO: .long 0x40000000,0x00000000
|
| 102 |
|
|
|
| 103 |
|
|
LTHOLD: .long 0x3f990000,0x80000000,0x00000000,0x00000000
|
| 104 |
|
|
|
| 105 |
|
|
LOGTBL:
|
| 106 |
|
|
.long 0x3FFE0000,0xFE03F80F,0xE03F80FE,0x00000000
|
| 107 |
|
|
.long 0x3FF70000,0xFF015358,0x833C47E2,0x00000000
|
| 108 |
|
|
.long 0x3FFE0000,0xFA232CF2,0x52138AC0,0x00000000
|
| 109 |
|
|
.long 0x3FF90000,0xBDC8D83E,0xAD88D549,0x00000000
|
| 110 |
|
|
.long 0x3FFE0000,0xF6603D98,0x0F6603DA,0x00000000
|
| 111 |
|
|
.long 0x3FFA0000,0x9CF43DCF,0xF5EAFD48,0x00000000
|
| 112 |
|
|
.long 0x3FFE0000,0xF2B9D648,0x0F2B9D65,0x00000000
|
| 113 |
|
|
.long 0x3FFA0000,0xDA16EB88,0xCB8DF614,0x00000000
|
| 114 |
|
|
.long 0x3FFE0000,0xEF2EB71F,0xC4345238,0x00000000
|
| 115 |
|
|
.long 0x3FFB0000,0x8B29B775,0x1BD70743,0x00000000
|
| 116 |
|
|
.long 0x3FFE0000,0xEBBDB2A5,0xC1619C8C,0x00000000
|
| 117 |
|
|
.long 0x3FFB0000,0xA8D839F8,0x30C1FB49,0x00000000
|
| 118 |
|
|
.long 0x3FFE0000,0xE865AC7B,0x7603A197,0x00000000
|
| 119 |
|
|
.long 0x3FFB0000,0xC61A2EB1,0x8CD907AD,0x00000000
|
| 120 |
|
|
.long 0x3FFE0000,0xE525982A,0xF70C880E,0x00000000
|
| 121 |
|
|
.long 0x3FFB0000,0xE2F2A47A,0xDE3A18AF,0x00000000
|
| 122 |
|
|
.long 0x3FFE0000,0xE1FC780E,0x1FC780E2,0x00000000
|
| 123 |
|
|
.long 0x3FFB0000,0xFF64898E,0xDF55D551,0x00000000
|
| 124 |
|
|
.long 0x3FFE0000,0xDEE95C4C,0xA037BA57,0x00000000
|
| 125 |
|
|
.long 0x3FFC0000,0x8DB956A9,0x7B3D0148,0x00000000
|
| 126 |
|
|
.long 0x3FFE0000,0xDBEB61EE,0xD19C5958,0x00000000
|
| 127 |
|
|
.long 0x3FFC0000,0x9B8FE100,0xF47BA1DE,0x00000000
|
| 128 |
|
|
.long 0x3FFE0000,0xD901B203,0x6406C80E,0x00000000
|
| 129 |
|
|
.long 0x3FFC0000,0xA9372F1D,0x0DA1BD17,0x00000000
|
| 130 |
|
|
.long 0x3FFE0000,0xD62B80D6,0x2B80D62C,0x00000000
|
| 131 |
|
|
.long 0x3FFC0000,0xB6B07F38,0xCE90E46B,0x00000000
|
| 132 |
|
|
.long 0x3FFE0000,0xD3680D36,0x80D3680D,0x00000000
|
| 133 |
|
|
.long 0x3FFC0000,0xC3FD0329,0x06488481,0x00000000
|
| 134 |
|
|
.long 0x3FFE0000,0xD0B69FCB,0xD2580D0B,0x00000000
|
| 135 |
|
|
.long 0x3FFC0000,0xD11DE0FF,0x15AB18CA,0x00000000
|
| 136 |
|
|
.long 0x3FFE0000,0xCE168A77,0x25080CE1,0x00000000
|
| 137 |
|
|
.long 0x3FFC0000,0xDE1433A1,0x6C66B150,0x00000000
|
| 138 |
|
|
.long 0x3FFE0000,0xCB8727C0,0x65C393E0,0x00000000
|
| 139 |
|
|
.long 0x3FFC0000,0xEAE10B5A,0x7DDC8ADD,0x00000000
|
| 140 |
|
|
.long 0x3FFE0000,0xC907DA4E,0x871146AD,0x00000000
|
| 141 |
|
|
.long 0x3FFC0000,0xF7856E5E,0xE2C9B291,0x00000000
|
| 142 |
|
|
.long 0x3FFE0000,0xC6980C69,0x80C6980C,0x00000000
|
| 143 |
|
|
.long 0x3FFD0000,0x82012CA5,0xA68206D7,0x00000000
|
| 144 |
|
|
.long 0x3FFE0000,0xC4372F85,0x5D824CA6,0x00000000
|
| 145 |
|
|
.long 0x3FFD0000,0x882C5FCD,0x7256A8C5,0x00000000
|
| 146 |
|
|
.long 0x3FFE0000,0xC1E4BBD5,0x95F6E947,0x00000000
|
| 147 |
|
|
.long 0x3FFD0000,0x8E44C60B,0x4CCFD7DE,0x00000000
|
| 148 |
|
|
.long 0x3FFE0000,0xBFA02FE8,0x0BFA02FF,0x00000000
|
| 149 |
|
|
.long 0x3FFD0000,0x944AD09E,0xF4351AF6,0x00000000
|
| 150 |
|
|
.long 0x3FFE0000,0xBD691047,0x07661AA3,0x00000000
|
| 151 |
|
|
.long 0x3FFD0000,0x9A3EECD4,0xC3EAA6B2,0x00000000
|
| 152 |
|
|
.long 0x3FFE0000,0xBB3EE721,0xA54D880C,0x00000000
|
| 153 |
|
|
.long 0x3FFD0000,0xA0218434,0x353F1DE8,0x00000000
|
| 154 |
|
|
.long 0x3FFE0000,0xB92143FA,0x36F5E02E,0x00000000
|
| 155 |
|
|
.long 0x3FFD0000,0xA5F2FCAB,0xBBC506DA,0x00000000
|
| 156 |
|
|
.long 0x3FFE0000,0xB70FBB5A,0x19BE3659,0x00000000
|
| 157 |
|
|
.long 0x3FFD0000,0xABB3B8BA,0x2AD362A5,0x00000000
|
| 158 |
|
|
.long 0x3FFE0000,0xB509E68A,0x9B94821F,0x00000000
|
| 159 |
|
|
.long 0x3FFD0000,0xB1641795,0xCE3CA97B,0x00000000
|
| 160 |
|
|
.long 0x3FFE0000,0xB30F6352,0x8917C80B,0x00000000
|
| 161 |
|
|
.long 0x3FFD0000,0xB7047551,0x5D0F1C61,0x00000000
|
| 162 |
|
|
.long 0x3FFE0000,0xB11FD3B8,0x0B11FD3C,0x00000000
|
| 163 |
|
|
.long 0x3FFD0000,0xBC952AFE,0xEA3D13E1,0x00000000
|
| 164 |
|
|
.long 0x3FFE0000,0xAF3ADDC6,0x80AF3ADE,0x00000000
|
| 165 |
|
|
.long 0x3FFD0000,0xC2168ED0,0xF458BA4A,0x00000000
|
| 166 |
|
|
.long 0x3FFE0000,0xAD602B58,0x0AD602B6,0x00000000
|
| 167 |
|
|
.long 0x3FFD0000,0xC788F439,0xB3163BF1,0x00000000
|
| 168 |
|
|
.long 0x3FFE0000,0xAB8F69E2,0x8359CD11,0x00000000
|
| 169 |
|
|
.long 0x3FFD0000,0xCCECAC08,0xBF04565D,0x00000000
|
| 170 |
|
|
.long 0x3FFE0000,0xA9C84A47,0xA07F5638,0x00000000
|
| 171 |
|
|
.long 0x3FFD0000,0xD2420487,0x2DD85160,0x00000000
|
| 172 |
|
|
.long 0x3FFE0000,0xA80A80A8,0x0A80A80B,0x00000000
|
| 173 |
|
|
.long 0x3FFD0000,0xD7894992,0x3BC3588A,0x00000000
|
| 174 |
|
|
.long 0x3FFE0000,0xA655C439,0x2D7B73A8,0x00000000
|
| 175 |
|
|
.long 0x3FFD0000,0xDCC2C4B4,0x9887DACC,0x00000000
|
| 176 |
|
|
.long 0x3FFE0000,0xA4A9CF1D,0x96833751,0x00000000
|
| 177 |
|
|
.long 0x3FFD0000,0xE1EEBD3E,0x6D6A6B9E,0x00000000
|
| 178 |
|
|
.long 0x3FFE0000,0xA3065E3F,0xAE7CD0E0,0x00000000
|
| 179 |
|
|
.long 0x3FFD0000,0xE70D785C,0x2F9F5BDC,0x00000000
|
| 180 |
|
|
.long 0x3FFE0000,0xA16B312E,0xA8FC377D,0x00000000
|
| 181 |
|
|
.long 0x3FFD0000,0xEC1F392C,0x5179F283,0x00000000
|
| 182 |
|
|
.long 0x3FFE0000,0x9FD809FD,0x809FD80A,0x00000000
|
| 183 |
|
|
.long 0x3FFD0000,0xF12440D3,0xE36130E6,0x00000000
|
| 184 |
|
|
.long 0x3FFE0000,0x9E4CAD23,0xDD5F3A20,0x00000000
|
| 185 |
|
|
.long 0x3FFD0000,0xF61CCE92,0x346600BB,0x00000000
|
| 186 |
|
|
.long 0x3FFE0000,0x9CC8E160,0xC3FB19B9,0x00000000
|
| 187 |
|
|
.long 0x3FFD0000,0xFB091FD3,0x8145630A,0x00000000
|
| 188 |
|
|
.long 0x3FFE0000,0x9B4C6F9E,0xF03A3CAA,0x00000000
|
| 189 |
|
|
.long 0x3FFD0000,0xFFE97042,0xBFA4C2AD,0x00000000
|
| 190 |
|
|
.long 0x3FFE0000,0x99D722DA,0xBDE58F06,0x00000000
|
| 191 |
|
|
.long 0x3FFE0000,0x825EFCED,0x49369330,0x00000000
|
| 192 |
|
|
.long 0x3FFE0000,0x9868C809,0x868C8098,0x00000000
|
| 193 |
|
|
.long 0x3FFE0000,0x84C37A7A,0xB9A905C9,0x00000000
|
| 194 |
|
|
.long 0x3FFE0000,0x97012E02,0x5C04B809,0x00000000
|
| 195 |
|
|
.long 0x3FFE0000,0x87224C2E,0x8E645FB7,0x00000000
|
| 196 |
|
|
.long 0x3FFE0000,0x95A02568,0x095A0257,0x00000000
|
| 197 |
|
|
.long 0x3FFE0000,0x897B8CAC,0x9F7DE298,0x00000000
|
| 198 |
|
|
.long 0x3FFE0000,0x94458094,0x45809446,0x00000000
|
| 199 |
|
|
.long 0x3FFE0000,0x8BCF55DE,0xC4CD05FE,0x00000000
|
| 200 |
|
|
.long 0x3FFE0000,0x92F11384,0x0497889C,0x00000000
|
| 201 |
|
|
.long 0x3FFE0000,0x8E1DC0FB,0x89E125E5,0x00000000
|
| 202 |
|
|
.long 0x3FFE0000,0x91A2B3C4,0xD5E6F809,0x00000000
|
| 203 |
|
|
.long 0x3FFE0000,0x9066E68C,0x955B6C9B,0x00000000
|
| 204 |
|
|
.long 0x3FFE0000,0x905A3863,0x3E06C43B,0x00000000
|
| 205 |
|
|
.long 0x3FFE0000,0x92AADE74,0xC7BE59E0,0x00000000
|
| 206 |
|
|
.long 0x3FFE0000,0x8F1779D9,0xFDC3A219,0x00000000
|
| 207 |
|
|
.long 0x3FFE0000,0x94E9BFF6,0x15845643,0x00000000
|
| 208 |
|
|
.long 0x3FFE0000,0x8DDA5202,0x37694809,0x00000000
|
| 209 |
|
|
.long 0x3FFE0000,0x9723A1B7,0x20134203,0x00000000
|
| 210 |
|
|
.long 0x3FFE0000,0x8CA29C04,0x6514E023,0x00000000
|
| 211 |
|
|
.long 0x3FFE0000,0x995899C8,0x90EB8990,0x00000000
|
| 212 |
|
|
.long 0x3FFE0000,0x8B70344A,0x139BC75A,0x00000000
|
| 213 |
|
|
.long 0x3FFE0000,0x9B88BDAA,0x3A3DAE2F,0x00000000
|
| 214 |
|
|
.long 0x3FFE0000,0x8A42F870,0x5669DB46,0x00000000
|
| 215 |
|
|
.long 0x3FFE0000,0x9DB4224F,0xFFE1157C,0x00000000
|
| 216 |
|
|
.long 0x3FFE0000,0x891AC73A,0xE9819B50,0x00000000
|
| 217 |
|
|
.long 0x3FFE0000,0x9FDADC26,0x8B7A12DA,0x00000000
|
| 218 |
|
|
.long 0x3FFE0000,0x87F78087,0xF78087F8,0x00000000
|
| 219 |
|
|
.long 0x3FFE0000,0xA1FCFF17,0xCE733BD4,0x00000000
|
| 220 |
|
|
.long 0x3FFE0000,0x86D90544,0x7A34ACC6,0x00000000
|
| 221 |
|
|
.long 0x3FFE0000,0xA41A9E8F,0x5446FB9F,0x00000000
|
| 222 |
|
|
.long 0x3FFE0000,0x85BF3761,0x2CEE3C9B,0x00000000
|
| 223 |
|
|
.long 0x3FFE0000,0xA633CD7E,0x6771CD8B,0x00000000
|
| 224 |
|
|
.long 0x3FFE0000,0x84A9F9C8,0x084A9F9D,0x00000000
|
| 225 |
|
|
.long 0x3FFE0000,0xA8489E60,0x0B435A5E,0x00000000
|
| 226 |
|
|
.long 0x3FFE0000,0x83993052,0x3FBE3368,0x00000000
|
| 227 |
|
|
.long 0x3FFE0000,0xAA59233C,0xCCA4BD49,0x00000000
|
| 228 |
|
|
.long 0x3FFE0000,0x828CBFBE,0xB9A020A3,0x00000000
|
| 229 |
|
|
.long 0x3FFE0000,0xAC656DAE,0x6BCC4985,0x00000000
|
| 230 |
|
|
.long 0x3FFE0000,0x81848DA8,0xFAF0D277,0x00000000
|
| 231 |
|
|
.long 0x3FFE0000,0xAE6D8EE3,0x60BB2468,0x00000000
|
| 232 |
|
|
.long 0x3FFE0000,0x80808080,0x80808081,0x00000000
|
| 233 |
|
|
.long 0x3FFE0000,0xB07197A2,0x3C46C654,0x00000000
|
| 234 |
|
|
|
| 235 |
|
|
.set ADJK,L_SCR1
|
| 236 |
|
|
|
| 237 |
|
|
.set X,FP_SCR1
|
| 238 |
|
|
.set XDCARE,X+2
|
| 239 |
|
|
.set XFRAC,X+4
|
| 240 |
|
|
|
| 241 |
|
|
.set F,FP_SCR2
|
| 242 |
|
|
.set FFRAC,F+4
|
| 243 |
|
|
|
| 244 |
|
|
.set KLOG2,FP_SCR3
|
| 245 |
|
|
|
| 246 |
|
|
.set SAVEU,FP_SCR4
|
| 247 |
|
|
|
| 248 |
|
|
| xref t_frcinx
|
| 249 |
|
|
|xref t_extdnrm
|
| 250 |
|
|
|xref t_operr
|
| 251 |
|
|
|xref t_dz
|
| 252 |
|
|
|
| 253 |
|
|
.global slognd
|
| 254 |
|
|
slognd:
|
| 255 |
|
|
|--ENTRY POINT FOR LOG(X) FOR DENORMALIZED INPUT
|
| 256 |
|
|
|
| 257 |
|
|
movel #-100,ADJK(%a6) | ...INPUT = 2^(ADJK) * FP0
|
| 258 |
|
|
|
| 259 |
|
|
|----normalize the input value by left shifting k bits (k to be determined
|
| 260 |
|
|
|----below), adjusting exponent and storing -k to ADJK
|
| 261 |
|
|
|----the value TWOTO100 is no longer needed.
|
| 262 |
|
|
|----Note that this code assumes the denormalized input is NON-ZERO.
|
| 263 |
|
|
|
| 264 |
|
|
moveml %d2-%d7,-(%a7) | ...save some registers
|
| 265 |
|
|
movel #0x00000000,%d3 | ...D3 is exponent of smallest norm. #
|
| 266 |
|
|
movel 4(%a0),%d4
|
| 267 |
|
|
movel 8(%a0),%d5 | ...(D4,D5) is (Hi_X,Lo_X)
|
| 268 |
|
|
clrl %d2 | ...D2 used for holding K
|
| 269 |
|
|
|
| 270 |
|
|
tstl %d4
|
| 271 |
|
|
bnes HiX_not0
|
| 272 |
|
|
|
| 273 |
|
|
HiX_0:
|
| 274 |
|
|
movel %d5,%d4
|
| 275 |
|
|
clrl %d5
|
| 276 |
|
|
movel #32,%d2
|
| 277 |
|
|
clrl %d6
|
| 278 |
|
|
bfffo %d4{#0:#32},%d6
|
| 279 |
|
|
lsll %d6,%d4
|
| 280 |
|
|
addl %d6,%d2 | ...(D3,D4,D5) is normalized
|
| 281 |
|
|
|
| 282 |
|
|
movel %d3,X(%a6)
|
| 283 |
|
|
movel %d4,XFRAC(%a6)
|
| 284 |
|
|
movel %d5,XFRAC+4(%a6)
|
| 285 |
|
|
negl %d2
|
| 286 |
|
|
movel %d2,ADJK(%a6)
|
| 287 |
|
|
fmovex X(%a6),%fp0
|
| 288 |
|
|
moveml (%a7)+,%d2-%d7 | ...restore registers
|
| 289 |
|
|
lea X(%a6),%a0
|
| 290 |
|
|
bras LOGBGN | ...begin regular log(X)
|
| 291 |
|
|
|
| 292 |
|
|
|
| 293 |
|
|
HiX_not0:
|
| 294 |
|
|
clrl %d6
|
| 295 |
|
|
bfffo %d4{#0:#32},%d6 | ...find first 1
|
| 296 |
|
|
movel %d6,%d2 | ...get k
|
| 297 |
|
|
lsll %d6,%d4
|
| 298 |
|
|
movel %d5,%d7 | ...a copy of D5
|
| 299 |
|
|
lsll %d6,%d5
|
| 300 |
|
|
negl %d6
|
| 301 |
|
|
addil #32,%d6
|
| 302 |
|
|
lsrl %d6,%d7
|
| 303 |
|
|
orl %d7,%d4 | ...(D3,D4,D5) normalized
|
| 304 |
|
|
|
| 305 |
|
|
movel %d3,X(%a6)
|
| 306 |
|
|
movel %d4,XFRAC(%a6)
|
| 307 |
|
|
movel %d5,XFRAC+4(%a6)
|
| 308 |
|
|
negl %d2
|
| 309 |
|
|
movel %d2,ADJK(%a6)
|
| 310 |
|
|
fmovex X(%a6),%fp0
|
| 311 |
|
|
moveml (%a7)+,%d2-%d7 | ...restore registers
|
| 312 |
|
|
lea X(%a6),%a0
|
| 313 |
|
|
bras LOGBGN | ...begin regular log(X)
|
| 314 |
|
|
|
| 315 |
|
|
|
| 316 |
|
|
.global slogn
|
| 317 |
|
|
slogn:
|
| 318 |
|
|
|--ENTRY POINT FOR LOG(X) FOR X FINITE, NON-ZERO, NOT NAN'S
|
| 319 |
|
|
|
| 320 |
|
|
fmovex (%a0),%fp0 | ...LOAD INPUT
|
| 321 |
|
|
movel #0x00000000,ADJK(%a6)
|
| 322 |
|
|
|
| 323 |
|
|
LOGBGN:
|
| 324 |
|
|
|--FPCR SAVED AND CLEARED, INPUT IS 2^(ADJK)*FP0, FP0 CONTAINS
|
| 325 |
|
|
|--A FINITE, NON-ZERO, NORMALIZED NUMBER.
|
| 326 |
|
|
|
| 327 |
|
|
movel (%a0),%d0
|
| 328 |
|
|
movew 4(%a0),%d0
|
| 329 |
|
|
|
| 330 |
|
|
movel (%a0),X(%a6)
|
| 331 |
|
|
movel 4(%a0),X+4(%a6)
|
| 332 |
|
|
movel 8(%a0),X+8(%a6)
|
| 333 |
|
|
|
| 334 |
|
|
cmpil #0,%d0 | ...CHECK IF X IS NEGATIVE
|
| 335 |
|
|
blt LOGNEG | ...LOG OF NEGATIVE ARGUMENT IS INVALID
|
| 336 |
|
|
cmp2l BOUNDS1,%d0 | ...X IS POSITIVE, CHECK IF X IS NEAR 1
|
| 337 |
|
|
bcc LOGNEAR1 | ...BOUNDS IS ROUGHLY [15/16, 17/16]
|
| 338 |
|
|
|
| 339 |
|
|
LOGMAIN:
|
| 340 |
|
|
|--THIS SHOULD BE THE USUAL CASE, X NOT VERY CLOSE TO 1
|
| 341 |
|
|
|
| 342 |
|
|
|--X = 2^(K) * Y, 1 <= Y < 2. THUS, Y = 1.XXXXXXXX....XX IN BINARY.
|
| 343 |
|
|
|--WE DEFINE F = 1.XXXXXX1, I.E. FIRST 7 BITS OF Y AND ATTACH A 1.
|
| 344 |
|
|
|--THE IDEA IS THAT LOG(X) = K*LOG2 + LOG(Y)
|
| 345 |
|
|
|-- = K*LOG2 + LOG(F) + LOG(1 + (Y-F)/F).
|
| 346 |
|
|
|--NOTE THAT U = (Y-F)/F IS VERY SMALL AND THUS APPROXIMATING
|
| 347 |
|
|
|--LOG(1+U) CAN BE VERY EFFICIENT.
|
| 348 |
|
|
|--ALSO NOTE THAT THE VALUE 1/F IS STORED IN A TABLE SO THAT NO
|
| 349 |
|
|
|--DIVISION IS NEEDED TO CALCULATE (Y-F)/F.
|
| 350 |
|
|
|
| 351 |
|
|
|--GET K, Y, F, AND ADDRESS OF 1/F.
|
| 352 |
|
|
asrl #8,%d0
|
| 353 |
|
|
asrl #8,%d0 | ...SHIFTED 16 BITS, BIASED EXPO. OF X
|
| 354 |
|
|
subil #0x3FFF,%d0 | ...THIS IS K
|
| 355 |
|
|
addl ADJK(%a6),%d0 | ...ADJUST K, ORIGINAL INPUT MAY BE DENORM.
|
| 356 |
|
|
lea LOGTBL,%a0 | ...BASE ADDRESS OF 1/F AND LOG(F)
|
| 357 |
|
|
fmovel %d0,%fp1 | ...CONVERT K TO FLOATING-POINT FORMAT
|
| 358 |
|
|
|
| 359 |
|
|
|--WHILE THE CONVERSION IS GOING ON, WE GET F AND ADDRESS OF 1/F
|
| 360 |
|
|
movel #0x3FFF0000,X(%a6) | ...X IS NOW Y, I.E. 2^(-K)*X
|
| 361 |
|
|
movel XFRAC(%a6),FFRAC(%a6)
|
| 362 |
|
|
andil #0xFE000000,FFRAC(%a6) | ...FIRST 7 BITS OF Y
|
| 363 |
|
|
oril #0x01000000,FFRAC(%a6) | ...GET F: ATTACH A 1 AT THE EIGHTH BIT
|
| 364 |
|
|
movel FFRAC(%a6),%d0 | ...READY TO GET ADDRESS OF 1/F
|
| 365 |
|
|
andil #0x7E000000,%d0
|
| 366 |
|
|
asrl #8,%d0
|
| 367 |
|
|
asrl #8,%d0
|
| 368 |
|
|
asrl #4,%d0 | ...SHIFTED 20, D0 IS THE DISPLACEMENT
|
| 369 |
|
|
addal %d0,%a0 | ...A0 IS THE ADDRESS FOR 1/F
|
| 370 |
|
|
|
| 371 |
|
|
fmovex X(%a6),%fp0
|
| 372 |
|
|
movel #0x3fff0000,F(%a6)
|
| 373 |
|
|
clrl F+8(%a6)
|
| 374 |
|
|
fsubx F(%a6),%fp0 | ...Y-F
|
| 375 |
|
|
fmovemx %fp2-%fp2/%fp3,-(%sp) | ...SAVE FP2 WHILE FP0 IS NOT READY
|
| 376 |
|
|
|--SUMMARY: FP0 IS Y-F, A0 IS ADDRESS OF 1/F, FP1 IS K
|
| 377 |
|
|
|--REGISTERS SAVED: FPCR, FP1, FP2
|
| 378 |
|
|
|
| 379 |
|
|
LP1CONT1:
|
| 380 |
|
|
|--AN RE-ENTRY POINT FOR LOGNP1
|
| 381 |
|
|
fmulx (%a0),%fp0 | ...FP0 IS U = (Y-F)/F
|
| 382 |
|
|
fmulx LOGOF2,%fp1 | ...GET K*LOG2 WHILE FP0 IS NOT READY
|
| 383 |
|
|
fmovex %fp0,%fp2
|
| 384 |
|
|
fmulx %fp2,%fp2 | ...FP2 IS V=U*U
|
| 385 |
|
|
fmovex %fp1,KLOG2(%a6) | ...PUT K*LOG2 IN MEMORY, FREE FP1
|
| 386 |
|
|
|
| 387 |
|
|
|--LOG(1+U) IS APPROXIMATED BY
|
| 388 |
|
|
|--U + V*(A1+U*(A2+U*(A3+U*(A4+U*(A5+U*A6))))) WHICH IS
|
| 389 |
|
|
|--[U + V*(A1+V*(A3+V*A5))] + [U*V*(A2+V*(A4+V*A6))]
|
| 390 |
|
|
|
| 391 |
|
|
fmovex %fp2,%fp3
|
| 392 |
|
|
fmovex %fp2,%fp1
|
| 393 |
|
|
|
| 394 |
|
|
fmuld LOGA6,%fp1 | ...V*A6
|
| 395 |
|
|
fmuld LOGA5,%fp2 | ...V*A5
|
| 396 |
|
|
|
| 397 |
|
|
faddd LOGA4,%fp1 | ...A4+V*A6
|
| 398 |
|
|
faddd LOGA3,%fp2 | ...A3+V*A5
|
| 399 |
|
|
|
| 400 |
|
|
fmulx %fp3,%fp1 | ...V*(A4+V*A6)
|
| 401 |
|
|
fmulx %fp3,%fp2 | ...V*(A3+V*A5)
|
| 402 |
|
|
|
| 403 |
|
|
faddd LOGA2,%fp1 | ...A2+V*(A4+V*A6)
|
| 404 |
|
|
faddd LOGA1,%fp2 | ...A1+V*(A3+V*A5)
|
| 405 |
|
|
|
| 406 |
|
|
fmulx %fp3,%fp1 | ...V*(A2+V*(A4+V*A6))
|
| 407 |
|
|
addal #16,%a0 | ...ADDRESS OF LOG(F)
|
| 408 |
|
|
fmulx %fp3,%fp2 | ...V*(A1+V*(A3+V*A5)), FP3 RELEASED
|
| 409 |
|
|
|
| 410 |
|
|
fmulx %fp0,%fp1 | ...U*V*(A2+V*(A4+V*A6))
|
| 411 |
|
|
faddx %fp2,%fp0 | ...U+V*(A1+V*(A3+V*A5)), FP2 RELEASED
|
| 412 |
|
|
|
| 413 |
|
|
faddx (%a0),%fp1 | ...LOG(F)+U*V*(A2+V*(A4+V*A6))
|
| 414 |
|
|
fmovemx (%sp)+,%fp2-%fp2/%fp3 | ...RESTORE FP2
|
| 415 |
|
|
faddx %fp1,%fp0 | ...FP0 IS LOG(F) + LOG(1+U)
|
| 416 |
|
|
|
| 417 |
|
|
fmovel %d1,%fpcr
|
| 418 |
|
|
faddx KLOG2(%a6),%fp0 | ...FINAL ADD
|
| 419 |
|
|
bra t_frcinx
|
| 420 |
|
|
|
| 421 |
|
|
|
| 422 |
|
|
LOGNEAR1:
|
| 423 |
|
|
|--REGISTERS SAVED: FPCR, FP1. FP0 CONTAINS THE INPUT.
|
| 424 |
|
|
fmovex %fp0,%fp1
|
| 425 |
|
|
fsubs one,%fp1 | ...FP1 IS X-1
|
| 426 |
|
|
fadds one,%fp0 | ...FP0 IS X+1
|
| 427 |
|
|
faddx %fp1,%fp1 | ...FP1 IS 2(X-1)
|
| 428 |
|
|
|--LOG(X) = LOG(1+U/2)-LOG(1-U/2) WHICH IS AN ODD POLYNOMIAL
|
| 429 |
|
|
|--IN U, U = 2(X-1)/(X+1) = FP1/FP0
|
| 430 |
|
|
|
| 431 |
|
|
LP1CONT2:
|
| 432 |
|
|
|--THIS IS AN RE-ENTRY POINT FOR LOGNP1
|
| 433 |
|
|
fdivx %fp0,%fp1 | ...FP1 IS U
|
| 434 |
|
|
fmovemx %fp2-%fp2/%fp3,-(%sp) | ...SAVE FP2
|
| 435 |
|
|
|--REGISTERS SAVED ARE NOW FPCR,FP1,FP2,FP3
|
| 436 |
|
|
|--LET V=U*U, W=V*V, CALCULATE
|
| 437 |
|
|
|--U + U*V*(B1 + V*(B2 + V*(B3 + V*(B4 + V*B5)))) BY
|
| 438 |
|
|
|--U + U*V*( [B1 + W*(B3 + W*B5)] + [V*(B2 + W*B4)] )
|
| 439 |
|
|
fmovex %fp1,%fp0
|
| 440 |
|
|
fmulx %fp0,%fp0 | ...FP0 IS V
|
| 441 |
|
|
fmovex %fp1,SAVEU(%a6) | ...STORE U IN MEMORY, FREE FP1
|
| 442 |
|
|
fmovex %fp0,%fp1
|
| 443 |
|
|
fmulx %fp1,%fp1 | ...FP1 IS W
|
| 444 |
|
|
|
| 445 |
|
|
fmoved LOGB5,%fp3
|
| 446 |
|
|
fmoved LOGB4,%fp2
|
| 447 |
|
|
|
| 448 |
|
|
fmulx %fp1,%fp3 | ...W*B5
|
| 449 |
|
|
fmulx %fp1,%fp2 | ...W*B4
|
| 450 |
|
|
|
| 451 |
|
|
faddd LOGB3,%fp3 | ...B3+W*B5
|
| 452 |
|
|
faddd LOGB2,%fp2 | ...B2+W*B4
|
| 453 |
|
|
|
| 454 |
|
|
fmulx %fp3,%fp1 | ...W*(B3+W*B5), FP3 RELEASED
|
| 455 |
|
|
|
| 456 |
|
|
fmulx %fp0,%fp2 | ...V*(B2+W*B4)
|
| 457 |
|
|
|
| 458 |
|
|
faddd LOGB1,%fp1 | ...B1+W*(B3+W*B5)
|
| 459 |
|
|
fmulx SAVEU(%a6),%fp0 | ...FP0 IS U*V
|
| 460 |
|
|
|
| 461 |
|
|
faddx %fp2,%fp1 | ...B1+W*(B3+W*B5) + V*(B2+W*B4), FP2 RELEASED
|
| 462 |
|
|
fmovemx (%sp)+,%fp2-%fp2/%fp3 | ...FP2 RESTORED
|
| 463 |
|
|
|
| 464 |
|
|
fmulx %fp1,%fp0 | ...U*V*( [B1+W*(B3+W*B5)] + [V*(B2+W*B4)] )
|
| 465 |
|
|
|
| 466 |
|
|
fmovel %d1,%fpcr
|
| 467 |
|
|
faddx SAVEU(%a6),%fp0
|
| 468 |
|
|
bra t_frcinx
|
| 469 |
|
|
rts
|
| 470 |
|
|
|
| 471 |
|
|
LOGNEG:
|
| 472 |
|
|
|--REGISTERS SAVED FPCR. LOG(-VE) IS INVALID
|
| 473 |
|
|
bra t_operr
|
| 474 |
|
|
|
| 475 |
|
|
.global slognp1d
|
| 476 |
|
|
slognp1d:
|
| 477 |
|
|
|--ENTRY POINT FOR LOG(1+Z) FOR DENORMALIZED INPUT
|
| 478 |
|
|
| Simply return the denorm
|
| 479 |
|
|
|
| 480 |
|
|
bra t_extdnrm
|
| 481 |
|
|
|
| 482 |
|
|
.global slognp1
|
| 483 |
|
|
slognp1:
|
| 484 |
|
|
|--ENTRY POINT FOR LOG(1+X) FOR X FINITE, NON-ZERO, NOT NAN'S
|
| 485 |
|
|
|
| 486 |
|
|
fmovex (%a0),%fp0 | ...LOAD INPUT
|
| 487 |
|
|
fabsx %fp0 |test magnitude
|
| 488 |
|
|
fcmpx LTHOLD,%fp0 |compare with min threshold
|
| 489 |
|
|
fbgt LP1REAL |if greater, continue
|
| 490 |
|
|
fmovel #0,%fpsr |clr N flag from compare
|
| 491 |
|
|
fmovel %d1,%fpcr
|
| 492 |
|
|
fmovex (%a0),%fp0 |return signed argument
|
| 493 |
|
|
bra t_frcinx
|
| 494 |
|
|
|
| 495 |
|
|
LP1REAL:
|
| 496 |
|
|
fmovex (%a0),%fp0 | ...LOAD INPUT
|
| 497 |
|
|
movel #0x00000000,ADJK(%a6)
|
| 498 |
|
|
fmovex %fp0,%fp1 | ...FP1 IS INPUT Z
|
| 499 |
|
|
fadds one,%fp0 | ...X := ROUND(1+Z)
|
| 500 |
|
|
fmovex %fp0,X(%a6)
|
| 501 |
|
|
movew XFRAC(%a6),XDCARE(%a6)
|
| 502 |
|
|
movel X(%a6),%d0
|
| 503 |
|
|
cmpil #0,%d0
|
| 504 |
|
|
ble LP1NEG0 | ...LOG OF ZERO OR -VE
|
| 505 |
|
|
cmp2l BOUNDS2,%d0
|
| 506 |
|
|
bcs LOGMAIN | ...BOUNDS2 IS [1/2,3/2]
|
| 507 |
|
|
|--IF 1+Z > 3/2 OR 1+Z < 1/2, THEN X, WHICH IS ROUNDING 1+Z,
|
| 508 |
|
|
|--CONTAINS AT LEAST 63 BITS OF INFORMATION OF Z. IN THAT CASE,
|
| 509 |
|
|
|--SIMPLY INVOKE LOG(X) FOR LOG(1+Z).
|
| 510 |
|
|
|
| 511 |
|
|
LP1NEAR1:
|
| 512 |
|
|
|--NEXT SEE IF EXP(-1/16) < X < EXP(1/16)
|
| 513 |
|
|
cmp2l BOUNDS1,%d0
|
| 514 |
|
|
bcss LP1CARE
|
| 515 |
|
|
|
| 516 |
|
|
LP1ONE16:
|
| 517 |
|
|
|--EXP(-1/16) < X < EXP(1/16). LOG(1+Z) = LOG(1+U/2) - LOG(1-U/2)
|
| 518 |
|
|
|--WHERE U = 2Z/(2+Z) = 2Z/(1+X).
|
| 519 |
|
|
faddx %fp1,%fp1 | ...FP1 IS 2Z
|
| 520 |
|
|
fadds one,%fp0 | ...FP0 IS 1+X
|
| 521 |
|
|
|--U = FP1/FP0
|
| 522 |
|
|
bra LP1CONT2
|
| 523 |
|
|
|
| 524 |
|
|
LP1CARE:
|
| 525 |
|
|
|--HERE WE USE THE USUAL TABLE DRIVEN APPROACH. CARE HAS TO BE
|
| 526 |
|
|
|--TAKEN BECAUSE 1+Z CAN HAVE 67 BITS OF INFORMATION AND WE MUST
|
| 527 |
|
|
|--PRESERVE ALL THE INFORMATION. BECAUSE 1+Z IS IN [1/2,3/2],
|
| 528 |
|
|
|--THERE ARE ONLY TWO CASES.
|
| 529 |
|
|
|--CASE 1: 1+Z < 1, THEN K = -1 AND Y-F = (2-F) + 2Z
|
| 530 |
|
|
|--CASE 2: 1+Z > 1, THEN K = 0 AND Y-F = (1-F) + Z
|
| 531 |
|
|
|--ON RETURNING TO LP1CONT1, WE MUST HAVE K IN FP1, ADDRESS OF
|
| 532 |
|
|
|--(1/F) IN A0, Y-F IN FP0, AND FP2 SAVED.
|
| 533 |
|
|
|
| 534 |
|
|
movel XFRAC(%a6),FFRAC(%a6)
|
| 535 |
|
|
andil #0xFE000000,FFRAC(%a6)
|
| 536 |
|
|
oril #0x01000000,FFRAC(%a6) | ...F OBTAINED
|
| 537 |
|
|
cmpil #0x3FFF8000,%d0 | ...SEE IF 1+Z > 1
|
| 538 |
|
|
bges KISZERO
|
| 539 |
|
|
|
| 540 |
|
|
KISNEG1:
|
| 541 |
|
|
fmoves TWO,%fp0
|
| 542 |
|
|
movel #0x3fff0000,F(%a6)
|
| 543 |
|
|
clrl F+8(%a6)
|
| 544 |
|
|
fsubx F(%a6),%fp0 | ...2-F
|
| 545 |
|
|
movel FFRAC(%a6),%d0
|
| 546 |
|
|
andil #0x7E000000,%d0
|
| 547 |
|
|
asrl #8,%d0
|
| 548 |
|
|
asrl #8,%d0
|
| 549 |
|
|
asrl #4,%d0 | ...D0 CONTAINS DISPLACEMENT FOR 1/F
|
| 550 |
|
|
faddx %fp1,%fp1 | ...GET 2Z
|
| 551 |
|
|
fmovemx %fp2-%fp2/%fp3,-(%sp) | ...SAVE FP2
|
| 552 |
|
|
faddx %fp1,%fp0 | ...FP0 IS Y-F = (2-F)+2Z
|
| 553 |
|
|
lea LOGTBL,%a0 | ...A0 IS ADDRESS OF 1/F
|
| 554 |
|
|
addal %d0,%a0
|
| 555 |
|
|
fmoves negone,%fp1 | ...FP1 IS K = -1
|
| 556 |
|
|
bra LP1CONT1
|
| 557 |
|
|
|
| 558 |
|
|
KISZERO:
|
| 559 |
|
|
fmoves one,%fp0
|
| 560 |
|
|
movel #0x3fff0000,F(%a6)
|
| 561 |
|
|
clrl F+8(%a6)
|
| 562 |
|
|
fsubx F(%a6),%fp0 | ...1-F
|
| 563 |
|
|
movel FFRAC(%a6),%d0
|
| 564 |
|
|
andil #0x7E000000,%d0
|
| 565 |
|
|
asrl #8,%d0
|
| 566 |
|
|
asrl #8,%d0
|
| 567 |
|
|
asrl #4,%d0
|
| 568 |
|
|
faddx %fp1,%fp0 | ...FP0 IS Y-F
|
| 569 |
|
|
fmovemx %fp2-%fp2/%fp3,-(%sp) | ...FP2 SAVED
|
| 570 |
|
|
lea LOGTBL,%a0
|
| 571 |
|
|
addal %d0,%a0 | ...A0 IS ADDRESS OF 1/F
|
| 572 |
|
|
fmoves zero,%fp1 | ...FP1 IS K = 0
|
| 573 |
|
|
bra LP1CONT1
|
| 574 |
|
|
|
| 575 |
|
|
LP1NEG0:
|
| 576 |
|
|
|--FPCR SAVED. D0 IS X IN COMPACT FORM.
|
| 577 |
|
|
cmpil #0,%d0
|
| 578 |
|
|
blts LP1NEG
|
| 579 |
|
|
LP1ZERO:
|
| 580 |
|
|
fmoves negone,%fp0
|
| 581 |
|
|
|
| 582 |
|
|
fmovel %d1,%fpcr
|
| 583 |
|
|
bra t_dz
|
| 584 |
|
|
|
| 585 |
|
|
LP1NEG:
|
| 586 |
|
|
fmoves zero,%fp0
|
| 587 |
|
|
|
| 588 |
|
|
fmovel %d1,%fpcr
|
| 589 |
|
|
bra t_operr
|
| 590 |
|
|
|
| 591 |
|
|
|end
|
