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Based on kernel version `4.2`. Page generated on `2015-09-09 12:14 EST`.

1 Introduction 2 ============ 3 4 Having looked at the linux mtd/nand driver and more specific at nand_ecc.c 5 I felt there was room for optimisation. I bashed the code for a few hours 6 performing tricks like table lookup removing superfluous code etc. 7 After that the speed was increased by 35-40%. 8 Still I was not too happy as I felt there was additional room for improvement. 9 10 Bad! I was hooked. 11 I decided to annotate my steps in this file. Perhaps it is useful to someone 12 or someone learns something from it. 13 14 15 The problem 16 =========== 17 18 NAND flash (at least SLC one) typically has sectors of 256 bytes. 19 However NAND flash is not extremely reliable so some error detection 20 (and sometimes correction) is needed. 21 22 This is done by means of a Hamming code. I'll try to explain it in 23 laymans terms (and apologies to all the pro's in the field in case I do 24 not use the right terminology, my coding theory class was almost 30 25 years ago, and I must admit it was not one of my favourites). 26 27 As I said before the ecc calculation is performed on sectors of 256 28 bytes. This is done by calculating several parity bits over the rows and 29 columns. The parity used is even parity which means that the parity bit = 1 30 if the data over which the parity is calculated is 1 and the parity bit = 0 31 if the data over which the parity is calculated is 0. So the total 32 number of bits over the data over which the parity is calculated + the 33 parity bit is even. (see wikipedia if you can't follow this). 34 Parity is often calculated by means of an exclusive or operation, 35 sometimes also referred to as xor. In C the operator for xor is ^ 36 37 Back to ecc. 38 Let's give a small figure: 39 40 byte 0: bit7 bit6 bit5 bit4 bit3 bit2 bit1 bit0 rp0 rp2 rp4 ... rp14 41 byte 1: bit7 bit6 bit5 bit4 bit3 bit2 bit1 bit0 rp1 rp2 rp4 ... rp14 42 byte 2: bit7 bit6 bit5 bit4 bit3 bit2 bit1 bit0 rp0 rp3 rp4 ... rp14 43 byte 3: bit7 bit6 bit5 bit4 bit3 bit2 bit1 bit0 rp1 rp3 rp4 ... rp14 44 byte 4: bit7 bit6 bit5 bit4 bit3 bit2 bit1 bit0 rp0 rp2 rp5 ... rp14 45 .... 46 byte 254: bit7 bit6 bit5 bit4 bit3 bit2 bit1 bit0 rp0 rp3 rp5 ... rp15 47 byte 255: bit7 bit6 bit5 bit4 bit3 bit2 bit1 bit0 rp1 rp3 rp5 ... rp15 48 cp1 cp0 cp1 cp0 cp1 cp0 cp1 cp0 49 cp3 cp3 cp2 cp2 cp3 cp3 cp2 cp2 50 cp5 cp5 cp5 cp5 cp4 cp4 cp4 cp4 51 52 This figure represents a sector of 256 bytes. 53 cp is my abbreviation for column parity, rp for row parity. 54 55 Let's start to explain column parity. 56 cp0 is the parity that belongs to all bit0, bit2, bit4, bit6. 57 so the sum of all bit0, bit2, bit4 and bit6 values + cp0 itself is even. 58 Similarly cp1 is the sum of all bit1, bit3, bit5 and bit7. 59 cp2 is the parity over bit0, bit1, bit4 and bit5 60 cp3 is the parity over bit2, bit3, bit6 and bit7. 61 cp4 is the parity over bit0, bit1, bit2 and bit3. 62 cp5 is the parity over bit4, bit5, bit6 and bit7. 63 Note that each of cp0 .. cp5 is exactly one bit. 64 65 Row parity actually works almost the same. 66 rp0 is the parity of all even bytes (0, 2, 4, 6, ... 252, 254) 67 rp1 is the parity of all odd bytes (1, 3, 5, 7, ..., 253, 255) 68 rp2 is the parity of all bytes 0, 1, 4, 5, 8, 9, ... 69 (so handle two bytes, then skip 2 bytes). 70 rp3 is covers the half rp2 does not cover (bytes 2, 3, 6, 7, 10, 11, ...) 71 for rp4 the rule is cover 4 bytes, skip 4 bytes, cover 4 bytes, skip 4 etc. 72 so rp4 calculates parity over bytes 0, 1, 2, 3, 8, 9, 10, 11, 16, ...) 73 and rp5 covers the other half, so bytes 4, 5, 6, 7, 12, 13, 14, 15, 20, .. 74 The story now becomes quite boring. I guess you get the idea. 75 rp6 covers 8 bytes then skips 8 etc 76 rp7 skips 8 bytes then covers 8 etc 77 rp8 covers 16 bytes then skips 16 etc 78 rp9 skips 16 bytes then covers 16 etc 79 rp10 covers 32 bytes then skips 32 etc 80 rp11 skips 32 bytes then covers 32 etc 81 rp12 covers 64 bytes then skips 64 etc 82 rp13 skips 64 bytes then covers 64 etc 83 rp14 covers 128 bytes then skips 128 84 rp15 skips 128 bytes then covers 128 85 86 In the end the parity bits are grouped together in three bytes as 87 follows: 88 ECC Bit 7 Bit 6 Bit 5 Bit 4 Bit 3 Bit 2 Bit 1 Bit 0 89 ECC 0 rp07 rp06 rp05 rp04 rp03 rp02 rp01 rp00 90 ECC 1 rp15 rp14 rp13 rp12 rp11 rp10 rp09 rp08 91 ECC 2 cp5 cp4 cp3 cp2 cp1 cp0 1 1 92 93 I detected after writing this that ST application note AN1823 94 (http://www.st.com/stonline/) gives a much 95 nicer picture.(but they use line parity as term where I use row parity) 96 Oh well, I'm graphically challenged, so suffer with me for a moment :-) 97 And I could not reuse the ST picture anyway for copyright reasons. 98 99 100 Attempt 0 101 ========= 102 103 Implementing the parity calculation is pretty simple. 104 In C pseudocode: 105 for (i = 0; i < 256; i++) 106 { 107 if (i & 0x01) 108 rp1 = bit7 ^ bit6 ^ bit5 ^ bit4 ^ bit3 ^ bit2 ^ bit1 ^ bit0 ^ rp1; 109 else 110 rp0 = bit7 ^ bit6 ^ bit5 ^ bit4 ^ bit3 ^ bit2 ^ bit1 ^ bit0 ^ rp1; 111 if (i & 0x02) 112 rp3 = bit7 ^ bit6 ^ bit5 ^ bit4 ^ bit3 ^ bit2 ^ bit1 ^ bit0 ^ rp3; 113 else 114 rp2 = bit7 ^ bit6 ^ bit5 ^ bit4 ^ bit3 ^ bit2 ^ bit1 ^ bit0 ^ rp2; 115 if (i & 0x04) 116 rp5 = bit7 ^ bit6 ^ bit5 ^ bit4 ^ bit3 ^ bit2 ^ bit1 ^ bit0 ^ rp5; 117 else 118 rp4 = bit7 ^ bit6 ^ bit5 ^ bit4 ^ bit3 ^ bit2 ^ bit1 ^ bit0 ^ rp4; 119 if (i & 0x08) 120 rp7 = bit7 ^ bit6 ^ bit5 ^ bit4 ^ bit3 ^ bit2 ^ bit1 ^ bit0 ^ rp7; 121 else 122 rp6 = bit7 ^ bit6 ^ bit5 ^ bit4 ^ bit3 ^ bit2 ^ bit1 ^ bit0 ^ rp6; 123 if (i & 0x10) 124 rp9 = bit7 ^ bit6 ^ bit5 ^ bit4 ^ bit3 ^ bit2 ^ bit1 ^ bit0 ^ rp9; 125 else 126 rp8 = bit7 ^ bit6 ^ bit5 ^ bit4 ^ bit3 ^ bit2 ^ bit1 ^ bit0 ^ rp8; 127 if (i & 0x20) 128 rp11 = bit7 ^ bit6 ^ bit5 ^ bit4 ^ bit3 ^ bit2 ^ bit1 ^ bit0 ^ rp11; 129 else 130 rp10 = bit7 ^ bit6 ^ bit5 ^ bit4 ^ bit3 ^ bit2 ^ bit1 ^ bit0 ^ rp10; 131 if (i & 0x40) 132 rp13 = bit7 ^ bit6 ^ bit5 ^ bit4 ^ bit3 ^ bit2 ^ bit1 ^ bit0 ^ rp13; 133 else 134 rp12 = bit7 ^ bit6 ^ bit5 ^ bit4 ^ bit3 ^ bit2 ^ bit1 ^ bit0 ^ rp12; 135 if (i & 0x80) 136 rp15 = bit7 ^ bit6 ^ bit5 ^ bit4 ^ bit3 ^ bit2 ^ bit1 ^ bit0 ^ rp15; 137 else 138 rp14 = bit7 ^ bit6 ^ bit5 ^ bit4 ^ bit3 ^ bit2 ^ bit1 ^ bit0 ^ rp14; 139 cp0 = bit6 ^ bit4 ^ bit2 ^ bit0 ^ cp0; 140 cp1 = bit7 ^ bit5 ^ bit3 ^ bit1 ^ cp1; 141 cp2 = bit5 ^ bit4 ^ bit1 ^ bit0 ^ cp2; 142 cp3 = bit7 ^ bit6 ^ bit3 ^ bit2 ^ cp3 143 cp4 = bit3 ^ bit2 ^ bit1 ^ bit0 ^ cp4 144 cp5 = bit7 ^ bit6 ^ bit5 ^ bit4 ^ cp5 145 } 146 147 148 Analysis 0 149 ========== 150 151 C does have bitwise operators but not really operators to do the above 152 efficiently (and most hardware has no such instructions either). 153 Therefore without implementing this it was clear that the code above was 154 not going to bring me a Nobel prize :-) 155 156 Fortunately the exclusive or operation is commutative, so we can combine 157 the values in any order. So instead of calculating all the bits 158 individually, let us try to rearrange things. 159 For the column parity this is easy. We can just xor the bytes and in the 160 end filter out the relevant bits. This is pretty nice as it will bring 161 all cp calculation out of the if loop. 162 163 Similarly we can first xor the bytes for the various rows. 164 This leads to: 165 166 167 Attempt 1 168 ========= 169 170 const char parity[256] = { 171 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 172 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 173 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 174 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 175 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 176 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 177 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 178 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 179 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 180 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 181 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 182 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 183 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 184 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 185 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 186 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0 187 }; 188 189 void ecc1(const unsigned char *buf, unsigned char *code) 190 { 191 int i; 192 const unsigned char *bp = buf; 193 unsigned char cur; 194 unsigned char rp0, rp1, rp2, rp3, rp4, rp5, rp6, rp7; 195 unsigned char rp8, rp9, rp10, rp11, rp12, rp13, rp14, rp15; 196 unsigned char par; 197 198 par = 0; 199 rp0 = 0; rp1 = 0; rp2 = 0; rp3 = 0; 200 rp4 = 0; rp5 = 0; rp6 = 0; rp7 = 0; 201 rp8 = 0; rp9 = 0; rp10 = 0; rp11 = 0; 202 rp12 = 0; rp13 = 0; rp14 = 0; rp15 = 0; 203 204 for (i = 0; i < 256; i++) 205 { 206 cur = *bp++; 207 par ^= cur; 208 if (i & 0x01) rp1 ^= cur; else rp0 ^= cur; 209 if (i & 0x02) rp3 ^= cur; else rp2 ^= cur; 210 if (i & 0x04) rp5 ^= cur; else rp4 ^= cur; 211 if (i & 0x08) rp7 ^= cur; else rp6 ^= cur; 212 if (i & 0x10) rp9 ^= cur; else rp8 ^= cur; 213 if (i & 0x20) rp11 ^= cur; else rp10 ^= cur; 214 if (i & 0x40) rp13 ^= cur; else rp12 ^= cur; 215 if (i & 0x80) rp15 ^= cur; else rp14 ^= cur; 216 } 217 code[0] = 218 (parity[rp7] << 7) | 219 (parity[rp6] << 6) | 220 (parity[rp5] << 5) | 221 (parity[rp4] << 4) | 222 (parity[rp3] << 3) | 223 (parity[rp2] << 2) | 224 (parity[rp1] << 1) | 225 (parity[rp0]); 226 code[1] = 227 (parity[rp15] << 7) | 228 (parity[rp14] << 6) | 229 (parity[rp13] << 5) | 230 (parity[rp12] << 4) | 231 (parity[rp11] << 3) | 232 (parity[rp10] << 2) | 233 (parity[rp9] << 1) | 234 (parity[rp8]); 235 code[2] = 236 (parity[par & 0xf0] << 7) | 237 (parity[par & 0x0f] << 6) | 238 (parity[par & 0xcc] << 5) | 239 (parity[par & 0x33] << 4) | 240 (parity[par & 0xaa] << 3) | 241 (parity[par & 0x55] << 2); 242 code[0] = ~code[0]; 243 code[1] = ~code[1]; 244 code[2] = ~code[2]; 245 } 246 247 Still pretty straightforward. The last three invert statements are there to 248 give a checksum of 0xff 0xff 0xff for an empty flash. In an empty flash 249 all data is 0xff, so the checksum then matches. 250 251 I also introduced the parity lookup. I expected this to be the fastest 252 way to calculate the parity, but I will investigate alternatives later 253 on. 254 255 256 Analysis 1 257 ========== 258 259 The code works, but is not terribly efficient. On my system it took 260 almost 4 times as much time as the linux driver code. But hey, if it was 261 *that* easy this would have been done long before. 262 No pain. no gain. 263 264 Fortunately there is plenty of room for improvement. 265 266 In step 1 we moved from bit-wise calculation to byte-wise calculation. 267 However in C we can also use the unsigned long data type and virtually 268 every modern microprocessor supports 32 bit operations, so why not try 269 to write our code in such a way that we process data in 32 bit chunks. 270 271 Of course this means some modification as the row parity is byte by 272 byte. A quick analysis: 273 for the column parity we use the par variable. When extending to 32 bits 274 we can in the end easily calculate p0 and p1 from it. 275 (because par now consists of 4 bytes, contributing to rp1, rp0, rp1, rp0 276 respectively) 277 also rp2 and rp3 can be easily retrieved from par as rp3 covers the 278 first two bytes and rp2 the last two bytes. 279 280 Note that of course now the loop is executed only 64 times (256/4). 281 And note that care must taken wrt byte ordering. The way bytes are 282 ordered in a long is machine dependent, and might affect us. 283 Anyway, if there is an issue: this code is developed on x86 (to be 284 precise: a DELL PC with a D920 Intel CPU) 285 286 And of course the performance might depend on alignment, but I expect 287 that the I/O buffers in the nand driver are aligned properly (and 288 otherwise that should be fixed to get maximum performance). 289 290 Let's give it a try... 291 292 293 Attempt 2 294 ========= 295 296 extern const char parity[256]; 297 298 void ecc2(const unsigned char *buf, unsigned char *code) 299 { 300 int i; 301 const unsigned long *bp = (unsigned long *)buf; 302 unsigned long cur; 303 unsigned long rp0, rp1, rp2, rp3, rp4, rp5, rp6, rp7; 304 unsigned long rp8, rp9, rp10, rp11, rp12, rp13, rp14, rp15; 305 unsigned long par; 306 307 par = 0; 308 rp0 = 0; rp1 = 0; rp2 = 0; rp3 = 0; 309 rp4 = 0; rp5 = 0; rp6 = 0; rp7 = 0; 310 rp8 = 0; rp9 = 0; rp10 = 0; rp11 = 0; 311 rp12 = 0; rp13 = 0; rp14 = 0; rp15 = 0; 312 313 for (i = 0; i < 64; i++) 314 { 315 cur = *bp++; 316 par ^= cur; 317 if (i & 0x01) rp5 ^= cur; else rp4 ^= cur; 318 if (i & 0x02) rp7 ^= cur; else rp6 ^= cur; 319 if (i & 0x04) rp9 ^= cur; else rp8 ^= cur; 320 if (i & 0x08) rp11 ^= cur; else rp10 ^= cur; 321 if (i & 0x10) rp13 ^= cur; else rp12 ^= cur; 322 if (i & 0x20) rp15 ^= cur; else rp14 ^= cur; 323 } 324 /* 325 we need to adapt the code generation for the fact that rp vars are now 326 long; also the column parity calculation needs to be changed. 327 we'll bring rp4 to 15 back to single byte entities by shifting and 328 xoring 329 */ 330 rp4 ^= (rp4 >> 16); rp4 ^= (rp4 >> 8); rp4 &= 0xff; 331 rp5 ^= (rp5 >> 16); rp5 ^= (rp5 >> 8); rp5 &= 0xff; 332 rp6 ^= (rp6 >> 16); rp6 ^= (rp6 >> 8); rp6 &= 0xff; 333 rp7 ^= (rp7 >> 16); rp7 ^= (rp7 >> 8); rp7 &= 0xff; 334 rp8 ^= (rp8 >> 16); rp8 ^= (rp8 >> 8); rp8 &= 0xff; 335 rp9 ^= (rp9 >> 16); rp9 ^= (rp9 >> 8); rp9 &= 0xff; 336 rp10 ^= (rp10 >> 16); rp10 ^= (rp10 >> 8); rp10 &= 0xff; 337 rp11 ^= (rp11 >> 16); rp11 ^= (rp11 >> 8); rp11 &= 0xff; 338 rp12 ^= (rp12 >> 16); rp12 ^= (rp12 >> 8); rp12 &= 0xff; 339 rp13 ^= (rp13 >> 16); rp13 ^= (rp13 >> 8); rp13 &= 0xff; 340 rp14 ^= (rp14 >> 16); rp14 ^= (rp14 >> 8); rp14 &= 0xff; 341 rp15 ^= (rp15 >> 16); rp15 ^= (rp15 >> 8); rp15 &= 0xff; 342 rp3 = (par >> 16); rp3 ^= (rp3 >> 8); rp3 &= 0xff; 343 rp2 = par & 0xffff; rp2 ^= (rp2 >> 8); rp2 &= 0xff; 344 par ^= (par >> 16); 345 rp1 = (par >> 8); rp1 &= 0xff; 346 rp0 = (par & 0xff); 347 par ^= (par >> 8); par &= 0xff; 348 349 code[0] = 350 (parity[rp7] << 7) | 351 (parity[rp6] << 6) | 352 (parity[rp5] << 5) | 353 (parity[rp4] << 4) | 354 (parity[rp3] << 3) | 355 (parity[rp2] << 2) | 356 (parity[rp1] << 1) | 357 (parity[rp0]); 358 code[1] = 359 (parity[rp15] << 7) | 360 (parity[rp14] << 6) | 361 (parity[rp13] << 5) | 362 (parity[rp12] << 4) | 363 (parity[rp11] << 3) | 364 (parity[rp10] << 2) | 365 (parity[rp9] << 1) | 366 (parity[rp8]); 367 code[2] = 368 (parity[par & 0xf0] << 7) | 369 (parity[par & 0x0f] << 6) | 370 (parity[par & 0xcc] << 5) | 371 (parity[par & 0x33] << 4) | 372 (parity[par & 0xaa] << 3) | 373 (parity[par & 0x55] << 2); 374 code[0] = ~code[0]; 375 code[1] = ~code[1]; 376 code[2] = ~code[2]; 377 } 378 379 The parity array is not shown any more. Note also that for these 380 examples I kinda deviated from my regular programming style by allowing 381 multiple statements on a line, not using { } in then and else blocks 382 with only a single statement and by using operators like ^= 383 384 385 Analysis 2 386 ========== 387 388 The code (of course) works, and hurray: we are a little bit faster than 389 the linux driver code (about 15%). But wait, don't cheer too quickly. 390 THere is more to be gained. 391 If we look at e.g. rp14 and rp15 we see that we either xor our data with 392 rp14 or with rp15. However we also have par which goes over all data. 393 This means there is no need to calculate rp14 as it can be calculated from 394 rp15 through rp14 = par ^ rp15; 395 (or if desired we can avoid calculating rp15 and calculate it from 396 rp14). That is why some places refer to inverse parity. 397 Of course the same thing holds for rp4/5, rp6/7, rp8/9, rp10/11 and rp12/13. 398 Effectively this means we can eliminate the else clause from the if 399 statements. Also we can optimise the calculation in the end a little bit 400 by going from long to byte first. Actually we can even avoid the table 401 lookups 402 403 Attempt 3 404 ========= 405 406 Odd replaced: 407 if (i & 0x01) rp5 ^= cur; else rp4 ^= cur; 408 if (i & 0x02) rp7 ^= cur; else rp6 ^= cur; 409 if (i & 0x04) rp9 ^= cur; else rp8 ^= cur; 410 if (i & 0x08) rp11 ^= cur; else rp10 ^= cur; 411 if (i & 0x10) rp13 ^= cur; else rp12 ^= cur; 412 if (i & 0x20) rp15 ^= cur; else rp14 ^= cur; 413 with 414 if (i & 0x01) rp5 ^= cur; 415 if (i & 0x02) rp7 ^= cur; 416 if (i & 0x04) rp9 ^= cur; 417 if (i & 0x08) rp11 ^= cur; 418 if (i & 0x10) rp13 ^= cur; 419 if (i & 0x20) rp15 ^= cur; 420 421 and outside the loop added: 422 rp4 = par ^ rp5; 423 rp6 = par ^ rp7; 424 rp8 = par ^ rp9; 425 rp10 = par ^ rp11; 426 rp12 = par ^ rp13; 427 rp14 = par ^ rp15; 428 429 And after that the code takes about 30% more time, although the number of 430 statements is reduced. This is also reflected in the assembly code. 431 432 433 Analysis 3 434 ========== 435 436 Very weird. Guess it has to do with caching or instruction parallellism 437 or so. I also tried on an eeePC (Celeron, clocked at 900 Mhz). Interesting 438 observation was that this one is only 30% slower (according to time) 439 executing the code as my 3Ghz D920 processor. 440 441 Well, it was expected not to be easy so maybe instead move to a 442 different track: let's move back to the code from attempt2 and do some 443 loop unrolling. This will eliminate a few if statements. I'll try 444 different amounts of unrolling to see what works best. 445 446 447 Attempt 4 448 ========= 449 450 Unrolled the loop 1, 2, 3 and 4 times. 451 For 4 the code starts with: 452 453 for (i = 0; i < 4; i++) 454 { 455 cur = *bp++; 456 par ^= cur; 457 rp4 ^= cur; 458 rp6 ^= cur; 459 rp8 ^= cur; 460 rp10 ^= cur; 461 if (i & 0x1) rp13 ^= cur; else rp12 ^= cur; 462 if (i & 0x2) rp15 ^= cur; else rp14 ^= cur; 463 cur = *bp++; 464 par ^= cur; 465 rp5 ^= cur; 466 rp6 ^= cur; 467 ... 468 469 470 Analysis 4 471 ========== 472 473 Unrolling once gains about 15% 474 Unrolling twice keeps the gain at about 15% 475 Unrolling three times gives a gain of 30% compared to attempt 2. 476 Unrolling four times gives a marginal improvement compared to unrolling 477 three times. 478 479 I decided to proceed with a four time unrolled loop anyway. It was my gut 480 feeling that in the next steps I would obtain additional gain from it. 481 482 The next step was triggered by the fact that par contains the xor of all 483 bytes and rp4 and rp5 each contain the xor of half of the bytes. 484 So in effect par = rp4 ^ rp5. But as xor is commutative we can also say 485 that rp5 = par ^ rp4. So no need to keep both rp4 and rp5 around. We can 486 eliminate rp5 (or rp4, but I already foresaw another optimisation). 487 The same holds for rp6/7, rp8/9, rp10/11 rp12/13 and rp14/15. 488 489 490 Attempt 5 491 ========= 492 493 Effectively so all odd digit rp assignments in the loop were removed. 494 This included the else clause of the if statements. 495 Of course after the loop we need to correct things by adding code like: 496 rp5 = par ^ rp4; 497 Also the initial assignments (rp5 = 0; etc) could be removed. 498 Along the line I also removed the initialisation of rp0/1/2/3. 499 500 501 Analysis 5 502 ========== 503 504 Measurements showed this was a good move. The run-time roughly halved 505 compared with attempt 4 with 4 times unrolled, and we only require 1/3rd 506 of the processor time compared to the current code in the linux kernel. 507 508 However, still I thought there was more. I didn't like all the if 509 statements. Why not keep a running parity and only keep the last if 510 statement. Time for yet another version! 511 512 513 Attempt 6 514 ========= 515 516 THe code within the for loop was changed to: 517 518 for (i = 0; i < 4; i++) 519 { 520 cur = *bp++; tmppar = cur; rp4 ^= cur; 521 cur = *bp++; tmppar ^= cur; rp6 ^= tmppar; 522 cur = *bp++; tmppar ^= cur; rp4 ^= cur; 523 cur = *bp++; tmppar ^= cur; rp8 ^= tmppar; 524 525 cur = *bp++; tmppar ^= cur; rp4 ^= cur; rp6 ^= cur; 526 cur = *bp++; tmppar ^= cur; rp6 ^= cur; 527 cur = *bp++; tmppar ^= cur; rp4 ^= cur; 528 cur = *bp++; tmppar ^= cur; rp10 ^= tmppar; 529 530 cur = *bp++; tmppar ^= cur; rp4 ^= cur; rp6 ^= cur; rp8 ^= cur; 531 cur = *bp++; tmppar ^= cur; rp6 ^= cur; rp8 ^= cur; 532 cur = *bp++; tmppar ^= cur; rp4 ^= cur; rp8 ^= cur; 533 cur = *bp++; tmppar ^= cur; rp8 ^= cur; 534 535 cur = *bp++; tmppar ^= cur; rp4 ^= cur; rp6 ^= cur; 536 cur = *bp++; tmppar ^= cur; rp6 ^= cur; 537 cur = *bp++; tmppar ^= cur; rp4 ^= cur; 538 cur = *bp++; tmppar ^= cur; 539 540 par ^= tmppar; 541 if ((i & 0x1) == 0) rp12 ^= tmppar; 542 if ((i & 0x2) == 0) rp14 ^= tmppar; 543 } 544 545 As you can see tmppar is used to accumulate the parity within a for 546 iteration. In the last 3 statements is added to par and, if needed, 547 to rp12 and rp14. 548 549 While making the changes I also found that I could exploit that tmppar 550 contains the running parity for this iteration. So instead of having: 551 rp4 ^= cur; rp6 = cur; 552 I removed the rp6 = cur; statement and did rp6 ^= tmppar; on next 553 statement. A similar change was done for rp8 and rp10 554 555 556 Analysis 6 557 ========== 558 559 Measuring this code again showed big gain. When executing the original 560 linux code 1 million times, this took about 1 second on my system. 561 (using time to measure the performance). After this iteration I was back 562 to 0.075 sec. Actually I had to decide to start measuring over 10 563 million iterations in order not to lose too much accuracy. This one 564 definitely seemed to be the jackpot! 565 566 There is a little bit more room for improvement though. There are three 567 places with statements: 568 rp4 ^= cur; rp6 ^= cur; 569 It seems more efficient to also maintain a variable rp4_6 in the while 570 loop; This eliminates 3 statements per loop. Of course after the loop we 571 need to correct by adding: 572 rp4 ^= rp4_6; 573 rp6 ^= rp4_6 574 Furthermore there are 4 sequential assignments to rp8. This can be 575 encoded slightly more efficiently by saving tmppar before those 4 lines 576 and later do rp8 = rp8 ^ tmppar ^ notrp8; 577 (where notrp8 is the value of rp8 before those 4 lines). 578 Again a use of the commutative property of xor. 579 Time for a new test! 580 581 582 Attempt 7 583 ========= 584 585 The new code now looks like: 586 587 for (i = 0; i < 4; i++) 588 { 589 cur = *bp++; tmppar = cur; rp4 ^= cur; 590 cur = *bp++; tmppar ^= cur; rp6 ^= tmppar; 591 cur = *bp++; tmppar ^= cur; rp4 ^= cur; 592 cur = *bp++; tmppar ^= cur; rp8 ^= tmppar; 593 594 cur = *bp++; tmppar ^= cur; rp4_6 ^= cur; 595 cur = *bp++; tmppar ^= cur; rp6 ^= cur; 596 cur = *bp++; tmppar ^= cur; rp4 ^= cur; 597 cur = *bp++; tmppar ^= cur; rp10 ^= tmppar; 598 599 notrp8 = tmppar; 600 cur = *bp++; tmppar ^= cur; rp4_6 ^= cur; 601 cur = *bp++; tmppar ^= cur; rp6 ^= cur; 602 cur = *bp++; tmppar ^= cur; rp4 ^= cur; 603 cur = *bp++; tmppar ^= cur; 604 rp8 = rp8 ^ tmppar ^ notrp8; 605 606 cur = *bp++; tmppar ^= cur; rp4_6 ^= cur; 607 cur = *bp++; tmppar ^= cur; rp6 ^= cur; 608 cur = *bp++; tmppar ^= cur; rp4 ^= cur; 609 cur = *bp++; tmppar ^= cur; 610 611 par ^= tmppar; 612 if ((i & 0x1) == 0) rp12 ^= tmppar; 613 if ((i & 0x2) == 0) rp14 ^= tmppar; 614 } 615 rp4 ^= rp4_6; 616 rp6 ^= rp4_6; 617 618 619 Not a big change, but every penny counts :-) 620 621 622 Analysis 7 623 ========== 624 625 Actually this made things worse. Not very much, but I don't want to move 626 into the wrong direction. Maybe something to investigate later. Could 627 have to do with caching again. 628 629 Guess that is what there is to win within the loop. Maybe unrolling one 630 more time will help. I'll keep the optimisations from 7 for now. 631 632 633 Attempt 8 634 ========= 635 636 Unrolled the loop one more time. 637 638 639 Analysis 8 640 ========== 641 642 This makes things worse. Let's stick with attempt 6 and continue from there. 643 Although it seems that the code within the loop cannot be optimised 644 further there is still room to optimize the generation of the ecc codes. 645 We can simply calculate the total parity. If this is 0 then rp4 = rp5 646 etc. If the parity is 1, then rp4 = !rp5; 647 But if rp4 = rp5 we do not need rp5 etc. We can just write the even bits 648 in the result byte and then do something like 649 code[0] |= (code[0] << 1); 650 Lets test this. 651 652 653 Attempt 9 654 ========= 655 656 Changed the code but again this slightly degrades performance. Tried all 657 kind of other things, like having dedicated parity arrays to avoid the 658 shift after parity[rp7] << 7; No gain. 659 Change the lookup using the parity array by using shift operators (e.g. 660 replace parity[rp7] << 7 with: 661 rp7 ^= (rp7 << 4); 662 rp7 ^= (rp7 << 2); 663 rp7 ^= (rp7 << 1); 664 rp7 &= 0x80; 665 No gain. 666 667 The only marginal change was inverting the parity bits, so we can remove 668 the last three invert statements. 669 670 Ah well, pity this does not deliver more. Then again 10 million 671 iterations using the linux driver code takes between 13 and 13.5 672 seconds, whereas my code now takes about 0.73 seconds for those 10 673 million iterations. So basically I've improved the performance by a 674 factor 18 on my system. Not that bad. Of course on different hardware 675 you will get different results. No warranties! 676 677 But of course there is no such thing as a free lunch. The codesize almost 678 tripled (from 562 bytes to 1434 bytes). Then again, it is not that much. 679 680 681 Correcting errors 682 ================= 683 684 For correcting errors I again used the ST application note as a starter, 685 but I also peeked at the existing code. 686 The algorithm itself is pretty straightforward. Just xor the given and 687 the calculated ecc. If all bytes are 0 there is no problem. If 11 bits 688 are 1 we have one correctable bit error. If there is 1 bit 1, we have an 689 error in the given ecc code. 690 It proved to be fastest to do some table lookups. Performance gain 691 introduced by this is about a factor 2 on my system when a repair had to 692 be done, and 1% or so if no repair had to be done. 693 Code size increased from 330 bytes to 686 bytes for this function. 694 (gcc 4.2, -O3) 695 696 697 Conclusion 698 ========== 699 700 The gain when calculating the ecc is tremendous. Om my development hardware 701 a speedup of a factor of 18 for ecc calculation was achieved. On a test on an 702 embedded system with a MIPS core a factor 7 was obtained. 703 On a test with a Linksys NSLU2 (ARMv5TE processor) the speedup was a factor 704 5 (big endian mode, gcc 4.1.2, -O3) 705 For correction not much gain could be obtained (as bitflips are rare). Then 706 again there are also much less cycles spent there. 707 708 It seems there is not much more gain possible in this, at least when 709 programmed in C. Of course it might be possible to squeeze something more 710 out of it with an assembler program, but due to pipeline behaviour etc 711 this is very tricky (at least for intel hw). 712 713 Author: Frans Meulenbroeks 714 Copyright (C) 2008 Koninklijke Philips Electronics NV.

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