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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 | | | stwotox.sa 3.1 12/10/90 | | stwotox --- 2**X | stwotoxd --- 2**X for denormalized X | stentox --- 10**X | stentoxd --- 10**X for denormalized X | | Input: Double-extended number X in location pointed to | by address register a0. | | Output: The function values are returned in Fp0. | | Accuracy and Monotonicity: The returned result is within 2 ulps in | 64 significant bit, i.e. within 0.5001 ulp to 53 bits if the | result is subsequently rounded to double precision. The | result is provably monotonic in double precision. | | Speed: The program stwotox takes approximately 190 cycles and the | program stentox takes approximately 200 cycles. | | Algorithm: | | twotox | 1. If |X| > 16480, go to ExpBig. | | 2. If |X| < 2**(-70), go to ExpSm. | | 3. Decompose X as X = N/64 + r where |r| <= 1/128. Furthermore | decompose N as | N = 64(M + M') + j, j = 0,1,2,...,63. | | 4. Overwrite r := r * log2. Then | 2**X = 2**(M') * 2**(M) * 2**(j/64) * exp(r). | Go to expr to compute that expression. | | tentox | 1. If |X| > 16480*log_10(2) (base 10 log of 2), go to ExpBig. | | 2. If |X| < 2**(-70), go to ExpSm. | | 3. Set y := X*log_2(10)*64 (base 2 log of 10). Set | N := round-to-int(y). Decompose N as | N = 64(M + M') + j, j = 0,1,2,...,63. | | 4. Define r as | r := ((X - N*L1)-N*L2) * L10 | where L1, L2 are the leading and trailing parts of log_10(2)/64 | and L10 is the natural log of 10. Then | 10**X = 2**(M') * 2**(M) * 2**(j/64) * exp(r). | Go to expr to compute that expression. | | expr | 1. Fetch 2**(j/64) from table as Fact1 and Fact2. | | 2. Overwrite Fact1 and Fact2 by | Fact1 := 2**(M) * Fact1 | Fact2 := 2**(M) * Fact2 | Thus Fact1 + Fact2 = 2**(M) * 2**(j/64). | | 3. Calculate P where 1 + P approximates exp(r): | P = r + r*r*(A1+r*(A2+...+r*A5)). | | 4. Let AdjFact := 2**(M'). Return | AdjFact * ( Fact1 + ((Fact1*P) + Fact2) ). | Exit. | | ExpBig | 1. Generate overflow by Huge * Huge if X > 0; otherwise, generate | underflow by Tiny * Tiny. | | ExpSm | 1. Return 1 + X. | | Copyright (C) Motorola, Inc. 1990 | All Rights Reserved | | THIS IS UNPUBLISHED PROPRIETARY SOURCE CODE OF MOTOROLA | The copyright notice above does not evidence any | actual or intended publication of such source code. |STWOTOX idnt 2,1 | Motorola 040 Floating Point Software Package |section 8 .include "fpsp.h" BOUNDS1: .long 0x3FB98000,0x400D80C0 | ... 2^(-70),16480 BOUNDS2: .long 0x3FB98000,0x400B9B07 | ... 2^(-70),16480 LOG2/LOG10 L2TEN64: .long 0x406A934F,0x0979A371 | ... 64LOG10/LOG2 L10TWO1: .long 0x3F734413,0x509F8000 | ... LOG2/64LOG10 L10TWO2: .long 0xBFCD0000,0xC0219DC1,0xDA994FD2,0x00000000 LOG10: .long 0x40000000,0x935D8DDD,0xAAA8AC17,0x00000000 LOG2: .long 0x3FFE0000,0xB17217F7,0xD1CF79AC,0x00000000 EXPA5: .long 0x3F56C16D,0x6F7BD0B2 EXPA4: .long 0x3F811112,0x302C712C EXPA3: .long 0x3FA55555,0x55554CC1 EXPA2: .long 0x3FC55555,0x55554A54 EXPA1: .long 0x3FE00000,0x00000000,0x00000000,0x00000000 HUGE: .long 0x7FFE0000,0xFFFFFFFF,0xFFFFFFFF,0x00000000 TINY: .long 0x00010000,0xFFFFFFFF,0xFFFFFFFF,0x00000000 EXPTBL: .long 0x3FFF0000,0x80000000,0x00000000,0x3F738000 .long 0x3FFF0000,0x8164D1F3,0xBC030773,0x3FBEF7CA .long 0x3FFF0000,0x82CD8698,0xAC2BA1D7,0x3FBDF8A9 .long 0x3FFF0000,0x843A28C3,0xACDE4046,0x3FBCD7C9 .long 0x3FFF0000,0x85AAC367,0xCC487B15,0xBFBDE8DA .long 0x3FFF0000,0x871F6196,0x9E8D1010,0x3FBDE85C .long 0x3FFF0000,0x88980E80,0x92DA8527,0x3FBEBBF1 .long 0x3FFF0000,0x8A14D575,0x496EFD9A,0x3FBB80CA .long 0x3FFF0000,0x8B95C1E3,0xEA8BD6E7,0xBFBA8373 .long 0x3FFF0000,0x8D1ADF5B,0x7E5BA9E6,0xBFBE9670 .long 0x3FFF0000,0x8EA4398B,0x45CD53C0,0x3FBDB700 .long 0x3FFF0000,0x9031DC43,0x1466B1DC,0x3FBEEEB0 .long 0x3FFF0000,0x91C3D373,0xAB11C336,0x3FBBFD6D .long 0x3FFF0000,0x935A2B2F,0x13E6E92C,0xBFBDB319 .long 0x3FFF0000,0x94F4EFA8,0xFEF70961,0x3FBDBA2B .long 0x3FFF0000,0x96942D37,0x20185A00,0x3FBE91D5 .long 0x3FFF0000,0x9837F051,0x8DB8A96F,0x3FBE8D5A .long 0x3FFF0000,0x99E04593,0x20B7FA65,0xBFBCDE7B .long 0x3FFF0000,0x9B8D39B9,0xD54E5539,0xBFBEBAAF .long 0x3FFF0000,0x9D3ED9A7,0x2CFFB751,0xBFBD86DA .long 0x3FFF0000,0x9EF53260,0x91A111AE,0xBFBEBEDD .long 0x3FFF0000,0xA0B0510F,0xB9714FC2,0x3FBCC96E .long 0x3FFF0000,0xA2704303,0x0C496819,0xBFBEC90B .long 0x3FFF0000,0xA43515AE,0x09E6809E,0x3FBBD1DB .long 0x3FFF0000,0xA5FED6A9,0xB15138EA,0x3FBCE5EB .long 0x3FFF0000,0xA7CD93B4,0xE965356A,0xBFBEC274 .long 0x3FFF0000,0xA9A15AB4,0xEA7C0EF8,0x3FBEA83C .long 0x3FFF0000,0xAB7A39B5,0xA93ED337,0x3FBECB00 .long 0x3FFF0000,0xAD583EEA,0x42A14AC6,0x3FBE9301 .long 0x3FFF0000,0xAF3B78AD,0x690A4375,0xBFBD8367 .long 0x3FFF0000,0xB123F581,0xD2AC2590,0xBFBEF05F .long 0x3FFF0000,0xB311C412,0xA9112489,0x3FBDFB3C .long 0x3FFF0000,0xB504F333,0xF9DE6484,0x3FBEB2FB .long 0x3FFF0000,0xB6FD91E3,0x28D17791,0x3FBAE2CB .long 0x3FFF0000,0xB8FBAF47,0x62FB9EE9,0x3FBCDC3C .long 0x3FFF0000,0xBAFF5AB2,0x133E45FB,0x3FBEE9AA .long 0x3FFF0000,0xBD08A39F,0x580C36BF,0xBFBEAEFD .long 0x3FFF0000,0xBF1799B6,0x7A731083,0xBFBCBF51 .long 0x3FFF0000,0xC12C4CCA,0x66709456,0x3FBEF88A .long 0x3FFF0000,0xC346CCDA,0x24976407,0x3FBD83B2 .long 0x3FFF0000,0xC5672A11,0x5506DADD,0x3FBDF8AB .long 0x3FFF0000,0xC78D74C8,0xABB9B15D,0xBFBDFB17 .long 0x3FFF0000,0xC9B9BD86,0x6E2F27A3,0xBFBEFE3C .long 0x3FFF0000,0xCBEC14FE,0xF2727C5D,0xBFBBB6F8 .long 0x3FFF0000,0xCE248C15,0x1F8480E4,0xBFBCEE53 .long 0x3FFF0000,0xD06333DA,0xEF2B2595,0xBFBDA4AE .long 0x3FFF0000,0xD2A81D91,0xF12AE45A,0x3FBC9124 .long 0x3FFF0000,0xD4F35AAB,0xCFEDFA1F,0x3FBEB243 .long 0x3FFF0000,0xD744FCCA,0xD69D6AF4,0x3FBDE69A .long 0x3FFF0000,0xD99D15C2,0x78AFD7B6,0xBFB8BC61 .long 0x3FFF0000,0xDBFBB797,0xDAF23755,0x3FBDF610 .long 0x3FFF0000,0xDE60F482,0x5E0E9124,0xBFBD8BE1 .long 0x3FFF0000,0xE0CCDEEC,0x2A94E111,0x3FBACB12 .long 0x3FFF0000,0xE33F8972,0xBE8A5A51,0x3FBB9BFE .long 0x3FFF0000,0xE5B906E7,0x7C8348A8,0x3FBCF2F4 .long 0x3FFF0000,0xE8396A50,0x3C4BDC68,0x3FBEF22F .long 0x3FFF0000,0xEAC0C6E7,0xDD24392F,0xBFBDBF4A .long 0x3FFF0000,0xED4F301E,0xD9942B84,0x3FBEC01A .long 0x3FFF0000,0xEFE4B99B,0xDCDAF5CB,0x3FBE8CAC .long 0x3FFF0000,0xF281773C,0x59FFB13A,0xBFBCBB3F .long 0x3FFF0000,0xF5257D15,0x2486CC2C,0x3FBEF73A .long 0x3FFF0000,0xF7D0DF73,0x0AD13BB9,0xBFB8B795 .long 0x3FFF0000,0xFA83B2DB,0x722A033A,0x3FBEF84B .long 0x3FFF0000,0xFD3E0C0C,0xF486C175,0xBFBEF581 .set N,L_SCR1 .set X,FP_SCR1 .set XDCARE,X+2 .set XFRAC,X+4 .set ADJFACT,FP_SCR2 .set FACT1,FP_SCR3 .set FACT1HI,FACT1+4 .set FACT1LOW,FACT1+8 .set FACT2,FP_SCR4 .set FACT2HI,FACT2+4 .set FACT2LOW,FACT2+8 | xref t_unfl |xref t_ovfl |xref t_frcinx .global stwotoxd stwotoxd: |--ENTRY POINT FOR 2**(X) FOR DENORMALIZED ARGUMENT fmovel %d1,%fpcr | ...set user's rounding mode/precision fmoves #0x3F800000,%fp0 | ...RETURN 1 + X movel (%a0),%d0 orl #0x00800001,%d0 fadds %d0,%fp0 bra t_frcinx .global stwotox stwotox: |--ENTRY POINT FOR 2**(X), HERE X IS FINITE, NON-ZERO, AND NOT NAN'S fmovemx (%a0),%fp0-%fp0 | ...LOAD INPUT, do not set cc's movel (%a0),%d0 movew 4(%a0),%d0 fmovex %fp0,X(%a6) andil #0x7FFFFFFF,%d0 cmpil #0x3FB98000,%d0 | ...|X| >= 2**(-70)? bges TWOOK1 bra EXPBORS TWOOK1: cmpil #0x400D80C0,%d0 | ...|X| > 16480? bles TWOMAIN bra EXPBORS TWOMAIN: |--USUAL CASE, 2^(-70) <= |X| <= 16480 fmovex %fp0,%fp1 fmuls #0x42800000,%fp1 | ...64 * X fmovel %fp1,N(%a6) | ...N = ROUND-TO-INT(64 X) movel %d2,-(%sp) lea EXPTBL,%a1 | ...LOAD ADDRESS OF TABLE OF 2^(J/64) fmovel N(%a6),%fp1 | ...N --> FLOATING FMT movel N(%a6),%d0 movel %d0,%d2 andil #0x3F,%d0 | ...D0 IS J asll #4,%d0 | ...DISPLACEMENT FOR 2^(J/64) addal %d0,%a1 | ...ADDRESS FOR 2^(J/64) asrl #6,%d2 | ...d2 IS L, N = 64L + J movel %d2,%d0 asrl #1,%d0 | ...D0 IS M subl %d0,%d2 | ...d2 IS M', N = 64(M+M') + J addil #0x3FFF,%d2 movew %d2,ADJFACT(%a6) | ...ADJFACT IS 2^(M') movel (%sp)+,%d2 |--SUMMARY: a1 IS ADDRESS FOR THE LEADING PORTION OF 2^(J/64), |--D0 IS M WHERE N = 64(M+M') + J. NOTE THAT |M| <= 16140 BY DESIGN. |--ADJFACT = 2^(M'). |--REGISTERS SAVED SO FAR ARE (IN ORDER) FPCR, D0, FP1, a1, AND FP2. fmuls #0x3C800000,%fp1 | ...(1/64)*N movel (%a1)+,FACT1(%a6) movel (%a1)+,FACT1HI(%a6) movel (%a1)+,FACT1LOW(%a6) movew (%a1)+,FACT2(%a6) clrw FACT2+2(%a6) fsubx %fp1,%fp0 | ...X - (1/64)*INT(64 X) movew (%a1)+,FACT2HI(%a6) clrw FACT2HI+2(%a6) clrl FACT2LOW(%a6) addw %d0,FACT1(%a6) fmulx LOG2,%fp0 | ...FP0 IS R addw %d0,FACT2(%a6) bra expr EXPBORS: |--FPCR, D0 SAVED cmpil #0x3FFF8000,%d0 bgts EXPBIG EXPSM: |--|X| IS SMALL, RETURN 1 + X fmovel %d1,%FPCR |restore users exceptions fadds #0x3F800000,%fp0 | ...RETURN 1 + X bra t_frcinx EXPBIG: |--|X| IS LARGE, GENERATE OVERFLOW IF X > 0; ELSE GENERATE UNDERFLOW |--REGISTERS SAVE SO FAR ARE FPCR AND D0 movel X(%a6),%d0 cmpil #0,%d0 blts EXPNEG bclrb #7,(%a0) |t_ovfl expects positive value bra t_ovfl EXPNEG: bclrb #7,(%a0) |t_unfl expects positive value bra t_unfl .global stentoxd stentoxd: |--ENTRY POINT FOR 10**(X) FOR DENORMALIZED ARGUMENT fmovel %d1,%fpcr | ...set user's rounding mode/precision fmoves #0x3F800000,%fp0 | ...RETURN 1 + X movel (%a0),%d0 orl #0x00800001,%d0 fadds %d0,%fp0 bra t_frcinx .global stentox stentox: |--ENTRY POINT FOR 10**(X), HERE X IS FINITE, NON-ZERO, AND NOT NAN'S fmovemx (%a0),%fp0-%fp0 | ...LOAD INPUT, do not set cc's movel (%a0),%d0 movew 4(%a0),%d0 fmovex %fp0,X(%a6) andil #0x7FFFFFFF,%d0 cmpil #0x3FB98000,%d0 | ...|X| >= 2**(-70)? bges TENOK1 bra EXPBORS TENOK1: cmpil #0x400B9B07,%d0 | ...|X| <= 16480*log2/log10 ? bles TENMAIN bra EXPBORS TENMAIN: |--USUAL CASE, 2^(-70) <= |X| <= 16480 LOG 2 / LOG 10 fmovex %fp0,%fp1 fmuld L2TEN64,%fp1 | ...X*64*LOG10/LOG2 fmovel %fp1,N(%a6) | ...N=INT(X*64*LOG10/LOG2) movel %d2,-(%sp) lea EXPTBL,%a1 | ...LOAD ADDRESS OF TABLE OF 2^(J/64) fmovel N(%a6),%fp1 | ...N --> FLOATING FMT movel N(%a6),%d0 movel %d0,%d2 andil #0x3F,%d0 | ...D0 IS J asll #4,%d0 | ...DISPLACEMENT FOR 2^(J/64) addal %d0,%a1 | ...ADDRESS FOR 2^(J/64) asrl #6,%d2 | ...d2 IS L, N = 64L + J movel %d2,%d0 asrl #1,%d0 | ...D0 IS M subl %d0,%d2 | ...d2 IS M', N = 64(M+M') + J addil #0x3FFF,%d2 movew %d2,ADJFACT(%a6) | ...ADJFACT IS 2^(M') movel (%sp)+,%d2 |--SUMMARY: a1 IS ADDRESS FOR THE LEADING PORTION OF 2^(J/64), |--D0 IS M WHERE N = 64(M+M') + J. NOTE THAT |M| <= 16140 BY DESIGN. |--ADJFACT = 2^(M'). |--REGISTERS SAVED SO FAR ARE (IN ORDER) FPCR, D0, FP1, a1, AND FP2. fmovex %fp1,%fp2 fmuld L10TWO1,%fp1 | ...N*(LOG2/64LOG10)_LEAD movel (%a1)+,FACT1(%a6) fmulx L10TWO2,%fp2 | ...N*(LOG2/64LOG10)_TRAIL movel (%a1)+,FACT1HI(%a6) movel (%a1)+,FACT1LOW(%a6) fsubx %fp1,%fp0 | ...X - N L_LEAD movew (%a1)+,FACT2(%a6) fsubx %fp2,%fp0 | ...X - N L_TRAIL clrw FACT2+2(%a6) movew (%a1)+,FACT2HI(%a6) clrw FACT2HI+2(%a6) clrl FACT2LOW(%a6) fmulx LOG10,%fp0 | ...FP0 IS R addw %d0,FACT1(%a6) addw %d0,FACT2(%a6) expr: |--FPCR, FP2, FP3 ARE SAVED IN ORDER AS SHOWN. |--ADJFACT CONTAINS 2**(M'), FACT1 + FACT2 = 2**(M) * 2**(J/64). |--FP0 IS R. THE FOLLOWING CODE COMPUTES |-- 2**(M'+M) * 2**(J/64) * EXP(R) fmovex %fp0,%fp1 fmulx %fp1,%fp1 | ...FP1 IS S = R*R fmoved EXPA5,%fp2 | ...FP2 IS A5 fmoved EXPA4,%fp3 | ...FP3 IS A4 fmulx %fp1,%fp2 | ...FP2 IS S*A5 fmulx %fp1,%fp3 | ...FP3 IS S*A4 faddd EXPA3,%fp2 | ...FP2 IS A3+S*A5 faddd EXPA2,%fp3 | ...FP3 IS A2+S*A4 fmulx %fp1,%fp2 | ...FP2 IS S*(A3+S*A5) fmulx %fp1,%fp3 | ...FP3 IS S*(A2+S*A4) faddd EXPA1,%fp2 | ...FP2 IS A1+S*(A3+S*A5) fmulx %fp0,%fp3 | ...FP3 IS R*S*(A2+S*A4) fmulx %fp1,%fp2 | ...FP2 IS S*(A1+S*(A3+S*A5)) faddx %fp3,%fp0 | ...FP0 IS R+R*S*(A2+S*A4) faddx %fp2,%fp0 | ...FP0 IS EXP(R) - 1 |--FINAL RECONSTRUCTION PROCESS |--EXP(X) = 2^M*2^(J/64) + 2^M*2^(J/64)*(EXP(R)-1) - (1 OR 0) fmulx FACT1(%a6),%fp0 faddx FACT2(%a6),%fp0 faddx FACT1(%a6),%fp0 fmovel %d1,%FPCR |restore users exceptions clrw ADJFACT+2(%a6) movel #0x80000000,ADJFACT+4(%a6) clrl ADJFACT+8(%a6) fmulx ADJFACT(%a6),%fp0 | ...FINAL ADJUSTMENT bra t_frcinx |end |