sysdeps/ia64/fpu/e_exp2.S

   1 .file "exp2.s"
   2
   3
   4 // Copyright (c) 2000 - 2005, Intel Corporation
   5 // All rights reserved.
   6 //
   7 //
   8 // Redistribution and use in source and binary forms, with or without
   9 // modification, are permitted provided that the following conditions are
  10 // met:
  11 //
  12 // * Redistributions of source code must retain the above copyright
  13 // notice, this list of conditions and the following disclaimer.
  14 //
  15 // * Redistributions in binary form must reproduce the above copyright
  16 // notice, this list of conditions and the following disclaimer in the
  17 // documentation and/or other materials provided with the distribution.
  18 //
  19 // * The name of Intel Corporation may not be used to endorse or promote
  20 // products derived from this software without specific prior written
  21 // permission.
  22
  23 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
  24 // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
  25 // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
  26 // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INTEL OR ITS
  27 // CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
  28 // EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
  29 // PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
  30 // PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY
  31 // OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY OR TORT (INCLUDING
  32 // NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
  33 // SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
  34 //
  35 // Intel Corporation is the author of this code, and requests that all
  36 // problem reports or change requests be submitted to it directly at
  37 // http://www.intel.com/software/products/opensource/libraries/num.htm.
  38 //
  39 // History
  40 //==============================================================
  41 // 08/25/00  Initial version
  42 // 05/20/02  Cleaned up namespace and sf0 syntax
  43 // 09/05/02  Improved performance
  44 // 01/17/03  Fixed to call error support when x=1024.0
  45 // 03/31/05  Reformatted delimiters between data tables
  46 //
  47 // API
  48 //==============================================================
  49 // double exp2(double)
  50 //
  51 // Overview of operation
  52 //==============================================================
  53 // Background
  54 //
  55 // Implementation
  56 //
  57 // Let x= (K + fh + fl + r), where
  58 // K is an integer, fh= 0.b1 b2 b3 b4 b5,
  59 // fl= 2^{-5}* 0.b6 b7 b8 b8 b10 (fh, fl >= 0),
  60 // and |r|<2^{-11}
  61 // Th is a table that stores 2^fh (32 entries) rounded to
  62 // double extended precision (only mantissa is stored)
  63 // Tl is a table that stores 2^fl (32 entries) rounded to
  64 // double extended precision (only mantissa is stored)
  65 //
  66 // 2^x is approximated as
  67 // 2^K * Th [ f ] * Tl [ f ] * (1+c1*r+c2*r^2+c3*r^3+c4*r^4)
  68
  69 // Note: We use the following trick to speed up conversion from FP to integer:
  70 //
  71 // Let  x = K + r, where K is an integer, and  |r| <= 0.5
  72 // Let N be the number of significand bits for the FP format used
  73 //   ( N=64 for double-extended, N=53 for double)
  74 //
  75 // Then let y = 1.5 * 2^(N-1)  +  x    for RN mode
  76 //          K = y -  1.5 * 2^(N-1)
  77 //          r  = x - K
  78 //
  79 // If we want to obtain the integer part and the first m fractional bits of x,
  80 // we can use the same trick, but with a constant of  1.5 * 2^(N-1-m):
  81 //
  82 // Let x = K + f + r
  83 // f = 0.b_1 b_2 ... b_m
  84 // |r| <= 2^(-m-1)
  85 //
  86 // Then let y = 1.5 * 2^(N-1-m)  +  x    for RN mode
  87 //          (K+f) = y -  1.5 * 2^(N-1-m)
  88 //          r  = x - K
  89
  90
  91 // Special values
  92 //==============================================================
  93 // exp2(0)= 1
  94 // exp2(+inf)= inf
  95 // exp2(-inf)= 0
  96 //
  97
  98 // Registers used
  99 //==============================================================
 100 // r2-r3, r14-r40
 101 // f6-f15, f32-f45
 102 // p6-p8, p12
 103 //
 104
 105
 106 GR_TBL_START        = r2
 107 GR_LOG_TBL          = r3
 108
 109 GR_OF_LIMIT         = r14
 110 GR_UF_LIMIT         = r15
 111 GR_EXP_CORR         = r16
 112 GR_F_low            = r17
 113 GR_F_high           = r18
 114 GR_K                = r19
 115 GR_Flow_ADDR        = r20
 116
 117 GR_BIAS             = r21
 118 GR_Fh               = r22
 119 GR_Fh_ADDR          = r23
 120 GR_EXPMAX           = r24
 121 GR_EMIN             = r25
 122
 123 GR_ROUNDVAL         = r26
 124 GR_MASK             = r27
 125 GR_KF0              = r28
 126 GR_MASK_low         = r29
 127 GR_COEFF_START      = r30
 128
 129 GR_SAVE_B0          = r33
 130 GR_SAVE_PFS         = r34
 131 GR_SAVE_GP          = r35
 132 GR_SAVE_SP          = r36
 133
 134 GR_Parameter_X      = r37
 135 GR_Parameter_Y      = r38
 136 GR_Parameter_RESULT = r39
 137 GR_Parameter_TAG    = r40
 138
 139
 140 FR_X                = f10
 141 FR_Y                = f1
 142 FR_RESULT           = f8
 143
 144
 145 FR_COEFF1           = f6
 146 FR_COEFF2           = f7
 147 FR_R                = f9
 148
 149 FR_KF0              = f12
 150 FR_COEFF3           = f13
 151 FR_COEFF4           = f14
 152 FR_UF_LIMIT         = f15
 153
 154 FR_OF_LIMIT         = f32
 155 FR_EXPMIN           = f33
 156 FR_ROUNDVAL         = f34
 157 FR_KF               = f35
 158
 159 FR_2_TO_K           = f36
 160 FR_T_low            = f37
 161 FR_T_high           = f38
 162 FR_P34              = f39
 163 FR_R2               = f40
 164
 165 FR_P12              = f41
 166 FR_T_low_K          = f42
 167 FR_P14              = f43
 168 FR_T                = f44
 169 FR_P                = f45
 170
 171
 172 // Data tables
 173 //==============================================================
 174
 175 RODATA
 176
 177 .align 16
 178
 179 LOCAL_OBJECT_START(poly_coeffs)
 180
 181 data8 0x3fac6b08d704a0c0, 0x3f83b2ab6fba4e77 // C_3 and C_4
 182 data8 0xb17217f7d1cf79ab, 0x00003ffe // C_1
 183 data8 0xf5fdeffc162c7541, 0x00003ffc // C_2
 184 LOCAL_OBJECT_END(poly_coeffs)
 185
 186
 187 LOCAL_OBJECT_START(T_table)
 188
 189 // 2^{0.00000 b6 b7 b8 b9 b10}
 190 data8 0x8000000000000000, 0x8016302f17467628
 191 data8 0x802c6436d0e04f50, 0x80429c17d77c18ed
 192 data8 0x8058d7d2d5e5f6b0, 0x806f17687707a7af
 193 data8 0x80855ad965e88b83, 0x809ba2264dada76a
 194 data8 0x80b1ed4fd999ab6c, 0x80c83c56b50cf77f
 195 data8 0x80de8f3b8b85a0af, 0x80f4e5ff089f763e
 196 data8 0x810b40a1d81406d4, 0x81219f24a5baa59d
 197 data8 0x813801881d886f7b, 0x814e67cceb90502c
 198 data8 0x8164d1f3bc030773, 0x817b3ffd3b2f2e47
 199 data8 0x8191b1ea15813bfd, 0x81a827baf7838b78
 200 data8 0x81bea1708dde6055, 0x81d51f0b8557ec1c
 201 data8 0x81eba08c8ad4536f, 0x820225f44b55b33b
 202 data8 0x8218af4373fc25eb, 0x822f3c7ab205c89a
 203 data8 0x8245cd9ab2cec048, 0x825c62a423d13f0c
 204 data8 0x8272fb97b2a5894c, 0x828998760d01faf3
 205 data8 0x82a0393fe0bb0ca8, 0x82b6ddf5dbc35906
 206 //
 207 // 2^{0.b1 b2 b3 b4 b5}
 208 data8 0x8000000000000000, 0x82cd8698ac2ba1d7
 209 data8 0x85aac367cc487b14, 0x88980e8092da8527
 210 data8 0x8b95c1e3ea8bd6e6, 0x8ea4398b45cd53c0
 211 data8 0x91c3d373ab11c336, 0x94f4efa8fef70961
 212 data8 0x9837f0518db8a96f, 0x9b8d39b9d54e5538
 213 data8 0x9ef5326091a111ad, 0xa27043030c496818
 214 data8 0xa5fed6a9b15138ea, 0xa9a15ab4ea7c0ef8
 215 data8 0xad583eea42a14ac6, 0xb123f581d2ac258f
 216 data8 0xb504f333f9de6484, 0xb8fbaf4762fb9ee9
 217 data8 0xbd08a39f580c36be, 0xc12c4cca66709456
 218 data8 0xc5672a115506dadd, 0xc9b9bd866e2f27a2
 219 data8 0xce248c151f8480e3, 0xd2a81d91f12ae45a
 220 data8 0xd744fccad69d6af4, 0xdbfbb797daf23755
 221 data8 0xe0ccdeec2a94e111, 0xe5b906e77c8348a8
 222 data8 0xeac0c6e7dd24392e, 0xefe4b99bdcdaf5cb
 223 data8 0xf5257d152486cc2c, 0xfa83b2db722a033a
 224 LOCAL_OBJECT_END(T_table)
 225
 226
 227
 228 .section .text
 229 WEAK_LIBM_ENTRY(exp2)
 230
 231
 232 {.mfi
 233        alloc r32= ar.pfs, 1, 4, 4, 0
 234        // will continue only for non-zero normal/denormal numbers
 235        fclass.nm p12, p0= f8, 0x1b
 236        // GR_TBL_START= pointer to C_1...C_4 followed by T_table
 237        addl GR_TBL_START= @ltoff(poly_coeffs), gp
 238 }
 239 {.mlx
 240        mov GR_OF_LIMIT= 0xffff + 10              // Exponent of overflow limit
 241        movl GR_ROUNDVAL= 0x5a400000              // 1.5*2^(63-10) (SP)
 242 }
 243 ;;
 244
 245 // Form special constant 1.5*2^(63-10) to give integer part and first 10
 246 // fractional bits of x
 247 {.mfi
 248        setf.s FR_ROUNDVAL= GR_ROUNDVAL           // Form special constant
 249        fcmp.lt.s1 p6, p8= f8, f0                 // X<0 ?
 250        nop.i 0
 251 }
 252 {.mfb
 253        ld8 GR_COEFF_START= [ GR_TBL_START ]      // Load pointer to coeff table
 254        nop.f 0
 255  (p12) br.cond.spnt SPECIAL_exp2                 // Branch if nan, inf, zero
 256 }
 257 ;;
 258
 259 {.mlx
 260        setf.exp FR_OF_LIMIT= GR_OF_LIMIT         // Set overflow limit
 261        movl GR_UF_LIMIT= 0xc4866000              // (-2^10-51) = -1075
 262 }
 263 ;;
 264
 265 {.mfi
 266        ldfpd FR_COEFF3, FR_COEFF4= [ GR_COEFF_START ], 16 // load C_3, C_4
 267        fma.s0 f8= f8, f1, f0                     // normalize x
 268        nop.i 0
 269 }
 270 ;;
 271
 272 {.mmi
 273        setf.s FR_UF_LIMIT= GR_UF_LIMIT           // Set underflow limit
 274        ldfe FR_COEFF1= [ GR_COEFF_START ], 16    // load C_1
 275        mov GR_EXP_CORR= 0xffff-126
 276 }
 277 ;;
 278
 279 {.mfi
 280        ldfe FR_COEFF2= [ GR_COEFF_START ], 16    // load C_2
 281        fma.s1 FR_KF0= f8, f1, FR_ROUNDVAL        // y= x + 1.5*2^(63-10)
 282        nop.i 0
 283 }
 284 ;;
 285
 286 {.mfi
 287        mov GR_MASK= 1023
 288        fms.s1 FR_KF= FR_KF0, f1, FR_ROUNDVAL     // (K+f)
 289        mov GR_MASK_low= 31
 290 }
 291 ;;
 292
 293 {.mfi
 294        getf.sig GR_KF0= FR_KF0                   // (K+f)*2^10= round_to_int(y)
 295        fcmp.ge.s1 p12, p7= f8, FR_OF_LIMIT       // x >= overflow threshold ?
 296        add GR_LOG_TBL= 256, GR_COEFF_START       // Pointer to high T_table
 297 }
 298 ;;
 299
 300 {.mmi
 301        and GR_F_low= GR_KF0, GR_MASK_low         // f_low
 302        and GR_F_high= GR_MASK, GR_KF0            // f_high*32
 303        shr GR_K= GR_KF0, 10                      // K
 304 }
 305 ;;
 306
 307 {.mmi
 308        shladd GR_Flow_ADDR= GR_F_low, 3, GR_COEFF_START // address of 2^{f_low}
 309        add GR_BIAS= GR_K, GR_EXP_CORR            // K= bias-2*63
 310        shr GR_Fh= GR_F_high, 5                   // f_high
 311 }
 312 ;;
 313
 314 {.mfi
 315        setf.exp FR_2_TO_K= GR_BIAS               // 2^{K-126}
 316        fnma.s1 FR_R= FR_KF, f1, f8               // r= x - (K+f)
 317        shladd GR_Fh_ADDR= GR_Fh, 3, GR_LOG_TBL   // address of 2^{f_high}
 318 }
 319 {.mlx
 320        ldf8 FR_T_low= [ GR_Flow_ADDR ]           // load T_low= 2^{f_low}
 321        movl GR_EMIN= 0xc47f8000                  // EMIN= -1022
 322 }
 323 ;;
 324
 325 {.mfi
 326        ldf8 FR_T_high= [ GR_Fh_ADDR ]            // load T_high= 2^{f_high}
 327  (p7)  fcmp.lt.s1 p12, p7= f8, FR_UF_LIMIT       // x<underflow threshold ?
 328        nop.i 0
 329 }
 330 ;;
 331
 332 {.mfi
 333        setf.s FR_EXPMIN= GR_EMIN                 // FR_EXPMIN= EMIN
 334        fma.s1 FR_P34= FR_COEFF4, FR_R, FR_COEFF3 // P34= C_3+C_4*r
 335        nop.i 0
 336 }
 337 {.mfb
 338        nop.m 0
 339        fma.s1 FR_R2= FR_R, FR_R, f0              // r*r
 340  (p12) br.cond.spnt OUT_RANGE_exp2
 341 }
 342 ;;
 343
 344 {.mfi
 345        nop.m 0
 346        fma.s1 FR_P12= FR_COEFF2, FR_R, FR_COEFF1 // P12= C_1+C_2*r
 347        nop.i 0
 348 }
 349 ;;
 350
 351 {.mfi
 352        nop.m 0
 353        fma.s1 FR_T_low_K= FR_T_low, FR_2_TO_K, f0 // T= 2^{K-126}*T_low
 354        nop.i 0
 355 }
 356 ;;
 357
 358 {.mfi
 359        nop.m 0
 360        fma.s1 FR_P14= FR_R2, FR_P34, FR_P12       // P14= P12+r2*P34
 361        nop.i 0
 362 }
 363 ;;
 364
 365 {.mfi
 366        nop.m 0
 367        fma.s1 FR_T= FR_T_low_K, FR_T_high, f0     // T= T*T_high
 368        nop.i 0
 369 }
 370 ;;
 371
 372 {.mfi
 373        nop.m 0
 374        fcmp.lt.s0 p6, p8= f8, FR_EXPMIN           // underflow (x<EMIN) ?
 375        nop.i 0
 376 }
 377 ;;
 378
 379 {.mfi
 380        nop.m 0
 381        fma.s1 FR_P= FR_P14, FR_R, f0              // P= P14*r
 382        nop.i 0
 383 }
 384 ;;
 385
 386 {.mfb
 387        nop.m 0
 388        fma.d.s0 f8= FR_P, FR_T, FR_T              // result= T+T*P
 389  (p8)  br.ret.sptk b0                             // return
 390 }
 391 ;;
 392
 393 {.mfb
 394  (p6)  mov GR_Parameter_TAG= 162
 395        nop.f 0
 396  (p6)  br.cond.sptk __libm_error_region
 397 }
 398 ;;
 399
 400
 401 SPECIAL_exp2:
 402 {.mfi
 403        nop.m 0
 404        fclass.m p6, p0= f8, 0x22                  // x= -Infinity ?
 405        nop.i 0
 406 }
 407 ;;
 408
 409 {.mfi
 410        nop.m 0
 411        fclass.m p7, p0= f8, 0x21                  // x= +Infinity ?
 412        nop.i 0
 413 }
 414 ;;
 415
 416 {.mfi
 417        nop.m 0
 418        fclass.m p8, p0= f8, 0x7                   // x= +/-Zero ?
 419        nop.i 0
 420 }
 421 {.mfb
 422        nop.m 0
 423  (p6)  mov f8= f0                                 // exp2(-Infinity)= 0
 424  (p6)  br.ret.spnt b0
 425 }
 426 ;;
 427
 428 {.mfb
 429        nop.m 0
 430        nop.f 0
 431  (p7)  br.ret.spnt b0                             // exp2(+Infinity)= +Infinity
 432 }
 433 ;;
 434
 435 {.mfb
 436        nop.m 0
 437  (p8)  mov f8= f1                                 // exp2(+/-0)= 1
 438  (p8)  br.ret.spnt b0
 439 }
 440 ;;
 441
 442 {.mfb
 443        nop.m 0
 444        fma.d.s0 f8= f8, f1, f0                    // Remaining cases: NaNs
 445        br.ret.sptk b0
 446 }
 447 ;;
 448
 449
 450 OUT_RANGE_exp2:
 451
 452 // overflow: p8= 1
 453
 454 {.mii
 455  (p8)  mov GR_EXPMAX= 0x1fffe
 456        nop.i 0
 457        nop.i 0
 458 }
 459 ;;
 460
 461 {.mmb
 462  (p8)  mov GR_Parameter_TAG= 161
 463  (p8)  setf.exp FR_R= GR_EXPMAX
 464        nop.b 999
 465 }
 466 ;;
 467
 468 {.mfi
 469        nop.m 999
 470  (p8)  fma.d.s0 f8= FR_R, FR_R, f0                // Create overflow
 471        nop.i 999
 472 }
 473 // underflow: p6= 1
 474 {.mii
 475  (p6)  mov GR_Parameter_TAG= 162
 476  (p6)  mov GR_EXPMAX= 1
 477        nop.i 0
 478 }
 479 ;;
 480
 481 {.mmb
 482        nop.m 0
 483  (p6)  setf.exp FR_R= GR_EXPMAX
 484        nop.b 999
 485 }
 486 ;;
 487
 488 {.mfb
 489        nop.m 999
 490  (p6)  fma.d.s0 f8= FR_R, FR_R, f0                // Create underflow
 491        nop.b 0
 492 }
 493 ;;
 494
 495 WEAK_LIBM_END(exp2)
 496 libm_alias_double_other (__exp2, exp2)
 497 #ifdef SHARED
 498 .symver exp2,exp2@@GLIBC_2.29
 499 .weak __exp2_compat
 500 .set __exp2_compat,__exp2
 501 .symver __exp2_compat,exp2@GLIBC_2.2
 502 #endif
 503
 504
 505 LOCAL_LIBM_ENTRY(__libm_error_region)
 506
 507 .prologue
 508 {.mfi
 509        add GR_Parameter_Y= -32, sp                // Parameter 2 value
 510        nop.f 0
 511 .save ar.pfs, GR_SAVE_PFS
 512        mov GR_SAVE_PFS= ar.pfs                    // Save ar.pfs
 513 }
 514
 515 {.mfi
 516 .fframe 64
 517        add sp= -64, sp                            // Create new stack
 518        nop.f 0
 519        mov GR_SAVE_GP= gp                         // Save gp
 520 }
 521 ;;
 522
 523 {.mmi
 524        stfd [ GR_Parameter_Y ]= FR_Y, 16          // STORE Parameter 2 on stack
 525        add GR_Parameter_X= 16, sp                 // Parameter 1 address
 526 .save b0, GR_SAVE_B0
 527        mov GR_SAVE_B0= b0                         // Save b0
 528 }
 529 ;;
 530
 531 .body
 532 {.mib
 533        stfd [ GR_Parameter_X ]= FR_X              // STORE Parameter 1 on stack
 534        add GR_Parameter_RESULT= 0, GR_Parameter_Y // Parameter 3 address
 535        nop.b 0
 536 }
 537 {.mib
 538        stfd [ GR_Parameter_Y ]= FR_RESULT         // STORE Parameter 3 on stack
 539        add GR_Parameter_Y= -16, GR_Parameter_Y
 540        br.call.sptk b0= __libm_error_support#    // Call error handling function
 541 }
 542 ;;
 543
 544 {.mmi
 545        add GR_Parameter_RESULT= 48, sp
 546        nop.m 0
 547        nop.i 0
 548 }
 549 ;;
 550
 551 {.mmi
 552        ldfd f8= [ GR_Parameter_RESULT ]          // Get return result off stack
 553 .restore sp
 554        add sp= 64, sp                            // Restore stack pointer
 555        mov b0= GR_SAVE_B0                        // Restore return address
 556 }
 557 ;;
 558
 559 {.mib
 560        mov gp= GR_SAVE_GP                        // Restore gp
 561        mov ar.pfs= GR_SAVE_PFS                   // Restore ar.pfs
 562        br.ret.sptk b0                            // Return
 563 }
 564 ;;
 565
 566
 567 LOCAL_LIBM_END(__libm_error_region)
 568
 569 .type __libm_error_support#, @function
 570 .global __libm_error_support#