1* $NetBSD: setox.sa,v 1.3 1994/10/26 07:49:42 cgd Exp $ 2 3* MOTOROLA MICROPROCESSOR & MEMORY TECHNOLOGY GROUP 4* M68000 Hi-Performance Microprocessor Division 5* M68040 Software Package 6* 7* M68040 Software Package Copyright (c) 1993, 1994 Motorola Inc. 8* All rights reserved. 9* 10* THE SOFTWARE is provided on an "AS IS" basis and without warranty. 11* To the maximum extent permitted by applicable law, 12* MOTOROLA DISCLAIMS ALL WARRANTIES WHETHER EXPRESS OR IMPLIED, 13* INCLUDING IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A 14* PARTICULAR PURPOSE and any warranty against infringement with 15* regard to the SOFTWARE (INCLUDING ANY MODIFIED VERSIONS THEREOF) 16* and any accompanying written materials. 17* 18* To the maximum extent permitted by applicable law, 19* IN NO EVENT SHALL MOTOROLA BE LIABLE FOR ANY DAMAGES WHATSOEVER 20* (INCLUDING WITHOUT LIMITATION, DAMAGES FOR LOSS OF BUSINESS 21* PROFITS, BUSINESS INTERRUPTION, LOSS OF BUSINESS INFORMATION, OR 22* OTHER PECUNIARY LOSS) ARISING OF THE USE OR INABILITY TO USE THE 23* SOFTWARE. Motorola assumes no responsibility for the maintenance 24* and support of the SOFTWARE. 25* 26* You are hereby granted a copyright license to use, modify, and 27* distribute the SOFTWARE so long as this entire notice is retained 28* without alteration in any modified and/or redistributed versions, 29* and that such modified versions are clearly identified as such. 30* No licenses are granted by implication, estoppel or otherwise 31* under any patents or trademarks of Motorola, Inc. 32 33* 34* setox.sa 3.1 12/10/90 35* 36* The entry point setox computes the exponential of a value. 37* setoxd does the same except the input value is a denormalized 38* number. setoxm1 computes exp(X)-1, and setoxm1d computes 39* exp(X)-1 for denormalized X. 40* 41* INPUT 42* ----- 43* Double-extended value in memory location pointed to by address 44* register a0. 45* 46* OUTPUT 47* ------ 48* exp(X) or exp(X)-1 returned in floating-point register fp0. 49* 50* ACCURACY and MONOTONICITY 51* ------------------------- 52* The returned result is within 0.85 ulps in 64 significant bit, i.e. 53* within 0.5001 ulp to 53 bits if the result is subsequently rounded 54* to double precision. The result is provably monotonic in double 55* precision. 56* 57* SPEED 58* ----- 59* Two timings are measured, both in the copy-back mode. The 60* first one is measured when the function is invoked the first time 61* (so the instructions and data are not in cache), and the 62* second one is measured when the function is reinvoked at the same 63* input argument. 64* 65* The program setox takes approximately 210/190 cycles for input 66* argument X whose magnitude is less than 16380 log2, which 67* is the usual situation. For the less common arguments, 68* depending on their values, the program may run faster or slower -- 69* but no worse than 10% slower even in the extreme cases. 70* 71* The program setoxm1 takes approximately ???/??? cycles for input 72* argument X, 0.25 <= |X| < 70log2. For |X| < 0.25, it takes 73* approximately ???/??? cycles. For the less common arguments, 74* depending on their values, the program may run faster or slower -- 75* but no worse than 10% slower even in the extreme cases. 76* 77* ALGORITHM and IMPLEMENTATION NOTES 78* ---------------------------------- 79* 80* setoxd 81* ------ 82* Step 1. Set ans := 1.0 83* 84* Step 2. Return ans := ans + sign(X)*2^(-126). Exit. 85* Notes: This will always generate one exception -- inexact. 86* 87* 88* setox 89* ----- 90* 91* Step 1. Filter out extreme cases of input argument. 92* 1.1 If |X| >= 2^(-65), go to Step 1.3. 93* 1.2 Go to Step 7. 94* 1.3 If |X| < 16380 log(2), go to Step 2. 95* 1.4 Go to Step 8. 96* Notes: The usual case should take the branches 1.1 -> 1.3 -> 2. 97* To avoid the use of floating-point comparisons, a 98* compact representation of |X| is used. This format is a 99* 32-bit integer, the upper (more significant) 16 bits are 100* the sign and biased exponent field of |X|; the lower 16 101* bits are the 16 most significant fraction (including the 102* explicit bit) bits of |X|. Consequently, the comparisons 103* in Steps 1.1 and 1.3 can be performed by integer comparison. 104* Note also that the constant 16380 log(2) used in Step 1.3 105* is also in the compact form. Thus taking the branch 106* to Step 2 guarantees |X| < 16380 log(2). There is no harm 107* to have a small number of cases where |X| is less than, 108* but close to, 16380 log(2) and the branch to Step 9 is 109* taken. 110* 111* Step 2. Calculate N = round-to-nearest-int( X * 64/log2 ). 112* 2.1 Set AdjFlag := 0 (indicates the branch 1.3 -> 2 was taken) 113* 2.2 N := round-to-nearest-integer( X * 64/log2 ). 114* 2.3 Calculate J = N mod 64; so J = 0,1,2,..., or 63. 115* 2.4 Calculate M = (N - J)/64; so N = 64M + J. 116* 2.5 Calculate the address of the stored value of 2^(J/64). 117* 2.6 Create the value Scale = 2^M. 118* Notes: The calculation in 2.2 is really performed by 119* 120* Z := X * constant 121* N := round-to-nearest-integer(Z) 122* 123* where 124* 125* constant := single-precision( 64/log 2 ). 126* 127* Using a single-precision constant avoids memory access. 128* Another effect of using a single-precision "constant" is 129* that the calculated value Z is 130* 131* Z = X*(64/log2)*(1+eps), |eps| <= 2^(-24). 132* 133* This error has to be considered later in Steps 3 and 4. 134* 135* Step 3. Calculate X - N*log2/64. 136* 3.1 R := X + N*L1, where L1 := single-precision(-log2/64). 137* 3.2 R := R + N*L2, L2 := extended-precision(-log2/64 - L1). 138* Notes: a) The way L1 and L2 are chosen ensures L1+L2 approximate 139* the value -log2/64 to 88 bits of accuracy. 140* b) N*L1 is exact because N is no longer than 22 bits and 141* L1 is no longer than 24 bits. 142* c) The calculation X+N*L1 is also exact due to cancellation. 143* Thus, R is practically X+N(L1+L2) to full 64 bits. 144* d) It is important to estimate how large can |R| be after 145* Step 3.2. 146* 147* N = rnd-to-int( X*64/log2 (1+eps) ), |eps|<=2^(-24) 148* X*64/log2 (1+eps) = N + f, |f| <= 0.5 149* X*64/log2 - N = f - eps*X 64/log2 150* X - N*log2/64 = f*log2/64 - eps*X 151* 152* 153* Now |X| <= 16446 log2, thus 154* 155* |X - N*log2/64| <= (0.5 + 16446/2^(18))*log2/64 156* <= 0.57 log2/64. 157* This bound will be used in Step 4. 158* 159* Step 4. Approximate exp(R)-1 by a polynomial 160* p = R + R*R*(A1 + R*(A2 + R*(A3 + R*(A4 + R*A5)))) 161* Notes: a) In order to reduce memory access, the coefficients are 162* made as "short" as possible: A1 (which is 1/2), A4 and A5 163* are single precision; A2 and A3 are double precision. 164* b) Even with the restrictions above, 165* |p - (exp(R)-1)| < 2^(-68.8) for all |R| <= 0.0062. 166* Note that 0.0062 is slightly bigger than 0.57 log2/64. 167* c) To fully utilize the pipeline, p is separated into 168* two independent pieces of roughly equal complexities 169* p = [ R + R*S*(A2 + S*A4) ] + 170* [ S*(A1 + S*(A3 + S*A5)) ] 171* where S = R*R. 172* 173* Step 5. Compute 2^(J/64)*exp(R) = 2^(J/64)*(1+p) by 174* ans := T + ( T*p + t) 175* where T and t are the stored values for 2^(J/64). 176* Notes: 2^(J/64) is stored as T and t where T+t approximates 177* 2^(J/64) to roughly 85 bits; T is in extended precision 178* and t is in single precision. Note also that T is rounded 179* to 62 bits so that the last two bits of T are zero. The 180* reason for such a special form is that T-1, T-2, and T-8 181* will all be exact --- a property that will give much 182* more accurate computation of the function EXPM1. 183* 184* Step 6. Reconstruction of exp(X) 185* exp(X) = 2^M * 2^(J/64) * exp(R). 186* 6.1 If AdjFlag = 0, go to 6.3 187* 6.2 ans := ans * AdjScale 188* 6.3 Restore the user FPCR 189* 6.4 Return ans := ans * Scale. Exit. 190* Notes: If AdjFlag = 0, we have X = Mlog2 + Jlog2/64 + R, 191* |M| <= 16380, and Scale = 2^M. Moreover, exp(X) will 192* neither overflow nor underflow. If AdjFlag = 1, that 193* means that 194* X = (M1+M)log2 + Jlog2/64 + R, |M1+M| >= 16380. 195* Hence, exp(X) may overflow or underflow or neither. 196* When that is the case, AdjScale = 2^(M1) where M1 is 197* approximately M. Thus 6.2 will never cause over/underflow. 198* Possible exception in 6.4 is overflow or underflow. 199* The inexact exception is not generated in 6.4. Although 200* one can argue that the inexact flag should always be 201* raised, to simulate that exception cost to much than the 202* flag is worth in practical uses. 203* 204* Step 7. Return 1 + X. 205* 7.1 ans := X 206* 7.2 Restore user FPCR. 207* 7.3 Return ans := 1 + ans. Exit 208* Notes: For non-zero X, the inexact exception will always be 209* raised by 7.3. That is the only exception raised by 7.3. 210* Note also that we use the FMOVEM instruction to move X 211* in Step 7.1 to avoid unnecessary trapping. (Although 212* the FMOVEM may not seem relevant since X is normalized, 213* the precaution will be useful in the library version of 214* this code where the separate entry for denormalized inputs 215* will be done away with.) 216* 217* Step 8. Handle exp(X) where |X| >= 16380log2. 218* 8.1 If |X| > 16480 log2, go to Step 9. 219* (mimic 2.2 - 2.6) 220* 8.2 N := round-to-integer( X * 64/log2 ) 221* 8.3 Calculate J = N mod 64, J = 0,1,...,63 222* 8.4 K := (N-J)/64, M1 := truncate(K/2), M = K-M1, AdjFlag := 1. 223* 8.5 Calculate the address of the stored value 2^(J/64). 224* 8.6 Create the values Scale = 2^M, AdjScale = 2^M1. 225* 8.7 Go to Step 3. 226* Notes: Refer to notes for 2.2 - 2.6. 227* 228* Step 9. Handle exp(X), |X| > 16480 log2. 229* 9.1 If X < 0, go to 9.3 230* 9.2 ans := Huge, go to 9.4 231* 9.3 ans := Tiny. 232* 9.4 Restore user FPCR. 233* 9.5 Return ans := ans * ans. Exit. 234* Notes: Exp(X) will surely overflow or underflow, depending on 235* X's sign. "Huge" and "Tiny" are respectively large/tiny 236* extended-precision numbers whose square over/underflow 237* with an inexact result. Thus, 9.5 always raises the 238* inexact together with either overflow or underflow. 239* 240* 241* setoxm1d 242* -------- 243* 244* Step 1. Set ans := 0 245* 246* Step 2. Return ans := X + ans. Exit. 247* Notes: This will return X with the appropriate rounding 248* precision prescribed by the user FPCR. 249* 250* setoxm1 251* ------- 252* 253* Step 1. Check |X| 254* 1.1 If |X| >= 1/4, go to Step 1.3. 255* 1.2 Go to Step 7. 256* 1.3 If |X| < 70 log(2), go to Step 2. 257* 1.4 Go to Step 10. 258* Notes: The usual case should take the branches 1.1 -> 1.3 -> 2. 259* However, it is conceivable |X| can be small very often 260* because EXPM1 is intended to evaluate exp(X)-1 accurately 261* when |X| is small. For further details on the comparisons, 262* see the notes on Step 1 of setox. 263* 264* Step 2. Calculate N = round-to-nearest-int( X * 64/log2 ). 265* 2.1 N := round-to-nearest-integer( X * 64/log2 ). 266* 2.2 Calculate J = N mod 64; so J = 0,1,2,..., or 63. 267* 2.3 Calculate M = (N - J)/64; so N = 64M + J. 268* 2.4 Calculate the address of the stored value of 2^(J/64). 269* 2.5 Create the values Sc = 2^M and OnebySc := -2^(-M). 270* Notes: See the notes on Step 2 of setox. 271* 272* Step 3. Calculate X - N*log2/64. 273* 3.1 R := X + N*L1, where L1 := single-precision(-log2/64). 274* 3.2 R := R + N*L2, L2 := extended-precision(-log2/64 - L1). 275* Notes: Applying the analysis of Step 3 of setox in this case 276* shows that |R| <= 0.0055 (note that |X| <= 70 log2 in 277* this case). 278* 279* Step 4. Approximate exp(R)-1 by a polynomial 280* p = R+R*R*(A1+R*(A2+R*(A3+R*(A4+R*(A5+R*A6))))) 281* Notes: a) In order to reduce memory access, the coefficients are 282* made as "short" as possible: A1 (which is 1/2), A5 and A6 283* are single precision; A2, A3 and A4 are double precision. 284* b) Even with the restriction above, 285* |p - (exp(R)-1)| < |R| * 2^(-72.7) 286* for all |R| <= 0.0055. 287* c) To fully utilize the pipeline, p is separated into 288* two independent pieces of roughly equal complexity 289* p = [ R*S*(A2 + S*(A4 + S*A6)) ] + 290* [ R + S*(A1 + S*(A3 + S*A5)) ] 291* where S = R*R. 292* 293* Step 5. Compute 2^(J/64)*p by 294* p := T*p 295* where T and t are the stored values for 2^(J/64). 296* Notes: 2^(J/64) is stored as T and t where T+t approximates 297* 2^(J/64) to roughly 85 bits; T is in extended precision 298* and t is in single precision. Note also that T is rounded 299* to 62 bits so that the last two bits of T are zero. The 300* reason for such a special form is that T-1, T-2, and T-8 301* will all be exact --- a property that will be exploited 302* in Step 6 below. The total relative error in p is no 303* bigger than 2^(-67.7) compared to the final result. 304* 305* Step 6. Reconstruction of exp(X)-1 306* exp(X)-1 = 2^M * ( 2^(J/64) + p - 2^(-M) ). 307* 6.1 If M <= 63, go to Step 6.3. 308* 6.2 ans := T + (p + (t + OnebySc)). Go to 6.6 309* 6.3 If M >= -3, go to 6.5. 310* 6.4 ans := (T + (p + t)) + OnebySc. Go to 6.6 311* 6.5 ans := (T + OnebySc) + (p + t). 312* 6.6 Restore user FPCR. 313* 6.7 Return ans := Sc * ans. Exit. 314* Notes: The various arrangements of the expressions give accurate 315* evaluations. 316* 317* Step 7. exp(X)-1 for |X| < 1/4. 318* 7.1 If |X| >= 2^(-65), go to Step 9. 319* 7.2 Go to Step 8. 320* 321* Step 8. Calculate exp(X)-1, |X| < 2^(-65). 322* 8.1 If |X| < 2^(-16312), goto 8.3 323* 8.2 Restore FPCR; return ans := X - 2^(-16382). Exit. 324* 8.3 X := X * 2^(140). 325* 8.4 Restore FPCR; ans := ans - 2^(-16382). 326* Return ans := ans*2^(140). Exit 327* Notes: The idea is to return "X - tiny" under the user 328* precision and rounding modes. To avoid unnecessary 329* inefficiency, we stay away from denormalized numbers the 330* best we can. For |X| >= 2^(-16312), the straightforward 331* 8.2 generates the inexact exception as the case warrants. 332* 333* Step 9. Calculate exp(X)-1, |X| < 1/4, by a polynomial 334* p = X + X*X*(B1 + X*(B2 + ... + X*B12)) 335* Notes: a) In order to reduce memory access, the coefficients are 336* made as "short" as possible: B1 (which is 1/2), B9 to B12 337* are single precision; B3 to B8 are double precision; and 338* B2 is double extended. 339* b) Even with the restriction above, 340* |p - (exp(X)-1)| < |X| 2^(-70.6) 341* for all |X| <= 0.251. 342* Note that 0.251 is slightly bigger than 1/4. 343* c) To fully preserve accuracy, the polynomial is computed 344* as X + ( S*B1 + Q ) where S = X*X and 345* Q = X*S*(B2 + X*(B3 + ... + X*B12)) 346* d) To fully utilize the pipeline, Q is separated into 347* two independent pieces of roughly equal complexity 348* Q = [ X*S*(B2 + S*(B4 + ... + S*B12)) ] + 349* [ S*S*(B3 + S*(B5 + ... + S*B11)) ] 350* 351* Step 10. Calculate exp(X)-1 for |X| >= 70 log 2. 352* 10.1 If X >= 70log2 , exp(X) - 1 = exp(X) for all practical 353* purposes. Therefore, go to Step 1 of setox. 354* 10.2 If X <= -70log2, exp(X) - 1 = -1 for all practical purposes. 355* ans := -1 356* Restore user FPCR 357* Return ans := ans + 2^(-126). Exit. 358* Notes: 10.2 will always create an inexact and return -1 + tiny 359* in the user rounding precision and mode. 360* 361 362setox IDNT 2,1 Motorola 040 Floating Point Software Package 363 364 section 8 365 366 include fpsp.h 367 368L2 DC.L $3FDC0000,$82E30865,$4361C4C6,$00000000 369 370EXPA3 DC.L $3FA55555,$55554431 371EXPA2 DC.L $3FC55555,$55554018 372 373HUGE DC.L $7FFE0000,$FFFFFFFF,$FFFFFFFF,$00000000 374TINY DC.L $00010000,$FFFFFFFF,$FFFFFFFF,$00000000 375 376EM1A4 DC.L $3F811111,$11174385 377EM1A3 DC.L $3FA55555,$55554F5A 378 379EM1A2 DC.L $3FC55555,$55555555,$00000000,$00000000 380 381EM1B8 DC.L $3EC71DE3,$A5774682 382EM1B7 DC.L $3EFA01A0,$19D7CB68 383 384EM1B6 DC.L $3F2A01A0,$1A019DF3 385EM1B5 DC.L $3F56C16C,$16C170E2 386 387EM1B4 DC.L $3F811111,$11111111 388EM1B3 DC.L $3FA55555,$55555555 389 390EM1B2 DC.L $3FFC0000,$AAAAAAAA,$AAAAAAAB 391 DC.L $00000000 392 393TWO140 DC.L $48B00000,$00000000 394TWON140 DC.L $37300000,$00000000 395 396EXPTBL 397 DC.L $3FFF0000,$80000000,$00000000,$00000000 398 DC.L $3FFF0000,$8164D1F3,$BC030774,$9F841A9B 399 DC.L $3FFF0000,$82CD8698,$AC2BA1D8,$9FC1D5B9 400 DC.L $3FFF0000,$843A28C3,$ACDE4048,$A0728369 401 DC.L $3FFF0000,$85AAC367,$CC487B14,$1FC5C95C 402 DC.L $3FFF0000,$871F6196,$9E8D1010,$1EE85C9F 403 DC.L $3FFF0000,$88980E80,$92DA8528,$9FA20729 404 DC.L $3FFF0000,$8A14D575,$496EFD9C,$A07BF9AF 405 DC.L $3FFF0000,$8B95C1E3,$EA8BD6E8,$A0020DCF 406 DC.L $3FFF0000,$8D1ADF5B,$7E5BA9E4,$205A63DA 407 DC.L $3FFF0000,$8EA4398B,$45CD53C0,$1EB70051 408 DC.L $3FFF0000,$9031DC43,$1466B1DC,$1F6EB029 409 DC.L $3FFF0000,$91C3D373,$AB11C338,$A0781494 410 DC.L $3FFF0000,$935A2B2F,$13E6E92C,$9EB319B0 411 DC.L $3FFF0000,$94F4EFA8,$FEF70960,$2017457D 412 DC.L $3FFF0000,$96942D37,$20185A00,$1F11D537 413 DC.L $3FFF0000,$9837F051,$8DB8A970,$9FB952DD 414 DC.L $3FFF0000,$99E04593,$20B7FA64,$1FE43087 415 DC.L $3FFF0000,$9B8D39B9,$D54E5538,$1FA2A818 416 DC.L $3FFF0000,$9D3ED9A7,$2CFFB750,$1FDE494D 417 DC.L $3FFF0000,$9EF53260,$91A111AC,$20504890 418 DC.L $3FFF0000,$A0B0510F,$B9714FC4,$A073691C 419 DC.L $3FFF0000,$A2704303,$0C496818,$1F9B7A05 420 DC.L $3FFF0000,$A43515AE,$09E680A0,$A0797126 421 DC.L $3FFF0000,$A5FED6A9,$B15138EC,$A071A140 422 DC.L $3FFF0000,$A7CD93B4,$E9653568,$204F62DA 423 DC.L $3FFF0000,$A9A15AB4,$EA7C0EF8,$1F283C4A 424 DC.L $3FFF0000,$AB7A39B5,$A93ED338,$9F9A7FDC 425 DC.L $3FFF0000,$AD583EEA,$42A14AC8,$A05B3FAC 426 DC.L $3FFF0000,$AF3B78AD,$690A4374,$1FDF2610 427 DC.L $3FFF0000,$B123F581,$D2AC2590,$9F705F90 428 DC.L $3FFF0000,$B311C412,$A9112488,$201F678A 429 DC.L $3FFF0000,$B504F333,$F9DE6484,$1F32FB13 430 DC.L $3FFF0000,$B6FD91E3,$28D17790,$20038B30 431 DC.L $3FFF0000,$B8FBAF47,$62FB9EE8,$200DC3CC 432 DC.L $3FFF0000,$BAFF5AB2,$133E45FC,$9F8B2AE6 433 DC.L $3FFF0000,$BD08A39F,$580C36C0,$A02BBF70 434 DC.L $3FFF0000,$BF1799B6,$7A731084,$A00BF518 435 DC.L $3FFF0000,$C12C4CCA,$66709458,$A041DD41 436 DC.L $3FFF0000,$C346CCDA,$24976408,$9FDF137B 437 DC.L $3FFF0000,$C5672A11,$5506DADC,$201F1568 438 DC.L $3FFF0000,$C78D74C8,$ABB9B15C,$1FC13A2E 439 DC.L $3FFF0000,$C9B9BD86,$6E2F27A4,$A03F8F03 440 DC.L $3FFF0000,$CBEC14FE,$F2727C5C,$1FF4907D 441 DC.L $3FFF0000,$CE248C15,$1F8480E4,$9E6E53E4 442 DC.L $3FFF0000,$D06333DA,$EF2B2594,$1FD6D45C 443 DC.L $3FFF0000,$D2A81D91,$F12AE45C,$A076EDB9 444 DC.L $3FFF0000,$D4F35AAB,$CFEDFA20,$9FA6DE21 445 DC.L $3FFF0000,$D744FCCA,$D69D6AF4,$1EE69A2F 446 DC.L $3FFF0000,$D99D15C2,$78AFD7B4,$207F439F 447 DC.L $3FFF0000,$DBFBB797,$DAF23754,$201EC207 448 DC.L $3FFF0000,$DE60F482,$5E0E9124,$9E8BE175 449 DC.L $3FFF0000,$E0CCDEEC,$2A94E110,$20032C4B 450 DC.L $3FFF0000,$E33F8972,$BE8A5A50,$2004DFF5 451 DC.L $3FFF0000,$E5B906E7,$7C8348A8,$1E72F47A 452 DC.L $3FFF0000,$E8396A50,$3C4BDC68,$1F722F22 453 DC.L $3FFF0000,$EAC0C6E7,$DD243930,$A017E945 454 DC.L $3FFF0000,$ED4F301E,$D9942B84,$1F401A5B 455 DC.L $3FFF0000,$EFE4B99B,$DCDAF5CC,$9FB9A9E3 456 DC.L $3FFF0000,$F281773C,$59FFB138,$20744C05 457 DC.L $3FFF0000,$F5257D15,$2486CC2C,$1F773A19 458 DC.L $3FFF0000,$F7D0DF73,$0AD13BB8,$1FFE90D5 459 DC.L $3FFF0000,$FA83B2DB,$722A033C,$A041ED22 460 DC.L $3FFF0000,$FD3E0C0C,$F486C174,$1F853F3A 461 462ADJFLAG equ L_SCR2 463SCALE equ FP_SCR1 464ADJSCALE equ FP_SCR2 465SC equ FP_SCR3 466ONEBYSC equ FP_SCR4 467 468 xref t_frcinx 469 xref t_extdnrm 470 xref t_unfl 471 xref t_ovfl 472 473 xdef setoxd 474setoxd: 475*--entry point for EXP(X), X is denormalized 476 MOVE.L (a0),d0 477 ANDI.L #$80000000,d0 478 ORI.L #$00800000,d0 ...sign(X)*2^(-126) 479 MOVE.L d0,-(sp) 480 FMOVE.S #:3F800000,fp0 481 fmove.l d1,fpcr 482 FADD.S (sp)+,fp0 483 bra t_frcinx 484 485 xdef setox 486setox: 487*--entry point for EXP(X), here X is finite, non-zero, and not NaN's 488 489*--Step 1. 490 MOVE.L (a0),d0 ...load part of input X 491 ANDI.L #$7FFF0000,d0 ...biased expo. of X 492 CMPI.L #$3FBE0000,d0 ...2^(-65) 493 BGE.B EXPC1 ...normal case 494 BRA.W EXPSM 495 496EXPC1: 497*--The case |X| >= 2^(-65) 498 MOVE.W 4(a0),d0 ...expo. and partial sig. of |X| 499 CMPI.L #$400CB167,d0 ...16380 log2 trunc. 16 bits 500 BLT.B EXPMAIN ...normal case 501 BRA.W EXPBIG 502 503EXPMAIN: 504*--Step 2. 505*--This is the normal branch: 2^(-65) <= |X| < 16380 log2. 506 FMOVE.X (a0),fp0 ...load input from (a0) 507 508 FMOVE.X fp0,fp1 509 FMUL.S #:42B8AA3B,fp0 ...64/log2 * X 510 fmovem.x fp2/fp3,-(a7) ...save fp2 511 CLR.L ADJFLAG(a6) 512 FMOVE.L fp0,d0 ...N = int( X * 64/log2 ) 513 LEA EXPTBL,a1 514 FMOVE.L d0,fp0 ...convert to floating-format 515 516 MOVE.L d0,L_SCR1(a6) ...save N temporarily 517 ANDI.L #$3F,d0 ...D0 is J = N mod 64 518 LSL.L #4,d0 519 ADDA.L d0,a1 ...address of 2^(J/64) 520 MOVE.L L_SCR1(a6),d0 521 ASR.L #6,d0 ...D0 is M 522 ADDI.W #$3FFF,d0 ...biased expo. of 2^(M) 523 MOVE.W L2,L_SCR1(a6) ...prefetch L2, no need in CB 524 525EXPCONT1: 526*--Step 3. 527*--fp1,fp2 saved on the stack. fp0 is N, fp1 is X, 528*--a0 points to 2^(J/64), D0 is biased expo. of 2^(M) 529 FMOVE.X fp0,fp2 530 FMUL.S #:BC317218,fp0 ...N * L1, L1 = lead(-log2/64) 531 FMUL.X L2,fp2 ...N * L2, L1+L2 = -log2/64 532 FADD.X fp1,fp0 ...X + N*L1 533 FADD.X fp2,fp0 ...fp0 is R, reduced arg. 534* MOVE.W #$3FA5,EXPA3 ...load EXPA3 in cache 535 536*--Step 4. 537*--WE NOW COMPUTE EXP(R)-1 BY A POLYNOMIAL 538*-- R + R*R*(A1 + R*(A2 + R*(A3 + R*(A4 + R*A5)))) 539*--TO FULLY UTILIZE THE PIPELINE, WE COMPUTE S = R*R 540*--[R+R*S*(A2+S*A4)] + [S*(A1+S*(A3+S*A5))] 541 542 FMOVE.X fp0,fp1 543 FMUL.X fp1,fp1 ...fp1 IS S = R*R 544 545 FMOVE.S #:3AB60B70,fp2 ...fp2 IS A5 546* CLR.W 2(a1) ...load 2^(J/64) in cache 547 548 FMUL.X fp1,fp2 ...fp2 IS S*A5 549 FMOVE.X fp1,fp3 550 FMUL.S #:3C088895,fp3 ...fp3 IS S*A4 551 552 FADD.D EXPA3,fp2 ...fp2 IS A3+S*A5 553 FADD.D EXPA2,fp3 ...fp3 IS A2+S*A4 554 555 FMUL.X fp1,fp2 ...fp2 IS S*(A3+S*A5) 556 MOVE.W d0,SCALE(a6) ...SCALE is 2^(M) in extended 557 clr.w SCALE+2(a6) 558 move.l #$80000000,SCALE+4(a6) 559 clr.l SCALE+8(a6) 560 561 FMUL.X fp1,fp3 ...fp3 IS S*(A2+S*A4) 562 563 FADD.S #:3F000000,fp2 ...fp2 IS A1+S*(A3+S*A5) 564 FMUL.X fp0,fp3 ...fp3 IS R*S*(A2+S*A4) 565 566 FMUL.X fp1,fp2 ...fp2 IS S*(A1+S*(A3+S*A5)) 567 FADD.X fp3,fp0 ...fp0 IS R+R*S*(A2+S*A4), 568* ...fp3 released 569 570 FMOVE.X (a1)+,fp1 ...fp1 is lead. pt. of 2^(J/64) 571 FADD.X fp2,fp0 ...fp0 is EXP(R) - 1 572* ...fp2 released 573 574*--Step 5 575*--final reconstruction process 576*--EXP(X) = 2^M * ( 2^(J/64) + 2^(J/64)*(EXP(R)-1) ) 577 578 FMUL.X fp1,fp0 ...2^(J/64)*(Exp(R)-1) 579 fmovem.x (a7)+,fp2/fp3 ...fp2 restored 580 FADD.S (a1),fp0 ...accurate 2^(J/64) 581 582 FADD.X fp1,fp0 ...2^(J/64) + 2^(J/64)*... 583 MOVE.L ADJFLAG(a6),d0 584 585*--Step 6 586 TST.L D0 587 BEQ.B NORMAL 588ADJUST: 589 FMUL.X ADJSCALE(a6),fp0 590NORMAL: 591 FMOVE.L d1,FPCR ...restore user FPCR 592 FMUL.X SCALE(a6),fp0 ...multiply 2^(M) 593 bra t_frcinx 594 595EXPSM: 596*--Step 7 597 FMOVEM.X (a0),fp0 ...in case X is denormalized 598 FMOVE.L d1,FPCR 599 FADD.S #:3F800000,fp0 ...1+X in user mode 600 bra t_frcinx 601 602EXPBIG: 603*--Step 8 604 CMPI.L #$400CB27C,d0 ...16480 log2 605 BGT.B EXP2BIG 606*--Steps 8.2 -- 8.6 607 FMOVE.X (a0),fp0 ...load input from (a0) 608 609 FMOVE.X fp0,fp1 610 FMUL.S #:42B8AA3B,fp0 ...64/log2 * X 611 fmovem.x fp2/fp3,-(a7) ...save fp2 612 MOVE.L #1,ADJFLAG(a6) 613 FMOVE.L fp0,d0 ...N = int( X * 64/log2 ) 614 LEA EXPTBL,a1 615 FMOVE.L d0,fp0 ...convert to floating-format 616 MOVE.L d0,L_SCR1(a6) ...save N temporarily 617 ANDI.L #$3F,d0 ...D0 is J = N mod 64 618 LSL.L #4,d0 619 ADDA.L d0,a1 ...address of 2^(J/64) 620 MOVE.L L_SCR1(a6),d0 621 ASR.L #6,d0 ...D0 is K 622 MOVE.L d0,L_SCR1(a6) ...save K temporarily 623 ASR.L #1,d0 ...D0 is M1 624 SUB.L d0,L_SCR1(a6) ...a1 is M 625 ADDI.W #$3FFF,d0 ...biased expo. of 2^(M1) 626 MOVE.W d0,ADJSCALE(a6) ...ADJSCALE := 2^(M1) 627 clr.w ADJSCALE+2(a6) 628 move.l #$80000000,ADJSCALE+4(a6) 629 clr.l ADJSCALE+8(a6) 630 MOVE.L L_SCR1(a6),d0 ...D0 is M 631 ADDI.W #$3FFF,d0 ...biased expo. of 2^(M) 632 BRA.W EXPCONT1 ...go back to Step 3 633 634EXP2BIG: 635*--Step 9 636 FMOVE.L d1,FPCR 637 MOVE.L (a0),d0 638 bclr.b #sign_bit,(a0) ...setox always returns positive 639 TST.L d0 640 BLT t_unfl 641 BRA t_ovfl 642 643 xdef setoxm1d 644setoxm1d: 645*--entry point for EXPM1(X), here X is denormalized 646*--Step 0. 647 bra t_extdnrm 648 649 650 xdef setoxm1 651setoxm1: 652*--entry point for EXPM1(X), here X is finite, non-zero, non-NaN 653 654*--Step 1. 655*--Step 1.1 656 MOVE.L (a0),d0 ...load part of input X 657 ANDI.L #$7FFF0000,d0 ...biased expo. of X 658 CMPI.L #$3FFD0000,d0 ...1/4 659 BGE.B EM1CON1 ...|X| >= 1/4 660 BRA.W EM1SM 661 662EM1CON1: 663*--Step 1.3 664*--The case |X| >= 1/4 665 MOVE.W 4(a0),d0 ...expo. and partial sig. of |X| 666 CMPI.L #$4004C215,d0 ...70log2 rounded up to 16 bits 667 BLE.B EM1MAIN ...1/4 <= |X| <= 70log2 668 BRA.W EM1BIG 669 670EM1MAIN: 671*--Step 2. 672*--This is the case: 1/4 <= |X| <= 70 log2. 673 FMOVE.X (a0),fp0 ...load input from (a0) 674 675 FMOVE.X fp0,fp1 676 FMUL.S #:42B8AA3B,fp0 ...64/log2 * X 677 fmovem.x fp2/fp3,-(a7) ...save fp2 678* MOVE.W #$3F81,EM1A4 ...prefetch in CB mode 679 FMOVE.L fp0,d0 ...N = int( X * 64/log2 ) 680 LEA EXPTBL,a1 681 FMOVE.L d0,fp0 ...convert to floating-format 682 683 MOVE.L d0,L_SCR1(a6) ...save N temporarily 684 ANDI.L #$3F,d0 ...D0 is J = N mod 64 685 LSL.L #4,d0 686 ADDA.L d0,a1 ...address of 2^(J/64) 687 MOVE.L L_SCR1(a6),d0 688 ASR.L #6,d0 ...D0 is M 689 MOVE.L d0,L_SCR1(a6) ...save a copy of M 690* MOVE.W #$3FDC,L2 ...prefetch L2 in CB mode 691 692*--Step 3. 693*--fp1,fp2 saved on the stack. fp0 is N, fp1 is X, 694*--a0 points to 2^(J/64), D0 and a1 both contain M 695 FMOVE.X fp0,fp2 696 FMUL.S #:BC317218,fp0 ...N * L1, L1 = lead(-log2/64) 697 FMUL.X L2,fp2 ...N * L2, L1+L2 = -log2/64 698 FADD.X fp1,fp0 ...X + N*L1 699 FADD.X fp2,fp0 ...fp0 is R, reduced arg. 700* MOVE.W #$3FC5,EM1A2 ...load EM1A2 in cache 701 ADDI.W #$3FFF,d0 ...D0 is biased expo. of 2^M 702 703*--Step 4. 704*--WE NOW COMPUTE EXP(R)-1 BY A POLYNOMIAL 705*-- R + R*R*(A1 + R*(A2 + R*(A3 + R*(A4 + R*(A5 + R*A6))))) 706*--TO FULLY UTILIZE THE PIPELINE, WE COMPUTE S = R*R 707*--[R*S*(A2+S*(A4+S*A6))] + [R+S*(A1+S*(A3+S*A5))] 708 709 FMOVE.X fp0,fp1 710 FMUL.X fp1,fp1 ...fp1 IS S = R*R 711 712 FMOVE.S #:3950097B,fp2 ...fp2 IS a6 713* CLR.W 2(a1) ...load 2^(J/64) in cache 714 715 FMUL.X fp1,fp2 ...fp2 IS S*A6 716 FMOVE.X fp1,fp3 717 FMUL.S #:3AB60B6A,fp3 ...fp3 IS S*A5 718 719 FADD.D EM1A4,fp2 ...fp2 IS A4+S*A6 720 FADD.D EM1A3,fp3 ...fp3 IS A3+S*A5 721 MOVE.W d0,SC(a6) ...SC is 2^(M) in extended 722 clr.w SC+2(a6) 723 move.l #$80000000,SC+4(a6) 724 clr.l SC+8(a6) 725 726 FMUL.X fp1,fp2 ...fp2 IS S*(A4+S*A6) 727 MOVE.L L_SCR1(a6),d0 ...D0 is M 728 NEG.W D0 ...D0 is -M 729 FMUL.X fp1,fp3 ...fp3 IS S*(A3+S*A5) 730 ADDI.W #$3FFF,d0 ...biased expo. of 2^(-M) 731 FADD.D EM1A2,fp2 ...fp2 IS A2+S*(A4+S*A6) 732 FADD.S #:3F000000,fp3 ...fp3 IS A1+S*(A3+S*A5) 733 734 FMUL.X fp1,fp2 ...fp2 IS S*(A2+S*(A4+S*A6)) 735 ORI.W #$8000,d0 ...signed/expo. of -2^(-M) 736 MOVE.W d0,ONEBYSC(a6) ...OnebySc is -2^(-M) 737 clr.w ONEBYSC+2(a6) 738 move.l #$80000000,ONEBYSC+4(a6) 739 clr.l ONEBYSC+8(a6) 740 FMUL.X fp3,fp1 ...fp1 IS S*(A1+S*(A3+S*A5)) 741* ...fp3 released 742 743 FMUL.X fp0,fp2 ...fp2 IS R*S*(A2+S*(A4+S*A6)) 744 FADD.X fp1,fp0 ...fp0 IS R+S*(A1+S*(A3+S*A5)) 745* ...fp1 released 746 747 FADD.X fp2,fp0 ...fp0 IS EXP(R)-1 748* ...fp2 released 749 fmovem.x (a7)+,fp2/fp3 ...fp2 restored 750 751*--Step 5 752*--Compute 2^(J/64)*p 753 754 FMUL.X (a1),fp0 ...2^(J/64)*(Exp(R)-1) 755 756*--Step 6 757*--Step 6.1 758 MOVE.L L_SCR1(a6),d0 ...retrieve M 759 CMPI.L #63,d0 760 BLE.B MLE63 761*--Step 6.2 M >= 64 762 FMOVE.S 12(a1),fp1 ...fp1 is t 763 FADD.X ONEBYSC(a6),fp1 ...fp1 is t+OnebySc 764 FADD.X fp1,fp0 ...p+(t+OnebySc), fp1 released 765 FADD.X (a1),fp0 ...T+(p+(t+OnebySc)) 766 BRA.B EM1SCALE 767MLE63: 768*--Step 6.3 M <= 63 769 CMPI.L #-3,d0 770 BGE.B MGEN3 771MLTN3: 772*--Step 6.4 M <= -4 773 FADD.S 12(a1),fp0 ...p+t 774 FADD.X (a1),fp0 ...T+(p+t) 775 FADD.X ONEBYSC(a6),fp0 ...OnebySc + (T+(p+t)) 776 BRA.B EM1SCALE 777MGEN3: 778*--Step 6.5 -3 <= M <= 63 779 FMOVE.X (a1)+,fp1 ...fp1 is T 780 FADD.S (a1),fp0 ...fp0 is p+t 781 FADD.X ONEBYSC(a6),fp1 ...fp1 is T+OnebySc 782 FADD.X fp1,fp0 ...(T+OnebySc)+(p+t) 783 784EM1SCALE: 785*--Step 6.6 786 FMOVE.L d1,FPCR 787 FMUL.X SC(a6),fp0 788 789 bra t_frcinx 790 791EM1SM: 792*--Step 7 |X| < 1/4. 793 CMPI.L #$3FBE0000,d0 ...2^(-65) 794 BGE.B EM1POLY 795 796EM1TINY: 797*--Step 8 |X| < 2^(-65) 798 CMPI.L #$00330000,d0 ...2^(-16312) 799 BLT.B EM12TINY 800*--Step 8.2 801 MOVE.L #$80010000,SC(a6) ...SC is -2^(-16382) 802 move.l #$80000000,SC+4(a6) 803 clr.l SC+8(a6) 804 FMOVE.X (a0),fp0 805 FMOVE.L d1,FPCR 806 FADD.X SC(a6),fp0 807 808 bra t_frcinx 809 810EM12TINY: 811*--Step 8.3 812 FMOVE.X (a0),fp0 813 FMUL.D TWO140,fp0 814 MOVE.L #$80010000,SC(a6) 815 move.l #$80000000,SC+4(a6) 816 clr.l SC+8(a6) 817 FADD.X SC(a6),fp0 818 FMOVE.L d1,FPCR 819 FMUL.D TWON140,fp0 820 821 bra t_frcinx 822 823EM1POLY: 824*--Step 9 exp(X)-1 by a simple polynomial 825 FMOVE.X (a0),fp0 ...fp0 is X 826 FMUL.X fp0,fp0 ...fp0 is S := X*X 827 fmovem.x fp2/fp3,-(a7) ...save fp2 828 FMOVE.S #:2F30CAA8,fp1 ...fp1 is B12 829 FMUL.X fp0,fp1 ...fp1 is S*B12 830 FMOVE.S #:310F8290,fp2 ...fp2 is B11 831 FADD.S #:32D73220,fp1 ...fp1 is B10+S*B12 832 833 FMUL.X fp0,fp2 ...fp2 is S*B11 834 FMUL.X fp0,fp1 ...fp1 is S*(B10 + ... 835 836 FADD.S #:3493F281,fp2 ...fp2 is B9+S*... 837 FADD.D EM1B8,fp1 ...fp1 is B8+S*... 838 839 FMUL.X fp0,fp2 ...fp2 is S*(B9+... 840 FMUL.X fp0,fp1 ...fp1 is S*(B8+... 841 842 FADD.D EM1B7,fp2 ...fp2 is B7+S*... 843 FADD.D EM1B6,fp1 ...fp1 is B6+S*... 844 845 FMUL.X fp0,fp2 ...fp2 is S*(B7+... 846 FMUL.X fp0,fp1 ...fp1 is S*(B6+... 847 848 FADD.D EM1B5,fp2 ...fp2 is B5+S*... 849 FADD.D EM1B4,fp1 ...fp1 is B4+S*... 850 851 FMUL.X fp0,fp2 ...fp2 is S*(B5+... 852 FMUL.X fp0,fp1 ...fp1 is S*(B4+... 853 854 FADD.D EM1B3,fp2 ...fp2 is B3+S*... 855 FADD.X EM1B2,fp1 ...fp1 is B2+S*... 856 857 FMUL.X fp0,fp2 ...fp2 is S*(B3+... 858 FMUL.X fp0,fp1 ...fp1 is S*(B2+... 859 860 FMUL.X fp0,fp2 ...fp2 is S*S*(B3+...) 861 FMUL.X (a0),fp1 ...fp1 is X*S*(B2... 862 863 FMUL.S #:3F000000,fp0 ...fp0 is S*B1 864 FADD.X fp2,fp1 ...fp1 is Q 865* ...fp2 released 866 867 fmovem.x (a7)+,fp2/fp3 ...fp2 restored 868 869 FADD.X fp1,fp0 ...fp0 is S*B1+Q 870* ...fp1 released 871 872 FMOVE.L d1,FPCR 873 FADD.X (a0),fp0 874 875 bra t_frcinx 876 877EM1BIG: 878*--Step 10 |X| > 70 log2 879 MOVE.L (a0),d0 880 TST.L d0 881 BGT.W EXPC1 882*--Step 10.2 883 FMOVE.S #:BF800000,fp0 ...fp0 is -1 884 FMOVE.L d1,FPCR 885 FADD.S #:00800000,fp0 ...-1 + 2^(-126) 886 887 bra t_frcinx 888 889 end 890