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 | THIS IS UNPUBLISHED PROPRIETARY SOURCE CODE OF MOTOROLA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