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