forked from torvalds/linux
-
Notifications
You must be signed in to change notification settings - Fork 0
/
stan.S
454 lines (359 loc) · 13 KB
/
stan.S
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
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
|
| stan.sa 3.3 7/29/91
|
| The entry point stan computes the tangent of
| an input argument;
| stand does the same except for denormalized input.
|
| Input: Double-extended number X in location pointed to
| by address register a0.
|
| Output: The value tan(X) returned in floating-point register Fp0.
|
| Accuracy and Monotonicity: The returned result is within 3 ulp 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 sTAN takes approximately 170 cycles for
| input argument X such that |X| < 15Pi, which is the usual
| situation.
|
| Algorithm:
|
| 1. If |X| >= 15Pi or |X| < 2**(-40), go to 6.
|
| 2. Decompose X as X = N(Pi/2) + r where |r| <= Pi/4. Let
| k = N mod 2, so in particular, k = 0 or 1.
|
| 3. If k is odd, go to 5.
|
| 4. (k is even) Tan(X) = tan(r) and tan(r) is approximated by a
| rational function U/V where
| U = r + r*s*(P1 + s*(P2 + s*P3)), and
| V = 1 + s*(Q1 + s*(Q2 + s*(Q3 + s*Q4))), s = r*r.
| Exit.
|
| 4. (k is odd) Tan(X) = -cot(r). Since tan(r) is approximated by a
| rational function U/V where
| U = r + r*s*(P1 + s*(P2 + s*P3)), and
| V = 1 + s*(Q1 + s*(Q2 + s*(Q3 + s*Q4))), s = r*r,
| -Cot(r) = -V/U. Exit.
|
| 6. If |X| > 1, go to 8.
|
| 7. (|X|<2**(-40)) Tan(X) = X. Exit.
|
| 8. Overwrite X by X := X rem 2Pi. Now that |X| <= Pi, go back to 2.
|
| Copyright (C) Motorola, Inc. 1990
| All Rights Reserved
|
| For details on the license for this file, please see the
| file, README, in this same directory.
|STAN idnt 2,1 | Motorola 040 Floating Point Software Package
|section 8
#include "fpsp.h"
BOUNDS1: .long 0x3FD78000,0x4004BC7E
TWOBYPI: .long 0x3FE45F30,0x6DC9C883
TANQ4: .long 0x3EA0B759,0xF50F8688
TANP3: .long 0xBEF2BAA5,0xA8924F04
TANQ3: .long 0xBF346F59,0xB39BA65F,0x00000000,0x00000000
TANP2: .long 0x3FF60000,0xE073D3FC,0x199C4A00,0x00000000
TANQ2: .long 0x3FF90000,0xD23CD684,0x15D95FA1,0x00000000
TANP1: .long 0xBFFC0000,0x8895A6C5,0xFB423BCA,0x00000000
TANQ1: .long 0xBFFD0000,0xEEF57E0D,0xA84BC8CE,0x00000000
INVTWOPI: .long 0x3FFC0000,0xA2F9836E,0x4E44152A,0x00000000
TWOPI1: .long 0x40010000,0xC90FDAA2,0x00000000,0x00000000
TWOPI2: .long 0x3FDF0000,0x85A308D4,0x00000000,0x00000000
|--N*PI/2, -32 <= N <= 32, IN A LEADING TERM IN EXT. AND TRAILING
|--TERM IN SGL. NOTE THAT PI IS 64-BIT LONG, THUS N*PI/2 IS AT
|--MOST 69 BITS LONG.
.global PITBL
PITBL:
.long 0xC0040000,0xC90FDAA2,0x2168C235,0x21800000
.long 0xC0040000,0xC2C75BCD,0x105D7C23,0xA0D00000
.long 0xC0040000,0xBC7EDCF7,0xFF523611,0xA1E80000
.long 0xC0040000,0xB6365E22,0xEE46F000,0x21480000
.long 0xC0040000,0xAFEDDF4D,0xDD3BA9EE,0xA1200000
.long 0xC0040000,0xA9A56078,0xCC3063DD,0x21FC0000
.long 0xC0040000,0xA35CE1A3,0xBB251DCB,0x21100000
.long 0xC0040000,0x9D1462CE,0xAA19D7B9,0xA1580000
.long 0xC0040000,0x96CBE3F9,0x990E91A8,0x21E00000
.long 0xC0040000,0x90836524,0x88034B96,0x20B00000
.long 0xC0040000,0x8A3AE64F,0x76F80584,0xA1880000
.long 0xC0040000,0x83F2677A,0x65ECBF73,0x21C40000
.long 0xC0030000,0xFB53D14A,0xA9C2F2C2,0x20000000
.long 0xC0030000,0xEEC2D3A0,0x87AC669F,0x21380000
.long 0xC0030000,0xE231D5F6,0x6595DA7B,0xA1300000
.long 0xC0030000,0xD5A0D84C,0x437F4E58,0x9FC00000
.long 0xC0030000,0xC90FDAA2,0x2168C235,0x21000000
.long 0xC0030000,0xBC7EDCF7,0xFF523611,0xA1680000
.long 0xC0030000,0xAFEDDF4D,0xDD3BA9EE,0xA0A00000
.long 0xC0030000,0xA35CE1A3,0xBB251DCB,0x20900000
.long 0xC0030000,0x96CBE3F9,0x990E91A8,0x21600000
.long 0xC0030000,0x8A3AE64F,0x76F80584,0xA1080000
.long 0xC0020000,0xFB53D14A,0xA9C2F2C2,0x1F800000
.long 0xC0020000,0xE231D5F6,0x6595DA7B,0xA0B00000
.long 0xC0020000,0xC90FDAA2,0x2168C235,0x20800000
.long 0xC0020000,0xAFEDDF4D,0xDD3BA9EE,0xA0200000
.long 0xC0020000,0x96CBE3F9,0x990E91A8,0x20E00000
.long 0xC0010000,0xFB53D14A,0xA9C2F2C2,0x1F000000
.long 0xC0010000,0xC90FDAA2,0x2168C235,0x20000000
.long 0xC0010000,0x96CBE3F9,0x990E91A8,0x20600000
.long 0xC0000000,0xC90FDAA2,0x2168C235,0x1F800000
.long 0xBFFF0000,0xC90FDAA2,0x2168C235,0x1F000000
.long 0x00000000,0x00000000,0x00000000,0x00000000
.long 0x3FFF0000,0xC90FDAA2,0x2168C235,0x9F000000
.long 0x40000000,0xC90FDAA2,0x2168C235,0x9F800000
.long 0x40010000,0x96CBE3F9,0x990E91A8,0xA0600000
.long 0x40010000,0xC90FDAA2,0x2168C235,0xA0000000
.long 0x40010000,0xFB53D14A,0xA9C2F2C2,0x9F000000
.long 0x40020000,0x96CBE3F9,0x990E91A8,0xA0E00000
.long 0x40020000,0xAFEDDF4D,0xDD3BA9EE,0x20200000
.long 0x40020000,0xC90FDAA2,0x2168C235,0xA0800000
.long 0x40020000,0xE231D5F6,0x6595DA7B,0x20B00000
.long 0x40020000,0xFB53D14A,0xA9C2F2C2,0x9F800000
.long 0x40030000,0x8A3AE64F,0x76F80584,0x21080000
.long 0x40030000,0x96CBE3F9,0x990E91A8,0xA1600000
.long 0x40030000,0xA35CE1A3,0xBB251DCB,0xA0900000
.long 0x40030000,0xAFEDDF4D,0xDD3BA9EE,0x20A00000
.long 0x40030000,0xBC7EDCF7,0xFF523611,0x21680000
.long 0x40030000,0xC90FDAA2,0x2168C235,0xA1000000
.long 0x40030000,0xD5A0D84C,0x437F4E58,0x1FC00000
.long 0x40030000,0xE231D5F6,0x6595DA7B,0x21300000
.long 0x40030000,0xEEC2D3A0,0x87AC669F,0xA1380000
.long 0x40030000,0xFB53D14A,0xA9C2F2C2,0xA0000000
.long 0x40040000,0x83F2677A,0x65ECBF73,0xA1C40000
.long 0x40040000,0x8A3AE64F,0x76F80584,0x21880000
.long 0x40040000,0x90836524,0x88034B96,0xA0B00000
.long 0x40040000,0x96CBE3F9,0x990E91A8,0xA1E00000
.long 0x40040000,0x9D1462CE,0xAA19D7B9,0x21580000
.long 0x40040000,0xA35CE1A3,0xBB251DCB,0xA1100000
.long 0x40040000,0xA9A56078,0xCC3063DD,0xA1FC0000
.long 0x40040000,0xAFEDDF4D,0xDD3BA9EE,0x21200000
.long 0x40040000,0xB6365E22,0xEE46F000,0xA1480000
.long 0x40040000,0xBC7EDCF7,0xFF523611,0x21E80000
.long 0x40040000,0xC2C75BCD,0x105D7C23,0x20D00000
.long 0x40040000,0xC90FDAA2,0x2168C235,0xA1800000
.set INARG,FP_SCR4
.set TWOTO63,L_SCR1
.set ENDFLAG,L_SCR2
.set N,L_SCR3
| xref t_frcinx
|xref t_extdnrm
.global stand
stand:
|--TAN(X) = X FOR DENORMALIZED X
bra t_extdnrm
.global stan
stan:
fmovex (%a0),%fp0 | ...LOAD INPUT
movel (%a0),%d0
movew 4(%a0),%d0
andil #0x7FFFFFFF,%d0
cmpil #0x3FD78000,%d0 | ...|X| >= 2**(-40)?
bges TANOK1
bra TANSM
TANOK1:
cmpil #0x4004BC7E,%d0 | ...|X| < 15 PI?
blts TANMAIN
bra REDUCEX
TANMAIN:
|--THIS IS THE USUAL CASE, |X| <= 15 PI.
|--THE ARGUMENT REDUCTION IS DONE BY TABLE LOOK UP.
fmovex %fp0,%fp1
fmuld TWOBYPI,%fp1 | ...X*2/PI
|--HIDE THE NEXT TWO INSTRUCTIONS
leal PITBL+0x200,%a1 | ...TABLE OF N*PI/2, N = -32,...,32
|--FP1 IS NOW READY
fmovel %fp1,%d0 | ...CONVERT TO INTEGER
asll #4,%d0
addal %d0,%a1 | ...ADDRESS N*PIBY2 IN Y1, Y2
fsubx (%a1)+,%fp0 | ...X-Y1
|--HIDE THE NEXT ONE
fsubs (%a1),%fp0 | ...FP0 IS R = (X-Y1)-Y2
rorl #5,%d0
andil #0x80000000,%d0 | ...D0 WAS ODD IFF D0 < 0
TANCONT:
cmpil #0,%d0
blt NODD
fmovex %fp0,%fp1
fmulx %fp1,%fp1 | ...S = R*R
fmoved TANQ4,%fp3
fmoved TANP3,%fp2
fmulx %fp1,%fp3 | ...SQ4
fmulx %fp1,%fp2 | ...SP3
faddd TANQ3,%fp3 | ...Q3+SQ4
faddx TANP2,%fp2 | ...P2+SP3
fmulx %fp1,%fp3 | ...S(Q3+SQ4)
fmulx %fp1,%fp2 | ...S(P2+SP3)
faddx TANQ2,%fp3 | ...Q2+S(Q3+SQ4)
faddx TANP1,%fp2 | ...P1+S(P2+SP3)
fmulx %fp1,%fp3 | ...S(Q2+S(Q3+SQ4))
fmulx %fp1,%fp2 | ...S(P1+S(P2+SP3))
faddx TANQ1,%fp3 | ...Q1+S(Q2+S(Q3+SQ4))
fmulx %fp0,%fp2 | ...RS(P1+S(P2+SP3))
fmulx %fp3,%fp1 | ...S(Q1+S(Q2+S(Q3+SQ4)))
faddx %fp2,%fp0 | ...R+RS(P1+S(P2+SP3))
fadds #0x3F800000,%fp1 | ...1+S(Q1+...)
fmovel %d1,%fpcr |restore users exceptions
fdivx %fp1,%fp0 |last inst - possible exception set
bra t_frcinx
NODD:
fmovex %fp0,%fp1
fmulx %fp0,%fp0 | ...S = R*R
fmoved TANQ4,%fp3
fmoved TANP3,%fp2
fmulx %fp0,%fp3 | ...SQ4
fmulx %fp0,%fp2 | ...SP3
faddd TANQ3,%fp3 | ...Q3+SQ4
faddx TANP2,%fp2 | ...P2+SP3
fmulx %fp0,%fp3 | ...S(Q3+SQ4)
fmulx %fp0,%fp2 | ...S(P2+SP3)
faddx TANQ2,%fp3 | ...Q2+S(Q3+SQ4)
faddx TANP1,%fp2 | ...P1+S(P2+SP3)
fmulx %fp0,%fp3 | ...S(Q2+S(Q3+SQ4))
fmulx %fp0,%fp2 | ...S(P1+S(P2+SP3))
faddx TANQ1,%fp3 | ...Q1+S(Q2+S(Q3+SQ4))
fmulx %fp1,%fp2 | ...RS(P1+S(P2+SP3))
fmulx %fp3,%fp0 | ...S(Q1+S(Q2+S(Q3+SQ4)))
faddx %fp2,%fp1 | ...R+RS(P1+S(P2+SP3))
fadds #0x3F800000,%fp0 | ...1+S(Q1+...)
fmovex %fp1,-(%sp)
eoril #0x80000000,(%sp)
fmovel %d1,%fpcr |restore users exceptions
fdivx (%sp)+,%fp0 |last inst - possible exception set
bra t_frcinx
TANBORS:
|--IF |X| > 15PI, WE USE THE GENERAL ARGUMENT REDUCTION.
|--IF |X| < 2**(-40), RETURN X OR 1.
cmpil #0x3FFF8000,%d0
bgts REDUCEX
TANSM:
fmovex %fp0,-(%sp)
fmovel %d1,%fpcr |restore users exceptions
fmovex (%sp)+,%fp0 |last inst - possible exception set
bra t_frcinx
REDUCEX:
|--WHEN REDUCEX IS USED, THE CODE WILL INEVITABLY BE SLOW.
|--THIS REDUCTION METHOD, HOWEVER, IS MUCH FASTER THAN USING
|--THE REMAINDER INSTRUCTION WHICH IS NOW IN SOFTWARE.
fmovemx %fp2-%fp5,-(%a7) | ...save FP2 through FP5
movel %d2,-(%a7)
fmoves #0x00000000,%fp1
|--If compact form of abs(arg) in d0=$7ffeffff, argument is so large that
|--there is a danger of unwanted overflow in first LOOP iteration. In this
|--case, reduce argument by one remainder step to make subsequent reduction
|--safe.
cmpil #0x7ffeffff,%d0 |is argument dangerously large?
bnes LOOP
movel #0x7ffe0000,FP_SCR2(%a6) |yes
| ;create 2**16383*PI/2
movel #0xc90fdaa2,FP_SCR2+4(%a6)
clrl FP_SCR2+8(%a6)
ftstx %fp0 |test sign of argument
movel #0x7fdc0000,FP_SCR3(%a6) |create low half of 2**16383*
| ;PI/2 at FP_SCR3
movel #0x85a308d3,FP_SCR3+4(%a6)
clrl FP_SCR3+8(%a6)
fblt red_neg
orw #0x8000,FP_SCR2(%a6) |positive arg
orw #0x8000,FP_SCR3(%a6)
red_neg:
faddx FP_SCR2(%a6),%fp0 |high part of reduction is exact
fmovex %fp0,%fp1 |save high result in fp1
faddx FP_SCR3(%a6),%fp0 |low part of reduction
fsubx %fp0,%fp1 |determine low component of result
faddx FP_SCR3(%a6),%fp1 |fp0/fp1 are reduced argument.
|--ON ENTRY, FP0 IS X, ON RETURN, FP0 IS X REM PI/2, |X| <= PI/4.
|--integer quotient will be stored in N
|--Intermediate remainder is 66-bit long; (R,r) in (FP0,FP1)
LOOP:
fmovex %fp0,INARG(%a6) | ...+-2**K * F, 1 <= F < 2
movew INARG(%a6),%d0
movel %d0,%a1 | ...save a copy of D0
andil #0x00007FFF,%d0
subil #0x00003FFF,%d0 | ...D0 IS K
cmpil #28,%d0
bles LASTLOOP
CONTLOOP:
subil #27,%d0 | ...D0 IS L := K-27
movel #0,ENDFLAG(%a6)
bras WORK
LASTLOOP:
clrl %d0 | ...D0 IS L := 0
movel #1,ENDFLAG(%a6)
WORK:
|--FIND THE REMAINDER OF (R,r) W.R.T. 2**L * (PI/2). L IS SO CHOSEN
|--THAT INT( X * (2/PI) / 2**(L) ) < 2**29.
|--CREATE 2**(-L) * (2/PI), SIGN(INARG)*2**(63),
|--2**L * (PIby2_1), 2**L * (PIby2_2)
movel #0x00003FFE,%d2 | ...BIASED EXPO OF 2/PI
subl %d0,%d2 | ...BIASED EXPO OF 2**(-L)*(2/PI)
movel #0xA2F9836E,FP_SCR1+4(%a6)
movel #0x4E44152A,FP_SCR1+8(%a6)
movew %d2,FP_SCR1(%a6) | ...FP_SCR1 is 2**(-L)*(2/PI)
fmovex %fp0,%fp2
fmulx FP_SCR1(%a6),%fp2
|--WE MUST NOW FIND INT(FP2). SINCE WE NEED THIS VALUE IN
|--FLOATING POINT FORMAT, THE TWO FMOVE'S FMOVE.L FP <--> N
|--WILL BE TOO INEFFICIENT. THE WAY AROUND IT IS THAT
|--(SIGN(INARG)*2**63 + FP2) - SIGN(INARG)*2**63 WILL GIVE
|--US THE DESIRED VALUE IN FLOATING POINT.
|--HIDE SIX CYCLES OF INSTRUCTION
movel %a1,%d2
swap %d2
andil #0x80000000,%d2
oril #0x5F000000,%d2 | ...D2 IS SIGN(INARG)*2**63 IN SGL
movel %d2,TWOTO63(%a6)
movel %d0,%d2
addil #0x00003FFF,%d2 | ...BIASED EXPO OF 2**L * (PI/2)
|--FP2 IS READY
fadds TWOTO63(%a6),%fp2 | ...THE FRACTIONAL PART OF FP1 IS ROUNDED
|--HIDE 4 CYCLES OF INSTRUCTION; creating 2**(L)*Piby2_1 and 2**(L)*Piby2_2
movew %d2,FP_SCR2(%a6)
clrw FP_SCR2+2(%a6)
movel #0xC90FDAA2,FP_SCR2+4(%a6)
clrl FP_SCR2+8(%a6) | ...FP_SCR2 is 2**(L) * Piby2_1
|--FP2 IS READY
fsubs TWOTO63(%a6),%fp2 | ...FP2 is N
addil #0x00003FDD,%d0
movew %d0,FP_SCR3(%a6)
clrw FP_SCR3+2(%a6)
movel #0x85A308D3,FP_SCR3+4(%a6)
clrl FP_SCR3+8(%a6) | ...FP_SCR3 is 2**(L) * Piby2_2
movel ENDFLAG(%a6),%d0
|--We are now ready to perform (R+r) - N*P1 - N*P2, P1 = 2**(L) * Piby2_1 and
|--P2 = 2**(L) * Piby2_2
fmovex %fp2,%fp4
fmulx FP_SCR2(%a6),%fp4 | ...W = N*P1
fmovex %fp2,%fp5
fmulx FP_SCR3(%a6),%fp5 | ...w = N*P2
fmovex %fp4,%fp3
|--we want P+p = W+w but |p| <= half ulp of P
|--Then, we need to compute A := R-P and a := r-p
faddx %fp5,%fp3 | ...FP3 is P
fsubx %fp3,%fp4 | ...W-P
fsubx %fp3,%fp0 | ...FP0 is A := R - P
faddx %fp5,%fp4 | ...FP4 is p = (W-P)+w
fmovex %fp0,%fp3 | ...FP3 A
fsubx %fp4,%fp1 | ...FP1 is a := r - p
|--Now we need to normalize (A,a) to "new (R,r)" where R+r = A+a but
|--|r| <= half ulp of R.
faddx %fp1,%fp0 | ...FP0 is R := A+a
|--No need to calculate r if this is the last loop
cmpil #0,%d0
bgt RESTORE
|--Need to calculate r
fsubx %fp0,%fp3 | ...A-R
faddx %fp3,%fp1 | ...FP1 is r := (A-R)+a
bra LOOP
RESTORE:
fmovel %fp2,N(%a6)
movel (%a7)+,%d2
fmovemx (%a7)+,%fp2-%fp5
movel N(%a6),%d0
rorl #1,%d0
bra TANCONT
|end