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
|
|
| stwotox.sa 3.1 12/10/90
|
| stwotox --- 2**X
| stwotoxd --- 2**X for denormalized X
| stentox --- 10**X
| stentoxd --- 10**X for denormalized X
|
| Input: Double-extended number X in location pointed to
| by address register a0.
|
| Output: The function values are returned in Fp0.
|
| Accuracy and Monotonicity: The returned result is within 2 ulps 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 stwotox takes approximately 190 cycles and the
| program stentox takes approximately 200 cycles.
|
| Algorithm:
|
| twotox
| 1. If |X| > 16480, go to ExpBig.
|
| 2. If |X| < 2**(-70), go to ExpSm.
|
| 3. Decompose X as X = N/64 + r where |r| <= 1/128. Furthermore
| decompose N as
| N = 64(M + M') + j, j = 0,1,2,...,63.
|
| 4. Overwrite r := r * log2. Then
| 2**X = 2**(M') * 2**(M) * 2**(j/64) * exp(r).
| Go to expr to compute that expression.
|
| tentox
| 1. If |X| > 16480*log_10(2) (base 10 log of 2), go to ExpBig.
|
| 2. If |X| < 2**(-70), go to ExpSm.
|
| 3. Set y := X*log_2(10)*64 (base 2 log of 10). Set
| N := round-to-int(y). Decompose N as
| N = 64(M + M') + j, j = 0,1,2,...,63.
|
| 4. Define r as
| r := ((X - N*L1)-N*L2) * L10
| where L1, L2 are the leading and trailing parts of log_10(2)/64
| and L10 is the natural log of 10. Then
| 10**X = 2**(M') * 2**(M) * 2**(j/64) * exp(r).
| Go to expr to compute that expression.
|
| expr
| 1. Fetch 2**(j/64) from table as Fact1 and Fact2.
|
| 2. Overwrite Fact1 and Fact2 by
| Fact1 := 2**(M) * Fact1
| Fact2 := 2**(M) * Fact2
| Thus Fact1 + Fact2 = 2**(M) * 2**(j/64).
|
| 3. Calculate P where 1 + P approximates exp(r):
| P = r + r*r*(A1+r*(A2+...+r*A5)).
|
| 4. Let AdjFact := 2**(M'). Return
| AdjFact * ( Fact1 + ((Fact1*P) + Fact2) ).
| Exit.
|
| ExpBig
| 1. Generate overflow by Huge * Huge if X > 0; otherwise, generate
| underflow by Tiny * Tiny.
|
| ExpSm
| 1. Return 1 + X.
|
| 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.
|STWOTOX idnt 2,1 | Motorola 040 Floating Point Software Package
|section 8
#include "fpsp.h"
BOUNDS1: .long 0x3FB98000,0x400D80C0 | ... 2^(-70),16480
BOUNDS2: .long 0x3FB98000,0x400B9B07 | ... 2^(-70),16480 LOG2/LOG10
L2TEN64: .long 0x406A934F,0x0979A371 | ... 64LOG10/LOG2
L10TWO1: .long 0x3F734413,0x509F8000 | ... LOG2/64LOG10
L10TWO2: .long 0xBFCD0000,0xC0219DC1,0xDA994FD2,0x00000000
LOG10: .long 0x40000000,0x935D8DDD,0xAAA8AC17,0x00000000
LOG2: .long 0x3FFE0000,0xB17217F7,0xD1CF79AC,0x00000000
EXPA5: .long 0x3F56C16D,0x6F7BD0B2
EXPA4: .long 0x3F811112,0x302C712C
EXPA3: .long 0x3FA55555,0x55554CC1
EXPA2: .long 0x3FC55555,0x55554A54
EXPA1: .long 0x3FE00000,0x00000000,0x00000000,0x00000000
HUGE: .long 0x7FFE0000,0xFFFFFFFF,0xFFFFFFFF,0x00000000
TINY: .long 0x00010000,0xFFFFFFFF,0xFFFFFFFF,0x00000000
EXPTBL:
.long 0x3FFF0000,0x80000000,0x00000000,0x3F738000
.long 0x3FFF0000,0x8164D1F3,0xBC030773,0x3FBEF7CA
.long 0x3FFF0000,0x82CD8698,0xAC2BA1D7,0x3FBDF8A9
.long 0x3FFF0000,0x843A28C3,0xACDE4046,0x3FBCD7C9
.long 0x3FFF0000,0x85AAC367,0xCC487B15,0xBFBDE8DA
.long 0x3FFF0000,0x871F6196,0x9E8D1010,0x3FBDE85C
.long 0x3FFF0000,0x88980E80,0x92DA8527,0x3FBEBBF1
.long 0x3FFF0000,0x8A14D575,0x496EFD9A,0x3FBB80CA
.long 0x3FFF0000,0x8B95C1E3,0xEA8BD6E7,0xBFBA8373
.long 0x3FFF0000,0x8D1ADF5B,0x7E5BA9E6,0xBFBE9670
.long 0x3FFF0000,0x8EA4398B,0x45CD53C0,0x3FBDB700
.long 0x3FFF0000,0x9031DC43,0x1466B1DC,0x3FBEEEB0
.long 0x3FFF0000,0x91C3D373,0xAB11C336,0x3FBBFD6D
.long 0x3FFF0000,0x935A2B2F,0x13E6E92C,0xBFBDB319
.long 0x3FFF0000,0x94F4EFA8,0xFEF70961,0x3FBDBA2B
.long 0x3FFF0000,0x96942D37,0x20185A00,0x3FBE91D5
.long 0x3FFF0000,0x9837F051,0x8DB8A96F,0x3FBE8D5A
.long 0x3FFF0000,0x99E04593,0x20B7FA65,0xBFBCDE7B
.long 0x3FFF0000,0x9B8D39B9,0xD54E5539,0xBFBEBAAF
.long 0x3FFF0000,0x9D3ED9A7,0x2CFFB751,0xBFBD86DA
.long 0x3FFF0000,0x9EF53260,0x91A111AE,0xBFBEBEDD
.long 0x3FFF0000,0xA0B0510F,0xB9714FC2,0x3FBCC96E
.long 0x3FFF0000,0xA2704303,0x0C496819,0xBFBEC90B
.long 0x3FFF0000,0xA43515AE,0x09E6809E,0x3FBBD1DB
.long 0x3FFF0000,0xA5FED6A9,0xB15138EA,0x3FBCE5EB
.long 0x3FFF0000,0xA7CD93B4,0xE965356A,0xBFBEC274
.long 0x3FFF0000,0xA9A15AB4,0xEA7C0EF8,0x3FBEA83C
.long 0x3FFF0000,0xAB7A39B5,0xA93ED337,0x3FBECB00
.long 0x3FFF0000,0xAD583EEA,0x42A14AC6,0x3FBE9301
.long 0x3FFF0000,0xAF3B78AD,0x690A4375,0xBFBD8367
.long 0x3FFF0000,0xB123F581,0xD2AC2590,0xBFBEF05F
.long 0x3FFF0000,0xB311C412,0xA9112489,0x3FBDFB3C
.long 0x3FFF0000,0xB504F333,0xF9DE6484,0x3FBEB2FB
.long 0x3FFF0000,0xB6FD91E3,0x28D17791,0x3FBAE2CB
.long 0x3FFF0000,0xB8FBAF47,0x62FB9EE9,0x3FBCDC3C
.long 0x3FFF0000,0xBAFF5AB2,0x133E45FB,0x3FBEE9AA
.long 0x3FFF0000,0xBD08A39F,0x580C36BF,0xBFBEAEFD
.long 0x3FFF0000,0xBF1799B6,0x7A731083,0xBFBCBF51
.long 0x3FFF0000,0xC12C4CCA,0x66709456,0x3FBEF88A
.long 0x3FFF0000,0xC346CCDA,0x24976407,0x3FBD83B2
.long 0x3FFF0000,0xC5672A11,0x5506DADD,0x3FBDF8AB
.long 0x3FFF0000,0xC78D74C8,0xABB9B15D,0xBFBDFB17
.long 0x3FFF0000,0xC9B9BD86,0x6E2F27A3,0xBFBEFE3C
.long 0x3FFF0000,0xCBEC14FE,0xF2727C5D,0xBFBBB6F8
.long 0x3FFF0000,0xCE248C15,0x1F8480E4,0xBFBCEE53
.long 0x3FFF0000,0xD06333DA,0xEF2B2595,0xBFBDA4AE
.long 0x3FFF0000,0xD2A81D91,0xF12AE45A,0x3FBC9124
.long 0x3FFF0000,0xD4F35AAB,0xCFEDFA1F,0x3FBEB243
.long 0x3FFF0000,0xD744FCCA,0xD69D6AF4,0x3FBDE69A
.long 0x3FFF0000,0xD99D15C2,0x78AFD7B6,0xBFB8BC61
.long 0x3FFF0000,0xDBFBB797,0xDAF23755,0x3FBDF610
.long 0x3FFF0000,0xDE60F482,0x5E0E9124,0xBFBD8BE1
.long 0x3FFF0000,0xE0CCDEEC,0x2A94E111,0x3FBACB12
.long 0x3FFF0000,0xE33F8972,0xBE8A5A51,0x3FBB9BFE
.long 0x3FFF0000,0xE5B906E7,0x7C8348A8,0x3FBCF2F4
.long 0x3FFF0000,0xE8396A50,0x3C4BDC68,0x3FBEF22F
.long 0x3FFF0000,0xEAC0C6E7,0xDD24392F,0xBFBDBF4A
.long 0x3FFF0000,0xED4F301E,0xD9942B84,0x3FBEC01A
.long 0x3FFF0000,0xEFE4B99B,0xDCDAF5CB,0x3FBE8CAC
.long 0x3FFF0000,0xF281773C,0x59FFB13A,0xBFBCBB3F
.long 0x3FFF0000,0xF5257D15,0x2486CC2C,0x3FBEF73A
.long 0x3FFF0000,0xF7D0DF73,0x0AD13BB9,0xBFB8B795
.long 0x3FFF0000,0xFA83B2DB,0x722A033A,0x3FBEF84B
.long 0x3FFF0000,0xFD3E0C0C,0xF486C175,0xBFBEF581
.set N,L_SCR1
.set X,FP_SCR1
.set XDCARE,X+2
.set XFRAC,X+4
.set ADJFACT,FP_SCR2
.set FACT1,FP_SCR3
.set FACT1HI,FACT1+4
.set FACT1LOW,FACT1+8
.set FACT2,FP_SCR4
.set FACT2HI,FACT2+4
.set FACT2LOW,FACT2+8
| xref t_unfl
|xref t_ovfl
|xref t_frcinx
.global stwotoxd
stwotoxd:
|--ENTRY POINT FOR 2**(X) FOR DENORMALIZED ARGUMENT
fmovel %d1,%fpcr | ...set user's rounding mode/precision
fmoves #0x3F800000,%fp0 | ...RETURN 1 + X
movel (%a0),%d0
orl #0x00800001,%d0
fadds %d0,%fp0
bra t_frcinx
.global stwotox
stwotox:
|--ENTRY POINT FOR 2**(X), HERE X IS FINITE, NON-ZERO, AND NOT NAN'S
fmovemx (%a0),%fp0-%fp0 | ...LOAD INPUT, do not set cc's
movel (%a0),%d0
movew 4(%a0),%d0
fmovex %fp0,X(%a6)
andil #0x7FFFFFFF,%d0
cmpil #0x3FB98000,%d0 | ...|X| >= 2**(-70)?
bges TWOOK1
bra EXPBORS
TWOOK1:
cmpil #0x400D80C0,%d0 | ...|X| > 16480?
bles TWOMAIN
bra EXPBORS
TWOMAIN:
|--USUAL CASE, 2^(-70) <= |X| <= 16480
fmovex %fp0,%fp1
fmuls #0x42800000,%fp1 | ...64 * X
fmovel %fp1,N(%a6) | ...N = ROUND-TO-INT(64 X)
movel %d2,-(%sp)
lea EXPTBL,%a1 | ...LOAD ADDRESS OF TABLE OF 2^(J/64)
fmovel N(%a6),%fp1 | ...N --> FLOATING FMT
movel N(%a6),%d0
movel %d0,%d2
andil #0x3F,%d0 | ...D0 IS J
asll #4,%d0 | ...DISPLACEMENT FOR 2^(J/64)
addal %d0,%a1 | ...ADDRESS FOR 2^(J/64)
asrl #6,%d2 | ...d2 IS L, N = 64L + J
movel %d2,%d0
asrl #1,%d0 | ...D0 IS M
subl %d0,%d2 | ...d2 IS M', N = 64(M+M') + J
addil #0x3FFF,%d2
movew %d2,ADJFACT(%a6) | ...ADJFACT IS 2^(M')
movel (%sp)+,%d2
|--SUMMARY: a1 IS ADDRESS FOR THE LEADING PORTION OF 2^(J/64),
|--D0 IS M WHERE N = 64(M+M') + J. NOTE THAT |M| <= 16140 BY DESIGN.
|--ADJFACT = 2^(M').
|--REGISTERS SAVED SO FAR ARE (IN ORDER) FPCR, D0, FP1, a1, AND FP2.
fmuls #0x3C800000,%fp1 | ...(1/64)*N
movel (%a1)+,FACT1(%a6)
movel (%a1)+,FACT1HI(%a6)
movel (%a1)+,FACT1LOW(%a6)
movew (%a1)+,FACT2(%a6)
clrw FACT2+2(%a6)
fsubx %fp1,%fp0 | ...X - (1/64)*INT(64 X)
movew (%a1)+,FACT2HI(%a6)
clrw FACT2HI+2(%a6)
clrl FACT2LOW(%a6)
addw %d0,FACT1(%a6)
fmulx LOG2,%fp0 | ...FP0 IS R
addw %d0,FACT2(%a6)
bra expr
EXPBORS:
|--FPCR, D0 SAVED
cmpil #0x3FFF8000,%d0
bgts EXPBIG
EXPSM:
|--|X| IS SMALL, RETURN 1 + X
fmovel %d1,%FPCR |restore users exceptions
fadds #0x3F800000,%fp0 | ...RETURN 1 + X
bra t_frcinx
EXPBIG:
|--|X| IS LARGE, GENERATE OVERFLOW IF X > 0; ELSE GENERATE UNDERFLOW
|--REGISTERS SAVE SO FAR ARE FPCR AND D0
movel X(%a6),%d0
cmpil #0,%d0
blts EXPNEG
bclrb #7,(%a0) |t_ovfl expects positive value
bra t_ovfl
EXPNEG:
bclrb #7,(%a0) |t_unfl expects positive value
bra t_unfl
.global stentoxd
stentoxd:
|--ENTRY POINT FOR 10**(X) FOR DENORMALIZED ARGUMENT
fmovel %d1,%fpcr | ...set user's rounding mode/precision
fmoves #0x3F800000,%fp0 | ...RETURN 1 + X
movel (%a0),%d0
orl #0x00800001,%d0
fadds %d0,%fp0
bra t_frcinx
.global stentox
stentox:
|--ENTRY POINT FOR 10**(X), HERE X IS FINITE, NON-ZERO, AND NOT NAN'S
fmovemx (%a0),%fp0-%fp0 | ...LOAD INPUT, do not set cc's
movel (%a0),%d0
movew 4(%a0),%d0
fmovex %fp0,X(%a6)
andil #0x7FFFFFFF,%d0
cmpil #0x3FB98000,%d0 | ...|X| >= 2**(-70)?
bges TENOK1
bra EXPBORS
TENOK1:
cmpil #0x400B9B07,%d0 | ...|X| <= 16480*log2/log10 ?
bles TENMAIN
bra EXPBORS
TENMAIN:
|--USUAL CASE, 2^(-70) <= |X| <= 16480 LOG 2 / LOG 10
fmovex %fp0,%fp1
fmuld L2TEN64,%fp1 | ...X*64*LOG10/LOG2
fmovel %fp1,N(%a6) | ...N=INT(X*64*LOG10/LOG2)
movel %d2,-(%sp)
lea EXPTBL,%a1 | ...LOAD ADDRESS OF TABLE OF 2^(J/64)
fmovel N(%a6),%fp1 | ...N --> FLOATING FMT
movel N(%a6),%d0
movel %d0,%d2
andil #0x3F,%d0 | ...D0 IS J
asll #4,%d0 | ...DISPLACEMENT FOR 2^(J/64)
addal %d0,%a1 | ...ADDRESS FOR 2^(J/64)
asrl #6,%d2 | ...d2 IS L, N = 64L + J
movel %d2,%d0
asrl #1,%d0 | ...D0 IS M
subl %d0,%d2 | ...d2 IS M', N = 64(M+M') + J
addil #0x3FFF,%d2
movew %d2,ADJFACT(%a6) | ...ADJFACT IS 2^(M')
movel (%sp)+,%d2
|--SUMMARY: a1 IS ADDRESS FOR THE LEADING PORTION OF 2^(J/64),
|--D0 IS M WHERE N = 64(M+M') + J. NOTE THAT |M| <= 16140 BY DESIGN.
|--ADJFACT = 2^(M').
|--REGISTERS SAVED SO FAR ARE (IN ORDER) FPCR, D0, FP1, a1, AND FP2.
fmovex %fp1,%fp2
fmuld L10TWO1,%fp1 | ...N*(LOG2/64LOG10)_LEAD
movel (%a1)+,FACT1(%a6)
fmulx L10TWO2,%fp2 | ...N*(LOG2/64LOG10)_TRAIL
movel (%a1)+,FACT1HI(%a6)
movel (%a1)+,FACT1LOW(%a6)
fsubx %fp1,%fp0 | ...X - N L_LEAD
movew (%a1)+,FACT2(%a6)
fsubx %fp2,%fp0 | ...X - N L_TRAIL
clrw FACT2+2(%a6)
movew (%a1)+,FACT2HI(%a6)
clrw FACT2HI+2(%a6)
clrl FACT2LOW(%a6)
fmulx LOG10,%fp0 | ...FP0 IS R
addw %d0,FACT1(%a6)
addw %d0,FACT2(%a6)
expr:
|--FPCR, FP2, FP3 ARE SAVED IN ORDER AS SHOWN.
|--ADJFACT CONTAINS 2**(M'), FACT1 + FACT2 = 2**(M) * 2**(J/64).
|--FP0 IS R. THE FOLLOWING CODE COMPUTES
|-- 2**(M'+M) * 2**(J/64) * EXP(R)
fmovex %fp0,%fp1
fmulx %fp1,%fp1 | ...FP1 IS S = R*R
fmoved EXPA5,%fp2 | ...FP2 IS A5
fmoved EXPA4,%fp3 | ...FP3 IS A4
fmulx %fp1,%fp2 | ...FP2 IS S*A5
fmulx %fp1,%fp3 | ...FP3 IS S*A4
faddd EXPA3,%fp2 | ...FP2 IS A3+S*A5
faddd EXPA2,%fp3 | ...FP3 IS A2+S*A4
fmulx %fp1,%fp2 | ...FP2 IS S*(A3+S*A5)
fmulx %fp1,%fp3 | ...FP3 IS S*(A2+S*A4)
faddd EXPA1,%fp2 | ...FP2 IS A1+S*(A3+S*A5)
fmulx %fp0,%fp3 | ...FP3 IS R*S*(A2+S*A4)
fmulx %fp1,%fp2 | ...FP2 IS S*(A1+S*(A3+S*A5))
faddx %fp3,%fp0 | ...FP0 IS R+R*S*(A2+S*A4)
faddx %fp2,%fp0 | ...FP0 IS EXP(R) - 1
|--FINAL RECONSTRUCTION PROCESS
|--EXP(X) = 2^M*2^(J/64) + 2^M*2^(J/64)*(EXP(R)-1) - (1 OR 0)
fmulx FACT1(%a6),%fp0
faddx FACT2(%a6),%fp0
faddx FACT1(%a6),%fp0
fmovel %d1,%FPCR |restore users exceptions
clrw ADJFACT+2(%a6)
movel #0x80000000,ADJFACT+4(%a6)
clrl ADJFACT+8(%a6)
fmulx ADJFACT(%a6),%fp0 | ...FINAL ADJUSTMENT
bra t_frcinx
|end
|
