1 |
2 | setox.sa 3.1 12/10/90
3 |
4 | The entry point setox computes the exponential of a value.
5 | setoxd does the same except the input value is a denormalized
6 | number. setoxm1 computes exp(X)-1, and setoxm1d computes
7 | exp(X)-1 for denormalized X.
8 |
9 | INPUT
10 | -----
11 | Double-extended value in memory location pointed to by address
12 | register a0.
13 |
14 | OUTPUT
15 | ------
16 | exp(X) or exp(X)-1 returned in floating-point register fp0.
17 |
18 | ACCURACY and MONOTONICITY
19 | -------------------------
20 | The returned result is within 0.85 ulps in 64 significant bit, i.e.
21 | within 0.5001 ulp to 53 bits if the result is subsequently rounded
22 | to double precision. The result is provably monotonic in double
23 | precision.
24 |
25 | SPEED
26 | -----
27 | Two timings are measured, both in the copy-back mode. The
28 | first one is measured when the function is invoked the first time
29 | (so the instructions and data are not in cache), and the
30 | second one is measured when the function is reinvoked at the same
31 | input argument.
32 |
33 | The program setox takes approximately 210/190 cycles for input
34 | argument X whose magnitude is less than 16380 log2, which
35 | is the usual situation. For the less common arguments,
36 | depending on their values, the program may run faster or slower --
37 | but no worse than 10% slower even in the extreme cases.
38 |
39 | The program setoxm1 takes approximately ???/??? cycles for input
40 | argument X, 0.25 <= |X| < 70log2. For |X| < 0.25, it takes
41 | approximately ???/??? cycles. For the less common arguments,
42 | depending on their values, the program may run faster or slower --
43 | but no worse than 10% slower even in the extreme cases.
44 |
45 | ALGORITHM and IMPLEMENTATION NOTES
46 | ----------------------------------
47 |
48 | setoxd
49 | ------
50 | Step 1. Set ans := 1.0
51 |
52 | Step 2. Return ans := ans + sign(X)*2^(-126). Exit.
53 | Notes: This will always generate one exception -- inexact.
54 |
55 |
56 | setox
57 | -----
58 |
59 | Step 1. Filter out extreme cases of input argument.
60 | 1.1 If |X| >= 2^(-65), go to Step 1.3.
61 | 1.2 Go to Step 7.
62 | 1.3 If |X| < 16380 log(2), go to Step 2.
63 | 1.4 Go to Step 8.
64 | Notes: The usual case should take the branches 1.1 -> 1.3 -> 2.
65 | To avoid the use of floating-point comparisons, a
66 | compact representation of |X| is used. This format is a
67 | 32-bit integer, the upper (more significant) 16 bits are
68 | the sign and biased exponent field of |X|; the lower 16
69 | bits are the 16 most significant fraction (including the
70 | explicit bit) bits of |X|. Consequently, the comparisons
71 | in Steps 1.1 and 1.3 can be performed by integer comparison.
72 | Note also that the constant 16380 log(2) used in Step 1.3
73 | is also in the compact form. Thus taking the branch
74 | to Step 2 guarantees |X| < 16380 log(2). There is no harm
75 | to have a small number of cases where |X| is less than,
76 | but close to, 16380 log(2) and the branch to Step 9 is
77 | taken.
78 |
79 | Step 2. Calculate N = round-to-nearest-int( X * 64/log2 ).
80 | 2.1 Set AdjFlag := 0 (indicates the branch 1.3 -> 2 was taken)
81 | 2.2 N := round-to-nearest-integer( X * 64/log2 ).
82 | 2.3 Calculate J = N mod 64; so J = 0,1,2,..., or 63.
83 | 2.4 Calculate M = (N - J)/64; so N = 64M + J.
84 | 2.5 Calculate the address of the stored value of 2^(J/64).
85 | 2.6 Create the value Scale = 2^M.
86 | Notes: The calculation in 2.2 is really performed by
87 |
88 | Z := X * constant
89 | N := round-to-nearest-integer(Z)
90 |
91 | where
92 |
93 | constant := single-precision( 64/log 2 ).
94 |
95 | Using a single-precision constant avoids memory access.
96 | Another effect of using a single-precision "constant" is
97 | that the calculated value Z is
98 |
99 | Z = X*(64/log2)*(1+eps), |eps| <= 2^(-24).
100 |
101 | This error has to be considered later in Steps 3 and 4.
102 |
103 | Step 3. Calculate X - N*log2/64.
104 | 3.1 R := X + N*L1, where L1 := single-precision(-log2/64).
105 | 3.2 R := R + N*L2, L2 := extended-precision(-log2/64 - L1).
106 | Notes: a) The way L1 and L2 are chosen ensures L1+L2 approximate
107 | the value -log2/64 to 88 bits of accuracy.
108 | b) N*L1 is exact because N is no longer than 22 bits and
109 | L1 is no longer than 24 bits.
110 | c) The calculation X+N*L1 is also exact due to cancellation.
111 | Thus, R is practically X+N(L1+L2) to full 64 bits.
112 | d) It is important to estimate how large can |R| be after
113 | Step 3.2.
114 |
115 | N = rnd-to-int( X*64/log2 (1+eps) ), |eps|<=2^(-24)
116 | X*64/log2 (1+eps) = N + f, |f| <= 0.5
117 | X*64/log2 - N = f - eps*X 64/log2
118 | X - N*log2/64 = f*log2/64 - eps*X
119 |
120 |
121 | Now |X| <= 16446 log2, thus
122 |
123 | |X - N*log2/64| <= (0.5 + 16446/2^(18))*log2/64
124 | <= 0.57 log2/64.
125 | This bound will be used in Step 4.
126 |
127 | Step 4. Approximate exp(R)-1 by a polynomial
128 | p = R + R*R*(A1 + R*(A2 + R*(A3 + R*(A4 + R*A5))))
129 | Notes: a) In order to reduce memory access, the coefficients are
130 | made as "short" as possible: A1 (which is 1/2), A4 and A5
131 | are single precision; A2 and A3 are double precision.
132 | b) Even with the restrictions above,
133 | |p - (exp(R)-1)| < 2^(-68.8) for all |R| <= 0.0062.
134 | Note that 0.0062 is slightly bigger than 0.57 log2/64.
135 | c) To fully utilize the pipeline, p is separated into
136 | two independent pieces of roughly equal complexities
137 | p = [ R + R*S*(A2 + S*A4) ] +
138 | [ S*(A1 + S*(A3 + S*A5)) ]
139 | where S = R*R.
140 |
141 | Step 5. Compute 2^(J/64)*exp(R) = 2^(J/64)*(1+p) by
142 | ans := T + ( T*p + t)
143 | where T and t are the stored values for 2^(J/64).
144 | Notes: 2^(J/64) is stored as T and t where T+t approximates
145 | 2^(J/64) to roughly 85 bits; T is in extended precision
146 | and t is in single precision. Note also that T is rounded
147 | to 62 bits so that the last two bits of T are zero. The
148 | reason for such a special form is that T-1, T-2, and T-8
149 | will all be exact --- a property that will give much
150 | more accurate computation of the function EXPM1.
151 |
152 | Step 6. Reconstruction of exp(X)
153 | exp(X) = 2^M * 2^(J/64) * exp(R).
154 | 6.1 If AdjFlag = 0, go to 6.3
155 | 6.2 ans := ans * AdjScale
156 | 6.3 Restore the user FPCR
157 | 6.4 Return ans := ans * Scale. Exit.
158 | Notes: If AdjFlag = 0, we have X = Mlog2 + Jlog2/64 + R,
159 | |M| <= 16380, and Scale = 2^M. Moreover, exp(X) will
160 | neither overflow nor underflow. If AdjFlag = 1, that
161 | means that
162 | X = (M1+M)log2 + Jlog2/64 + R, |M1+M| >= 16380.
163 | Hence, exp(X) may overflow or underflow or neither.
164 | When that is the case, AdjScale = 2^(M1) where M1 is
165 | approximately M. Thus 6.2 will never cause over/underflow.
166 | Possible exception in 6.4 is overflow or underflow.
167 | The inexact exception is not generated in 6.4. Although
168 | one can argue that the inexact flag should always be
169 | raised, to simulate that exception cost to much than the
170 | flag is worth in practical uses.
171 |
172 | Step 7. Return 1 + X.
173 | 7.1 ans := X
174 | 7.2 Restore user FPCR.
175 | 7.3 Return ans := 1 + ans. Exit
176 | Notes: For non-zero X, the inexact exception will always be
177 | raised by 7.3. That is the only exception raised by 7.3.
178 | Note also that we use the FMOVEM instruction to move X
179 | in Step 7.1 to avoid unnecessary trapping. (Although
180 | the FMOVEM may not seem relevant since X is normalized,
181 | the precaution will be useful in the library version of
182 | this code where the separate entry for denormalized inputs
183 | will be done away with.)
184 |
185 | Step 8. Handle exp(X) where |X| >= 16380log2.
186 | 8.1 If |X| > 16480 log2, go to Step 9.
187 | (mimic 2.2 - 2.6)
188 | 8.2 N := round-to-integer( X * 64/log2 )
189 | 8.3 Calculate J = N mod 64, J = 0,1,...,63
190 | 8.4 K := (N-J)/64, M1 := truncate(K/2), M = K-M1, AdjFlag := 1.
191 | 8.5 Calculate the address of the stored value 2^(J/64).
192 | 8.6 Create the values Scale = 2^M, AdjScale = 2^M1.
193 | 8.7 Go to Step 3.
194 | Notes: Refer to notes for 2.2 - 2.6.
195 |
196 | Step 9. Handle exp(X), |X| > 16480 log2.
197 | 9.1 If X < 0, go to 9.3
198 | 9.2 ans := Huge, go to 9.4
199 | 9.3 ans := Tiny.
200 | 9.4 Restore user FPCR.
201 | 9.5 Return ans := ans * ans. Exit.
202 | Notes: Exp(X) will surely overflow or underflow, depending on
203 | X's sign. "Huge" and "Tiny" are respectively large/tiny
204 | extended-precision numbers whose square over/underflow
205 | with an inexact result. Thus, 9.5 always raises the
206 | inexact together with either overflow or underflow.
207 |
208 |
209 | setoxm1d
210 | --------
211 |
212 | Step 1. Set ans := 0
213 |
214 | Step 2. Return ans := X + ans. Exit.
215 | Notes: This will return X with the appropriate rounding
216 | precision prescribed by the user FPCR.
217 |
218 | setoxm1
219 | -------
220 |
221 | Step 1. Check |X|
222 | 1.1 If |X| >= 1/4, go to Step 1.3.
223 | 1.2 Go to Step 7.
224 | 1.3 If |X| < 70 log(2), go to Step 2.
225 | 1.4 Go to Step 10.
226 | Notes: The usual case should take the branches 1.1 -> 1.3 -> 2.
227 | However, it is conceivable |X| can be small very often
228 | because EXPM1 is intended to evaluate exp(X)-1 accurately
229 | when |X| is small. For further details on the comparisons,
230 | see the notes on Step 1 of setox.
231 |
232 | Step 2. Calculate N = round-to-nearest-int( X * 64/log2 ).
233 | 2.1 N := round-to-nearest-integer( X * 64/log2 ).
234 | 2.2 Calculate J = N mod 64; so J = 0,1,2,..., or 63.
235 | 2.3 Calculate M = (N - J)/64; so N = 64M + J.
236 | 2.4 Calculate the address of the stored value of 2^(J/64).
237 | 2.5 Create the values Sc = 2^M and OnebySc := -2^(-M).
238 | Notes: See the notes on Step 2 of setox.
239 |
240 | Step 3. Calculate X - N*log2/64.
241 | 3.1 R := X + N*L1, where L1 := single-precision(-log2/64).
242 | 3.2 R := R + N*L2, L2 := extended-precision(-log2/64 - L1).
243 | Notes: Applying the analysis of Step 3 of setox in this case
244 | shows that |R| <= 0.0055 (note that |X| <= 70 log2 in
245 | this case).
246 |
247 | Step 4. Approximate exp(R)-1 by a polynomial
248 | p = R+R*R*(A1+R*(A2+R*(A3+R*(A4+R*(A5+R*A6)))))
249 | Notes: a) In order to reduce memory access, the coefficients are
250 | made as "short" as possible: A1 (which is 1/2), A5 and A6
251 | are single precision; A2, A3 and A4 are double precision.
252 | b) Even with the restriction above,
253 | |p - (exp(R)-1)| < |R| * 2^(-72.7)
254 | for all |R| <= 0.0055.
255 | c) To fully utilize the pipeline, p is separated into
256 | two independent pieces of roughly equal complexity
257 | p = [ R*S*(A2 + S*(A4 + S*A6)) ] +
258 | [ R + S*(A1 + S*(A3 + S*A5)) ]
259 | where S = R*R.
260 |
261 | Step 5. Compute 2^(J/64)*p by
262 | p := T*p
263 | where T and t are the stored values for 2^(J/64).
264 | Notes: 2^(J/64) is stored as T and t where T+t approximates
265 | 2^(J/64) to roughly 85 bits; T is in extended precision
266 | and t is in single precision. Note also that T is rounded
267 | to 62 bits so that the last two bits of T are zero. The
268 | reason for such a special form is that T-1, T-2, and T-8
269 | will all be exact --- a property that will be exploited
270 | in Step 6 below. The total relative error in p is no
271 | bigger than 2^(-67.7) compared to the final result.
272 |
273 | Step 6. Reconstruction of exp(X)-1
274 | exp(X)-1 = 2^M * ( 2^(J/64) + p - 2^(-M) ).
275 | 6.1 If M <= 63, go to Step 6.3.
276 | 6.2 ans := T + (p + (t + OnebySc)). Go to 6.6
277 | 6.3 If M >= -3, go to 6.5.
278 | 6.4 ans := (T + (p + t)) + OnebySc. Go to 6.6
279 | 6.5 ans := (T + OnebySc) + (p + t).
280 | 6.6 Restore user FPCR.
281 | 6.7 Return ans := Sc * ans. Exit.
282 | Notes: The various arrangements of the expressions give accurate
283 | evaluations.
284 |
285 | Step 7. exp(X)-1 for |X| < 1/4.
286 | 7.1 If |X| >= 2^(-65), go to Step 9.
287 | 7.2 Go to Step 8.
288 |
289 | Step 8. Calculate exp(X)-1, |X| < 2^(-65).
290 | 8.1 If |X| < 2^(-16312), goto 8.3
291 | 8.2 Restore FPCR; return ans := X - 2^(-16382). Exit.
292 | 8.3 X := X * 2^(140).
293 | 8.4 Restore FPCR; ans := ans - 2^(-16382).
294 | Return ans := ans*2^(140). Exit
295 | Notes: The idea is to return "X - tiny" under the user
296 | precision and rounding modes. To avoid unnecessary
297 | inefficiency, we stay away from denormalized numbers the
298 | best we can. For |X| >= 2^(-16312), the straightforward
299 | 8.2 generates the inexact exception as the case warrants.
300 |
301 | Step 9. Calculate exp(X)-1, |X| < 1/4, by a polynomial
302 | p = X + X*X*(B1 + X*(B2 + ... + X*B12))
303 | Notes: a) In order to reduce memory access, the coefficients are
304 | made as "short" as possible: B1 (which is 1/2), B9 to B12
305 | are single precision; B3 to B8 are double precision; and
306 | B2 is double extended.
307 | b) Even with the restriction above,
308 | |p - (exp(X)-1)| < |X| 2^(-70.6)
309 | for all |X| <= 0.251.
310 | Note that 0.251 is slightly bigger than 1/4.
311 | c) To fully preserve accuracy, the polynomial is computed
312 | as X + ( S*B1 + Q ) where S = X*X and
313 | Q = X*S*(B2 + X*(B3 + ... + X*B12))
314 | d) To fully utilize the pipeline, Q is separated into
315 | two independent pieces of roughly equal complexity
316 | Q = [ X*S*(B2 + S*(B4 + ... + S*B12)) ] +
317 | [ S*S*(B3 + S*(B5 + ... + S*B11)) ]
318 |
319 | Step 10. Calculate exp(X)-1 for |X| >= 70 log 2.
320 | 10.1 If X >= 70log2 , exp(X) - 1 = exp(X) for all practical
321 | purposes. Therefore, go to Step 1 of setox.
322 | 10.2 If X <= -70log2, exp(X) - 1 = -1 for all practical purposes.
323 | ans := -1
324 | Restore user FPCR
325 | Return ans := ans + 2^(-126). Exit.
326 | Notes: 10.2 will always create an inexact and return -1 + tiny
327 | in the user rounding precision and mode.
328 |
329 |
330
331 | Copyright (C) Motorola, Inc. 1990
332 | All Rights Reserved
333 |
334 | THIS IS UNPUBLISHED PROPRIETARY SOURCE CODE OF MOTOROLA
335 | The copyright notice above does not evidence any
336 | actual or intended publication of such source code.
337
338 |setox idnt 2,1 | Motorola 040 Floating Point Software Package
339
340 |section 8
341
342 .include "fpsp.h"
343
344 L2: .long 0x3FDC0000,0x82E30865,0x4361C4C6,0x00000000
345
346 EXPA3: .long 0x3FA55555,0x55554431
347 EXPA2: .long 0x3FC55555,0x55554018
348
349 HUGE: .long 0x7FFE0000,0xFFFFFFFF,0xFFFFFFFF,0x00000000
350 TINY: .long 0x00010000,0xFFFFFFFF,0xFFFFFFFF,0x00000000
351
352 EM1A4: .long 0x3F811111,0x11174385
353 EM1A3: .long 0x3FA55555,0x55554F5A
354
355 EM1A2: .long 0x3FC55555,0x55555555,0x00000000,0x00000000
356
357 EM1B8: .long 0x3EC71DE3,0xA5774682
358 EM1B7: .long 0x3EFA01A0,0x19D7CB68
359
360 EM1B6: .long 0x3F2A01A0,0x1A019DF3
361 EM1B5: .long 0x3F56C16C,0x16C170E2
362
363 EM1B4: .long 0x3F811111,0x11111111
364 EM1B3: .long 0x3FA55555,0x55555555
365
366 EM1B2: .long 0x3FFC0000,0xAAAAAAAA,0xAAAAAAAB
367 .long 0x00000000
368
369 TWO140: .long 0x48B00000,0x00000000
370 TWON140: .long 0x37300000,0x00000000
371
372 EXPTBL:
373 .long 0x3FFF0000,0x80000000,0x00000000,0x00000000
374 .long 0x3FFF0000,0x8164D1F3,0xBC030774,0x9F841A9B
375 .long 0x3FFF0000,0x82CD8698,0xAC2BA1D8,0x9FC1D5B9
376 .long 0x3FFF0000,0x843A28C3,0xACDE4048,0xA0728369
377 .long 0x3FFF0000,0x85AAC367,0xCC487B14,0x1FC5C95C
378 .long 0x3FFF0000,0x871F6196,0x9E8D1010,0x1EE85C9F
379 .long 0x3FFF0000,0x88980E80,0x92DA8528,0x9FA20729
380 .long 0x3FFF0000,0x8A14D575,0x496EFD9C,0xA07BF9AF
381 .long 0x3FFF0000,0x8B95C1E3,0xEA8BD6E8,0xA0020DCF
382 .long 0x3FFF0000,0x8D1ADF5B,0x7E5BA9E4,0x205A63DA
383 .long 0x3FFF0000,0x8EA4398B,0x45CD53C0,0x1EB70051
384 .long 0x3FFF0000,0x9031DC43,0x1466B1DC,0x1F6EB029
385 .long 0x3FFF0000,0x91C3D373,0xAB11C338,0xA0781494
386 .long 0x3FFF0000,0x935A2B2F,0x13E6E92C,0x9EB319B0
387 .long 0x3FFF0000,0x94F4EFA8,0xFEF70960,0x2017457D
388 .long 0x3FFF0000,0x96942D37,0x20185A00,0x1F11D537
389 .long 0x3FFF0000,0x9837F051,0x8DB8A970,0x9FB952DD
390 .long 0x3FFF0000,0x99E04593,0x20B7FA64,0x1FE43087
391 .long 0x3FFF0000,0x9B8D39B9,0xD54E5538,0x1FA2A818
392 .long 0x3FFF0000,0x9D3ED9A7,0x2CFFB750,0x1FDE494D
393 .long 0x3FFF0000,0x9EF53260,0x91A111AC,0x20504890
394 .long 0x3FFF0000,0xA0B0510F,0xB9714FC4,0xA073691C
395 .long 0x3FFF0000,0xA2704303,0x0C496818,0x1F9B7A05
396 .long 0x3FFF0000,0xA43515AE,0x09E680A0,0xA0797126
397 .long 0x3FFF0000,0xA5FED6A9,0xB15138EC,0xA071A140
398 .long 0x3FFF0000,0xA7CD93B4,0xE9653568,0x204F62DA
399 .long 0x3FFF0000,0xA9A15AB4,0xEA7C0EF8,0x1F283C4A
400 .long 0x3FFF0000,0xAB7A39B5,0xA93ED338,0x9F9A7FDC
401 .long 0x3FFF0000,0xAD583EEA,0x42A14AC8,0xA05B3FAC
402 .long 0x3FFF0000,0xAF3B78AD,0x690A4374,0x1FDF2610
403 .long 0x3FFF0000,0xB123F581,0xD2AC2590,0x9F705F90
404 .long 0x3FFF0000,0xB311C412,0xA9112488,0x201F678A
405 .long 0x3FFF0000,0xB504F333,0xF9DE6484,0x1F32FB13
406 .long 0x3FFF0000,0xB6FD91E3,0x28D17790,0x20038B30
407 .long 0x3FFF0000,0xB8FBAF47,0x62FB9EE8,0x200DC3CC
408 .long 0x3FFF0000,0xBAFF5AB2,0x133E45FC,0x9F8B2AE6
409 .long 0x3FFF0000,0xBD08A39F,0x580C36C0,0xA02BBF70
410 .long 0x3FFF0000,0xBF1799B6,0x7A731084,0xA00BF518
411 .long 0x3FFF0000,0xC12C4CCA,0x66709458,0xA041DD41
412 .long 0x3FFF0000,0xC346CCDA,0x24976408,0x9FDF137B
413 .long 0x3FFF0000,0xC5672A11,0x5506DADC,0x201F1568
414 .long 0x3FFF0000,0xC78D74C8,0xABB9B15C,0x1FC13A2E
415 .long 0x3FFF0000,0xC9B9BD86,0x6E2F27A4,0xA03F8F03
416 .long 0x3FFF0000,0xCBEC14FE,0xF2727C5C,0x1FF4907D
417 .long 0x3FFF0000,0xCE248C15,0x1F8480E4,0x9E6E53E4
418 .long 0x3FFF0000,0xD06333DA,0xEF2B2594,0x1FD6D45C
419 .long 0x3FFF0000,0xD2A81D91,0xF12AE45C,0xA076EDB9
420 .long 0x3FFF0000,0xD4F35AAB,0xCFEDFA20,0x9FA6DE21
421 .long 0x3FFF0000,0xD744FCCA,0xD69D6AF4,0x1EE69A2F
422 .long 0x3FFF0000,0xD99D15C2,0x78AFD7B4,0x207F439F
423 .long 0x3FFF0000,0xDBFBB797,0xDAF23754,0x201EC207
424 .long 0x3FFF0000,0xDE60F482,0x5E0E9124,0x9E8BE175
425 .long 0x3FFF0000,0xE0CCDEEC,0x2A94E110,0x20032C4B
426 .long 0x3FFF0000,0xE33F8972,0xBE8A5A50,0x2004DFF5
427 .long 0x3FFF0000,0xE5B906E7,0x7C8348A8,0x1E72F47A
428 .long 0x3FFF0000,0xE8396A50,0x3C4BDC68,0x1F722F22
429 .long 0x3FFF0000,0xEAC0C6E7,0xDD243930,0xA017E945
430 .long 0x3FFF0000,0xED4F301E,0xD9942B84,0x1F401A5B
431 .long 0x3FFF0000,0xEFE4B99B,0xDCDAF5CC,0x9FB9A9E3
432 .long 0x3FFF0000,0xF281773C,0x59FFB138,0x20744C05
433 .long 0x3FFF0000,0xF5257D15,0x2486CC2C,0x1F773A19
434 .long 0x3FFF0000,0xF7D0DF73,0x0AD13BB8,0x1FFE90D5
435 .long 0x3FFF0000,0xFA83B2DB,0x722A033C,0xA041ED22
436 .long 0x3FFF0000,0xFD3E0C0C,0xF486C174,0x1F853F3A
437
438 .set ADJFLAG,L_SCR2
439 .set SCALE,FP_SCR1
440 .set ADJSCALE,FP_SCR2
441 .set SC,FP_SCR3
442 .set ONEBYSC,FP_SCR4
443
444 | xref t_frcinx
445 |xref t_extdnrm
446 |xref t_unfl
447 |xref t_ovfl
448
449 .global setoxd
450 setoxd:
451 |--entry point for EXP(X), X is denormalized
452 movel (%a0),%d0
453 andil #0x80000000,%d0
454 oril #0x00800000,%d0 | ...sign(X)*2^(-126)
455 movel %d0,-(%sp)
456 fmoves #0x3F800000,%fp0
457 fmovel %d1,%fpcr
458 fadds (%sp)+,%fp0
459 bra t_frcinx
460
461 .global setox
462 setox:
463 |--entry point for EXP(X), here X is finite, non-zero, and not NaN's
464
465 |--Step 1.
466 movel (%a0),%d0 | ...load part of input X
467 andil #0x7FFF0000,%d0 | ...biased expo. of X
468 cmpil #0x3FBE0000,%d0 | ...2^(-65)
469 bges EXPC1 | ...normal case
470 bra EXPSM
471
472 EXPC1:
473 |--The case |X| >= 2^(-65)
474 movew 4(%a0),%d0 | ...expo. and partial sig. of |X|
475 cmpil #0x400CB167,%d0 | ...16380 log2 trunc. 16 bits
476 blts EXPMAIN | ...normal case
477 bra EXPBIG
478
479 EXPMAIN:
480 |--Step 2.
481 |--This is the normal branch: 2^(-65) <= |X| < 16380 log2.
482 fmovex (%a0),%fp0 | ...load input from (a0)
483
484 fmovex %fp0,%fp1
485 fmuls #0x42B8AA3B,%fp0 | ...64/log2 * X
486 fmovemx %fp2-%fp2/%fp3,-(%a7) | ...save fp2
487 movel #0,ADJFLAG(%a6)
488 fmovel %fp0,%d0 | ...N = int( X * 64/log2 )
489 lea EXPTBL,%a1
490 fmovel %d0,%fp0 | ...convert to floating-format
491
492 movel %d0,L_SCR1(%a6) | ...save N temporarily
493 andil #0x3F,%d0 | ...D0 is J = N mod 64
494 lsll #4,%d0
495 addal %d0,%a1 | ...address of 2^(J/64)
496 movel L_SCR1(%a6),%d0
497 asrl #6,%d0 | ...D0 is M
498 addiw #0x3FFF,%d0 | ...biased expo. of 2^(M)
499 movew L2,L_SCR1(%a6) | ...prefetch L2, no need in CB
500
501 EXPCONT1:
502 |--Step 3.
503 |--fp1,fp2 saved on the stack. fp0 is N, fp1 is X,
504 |--a0 points to 2^(J/64), D0 is biased expo. of 2^(M)
505 fmovex %fp0,%fp2
506 fmuls #0xBC317218,%fp0 | ...N * L1, L1 = lead(-log2/64)
507 fmulx L2,%fp2 | ...N * L2, L1+L2 = -log2/64
508 faddx %fp1,%fp0 | ...X + N*L1
509 faddx %fp2,%fp0 | ...fp0 is R, reduced arg.
510 | MOVE.W #$3FA5,EXPA3 ...load EXPA3 in cache
511
512 |--Step 4.
513 |--WE NOW COMPUTE EXP(R)-1 BY A POLYNOMIAL
514 |-- R + R*R*(A1 + R*(A2 + R*(A3 + R*(A4 + R*A5))))
515 |--TO FULLY UTILIZE THE PIPELINE, WE COMPUTE S = R*R
516 |--[R+R*S*(A2+S*A4)] + [S*(A1+S*(A3+S*A5))]
517
518 fmovex %fp0,%fp1
519 fmulx %fp1,%fp1 | ...fp1 IS S = R*R
520
521 fmoves #0x3AB60B70,%fp2 | ...fp2 IS A5
522 | MOVE.W #0,2(%a1) ...load 2^(J/64) in cache
523
524 fmulx %fp1,%fp2 | ...fp2 IS S*A5
525 fmovex %fp1,%fp3
526 fmuls #0x3C088895,%fp3 | ...fp3 IS S*A4
527
528 faddd EXPA3,%fp2 | ...fp2 IS A3+S*A5
529 faddd EXPA2,%fp3 | ...fp3 IS A2+S*A4
530
531 fmulx %fp1,%fp2 | ...fp2 IS S*(A3+S*A5)
532 movew %d0,SCALE(%a6) | ...SCALE is 2^(M) in extended
533 clrw SCALE+2(%a6)
534 movel #0x80000000,SCALE+4(%a6)
535 clrl SCALE+8(%a6)
536
537 fmulx %fp1,%fp3 | ...fp3 IS S*(A2+S*A4)
538
539 fadds #0x3F000000,%fp2 | ...fp2 IS A1+S*(A3+S*A5)
540 fmulx %fp0,%fp3 | ...fp3 IS R*S*(A2+S*A4)
541
542 fmulx %fp1,%fp2 | ...fp2 IS S*(A1+S*(A3+S*A5))
543 faddx %fp3,%fp0 | ...fp0 IS R+R*S*(A2+S*A4),
544 | ...fp3 released
545
546 fmovex (%a1)+,%fp1 | ...fp1 is lead. pt. of 2^(J/64)
547 faddx %fp2,%fp0 | ...fp0 is EXP(R) - 1
548 | ...fp2 released
549
550 |--Step 5
551 |--final reconstruction process
552 |--EXP(X) = 2^M * ( 2^(J/64) + 2^(J/64)*(EXP(R)-1) )
553
554 fmulx %fp1,%fp0 | ...2^(J/64)*(Exp(R)-1)
555 fmovemx (%a7)+,%fp2-%fp2/%fp3 | ...fp2 restored
556 fadds (%a1),%fp0 | ...accurate 2^(J/64)
557
558 faddx %fp1,%fp0 | ...2^(J/64) + 2^(J/64)*...
559 movel ADJFLAG(%a6),%d0
560
561 |--Step 6
562 tstl %d0
563 beqs NORMAL
564 ADJUST:
565 fmulx ADJSCALE(%a6),%fp0
566 NORMAL:
567 fmovel %d1,%FPCR | ...restore user FPCR
568 fmulx SCALE(%a6),%fp0 | ...multiply 2^(M)
569 bra t_frcinx
570
571 EXPSM:
572 |--Step 7
573 fmovemx (%a0),%fp0-%fp0 | ...in case X is denormalized
574 fmovel %d1,%FPCR
575 fadds #0x3F800000,%fp0 | ...1+X in user mode
576 bra t_frcinx
577
578 EXPBIG:
579 |--Step 8
580 cmpil #0x400CB27C,%d0 | ...16480 log2
581 bgts EXP2BIG
582 |--Steps 8.2 -- 8.6
583 fmovex (%a0),%fp0 | ...load input from (a0)
584
585 fmovex %fp0,%fp1
586 fmuls #0x42B8AA3B,%fp0 | ...64/log2 * X
587 fmovemx %fp2-%fp2/%fp3,-(%a7) | ...save fp2
588 movel #1,ADJFLAG(%a6)
589 fmovel %fp0,%d0 | ...N = int( X * 64/log2 )
590 lea EXPTBL,%a1
591 fmovel %d0,%fp0 | ...convert to floating-format
592 movel %d0,L_SCR1(%a6) | ...save N temporarily
593 andil #0x3F,%d0 | ...D0 is J = N mod 64
594 lsll #4,%d0
595 addal %d0,%a1 | ...address of 2^(J/64)
596 movel L_SCR1(%a6),%d0
597 asrl #6,%d0 | ...D0 is K
598 movel %d0,L_SCR1(%a6) | ...save K temporarily
599 asrl #1,%d0 | ...D0 is M1
600 subl %d0,L_SCR1(%a6) | ...a1 is M
601 addiw #0x3FFF,%d0 | ...biased expo. of 2^(M1)
602 movew %d0,ADJSCALE(%a6) | ...ADJSCALE := 2^(M1)
603 clrw ADJSCALE+2(%a6)
604 movel #0x80000000,ADJSCALE+4(%a6)
605 clrl ADJSCALE+8(%a6)
606 movel L_SCR1(%a6),%d0 | ...D0 is M
607 addiw #0x3FFF,%d0 | ...biased expo. of 2^(M)
608 bra EXPCONT1 | ...go back to Step 3
609
610 EXP2BIG:
611 |--Step 9
612 fmovel %d1,%FPCR
613 movel (%a0),%d0
614 bclrb #sign_bit,(%a0) | ...setox always returns positive
615 cmpil #0,%d0
616 blt t_unfl
617 bra t_ovfl
618
619 .global setoxm1d
620 setoxm1d:
621 |--entry point for EXPM1(X), here X is denormalized
622 |--Step 0.
623 bra t_extdnrm
624
625
626 .global setoxm1
627 setoxm1:
628 |--entry point for EXPM1(X), here X is finite, non-zero, non-NaN
629
630 |--Step 1.
631 |--Step 1.1
632 movel (%a0),%d0 | ...load part of input X
633 andil #0x7FFF0000,%d0 | ...biased expo. of X
634 cmpil #0x3FFD0000,%d0 | ...1/4
635 bges EM1CON1 | ...|X| >= 1/4
636 bra EM1SM
637
638 EM1CON1:
639 |--Step 1.3
640 |--The case |X| >= 1/4
641 movew 4(%a0),%d0 | ...expo. and partial sig. of |X|
642 cmpil #0x4004C215,%d0 | ...70log2 rounded up to 16 bits
643 bles EM1MAIN | ...1/4 <= |X| <= 70log2
644 bra EM1BIG
645
646 EM1MAIN:
647 |--Step 2.
648 |--This is the case: 1/4 <= |X| <= 70 log2.
649 fmovex (%a0),%fp0 | ...load input from (a0)
650
651 fmovex %fp0,%fp1
652 fmuls #0x42B8AA3B,%fp0 | ...64/log2 * X
653 fmovemx %fp2-%fp2/%fp3,-(%a7) | ...save fp2
654 | MOVE.W #$3F81,EM1A4 ...prefetch in CB mode
655 fmovel %fp0,%d0 | ...N = int( X * 64/log2 )
656 lea EXPTBL,%a1
657 fmovel %d0,%fp0 | ...convert to floating-format
658
659 movel %d0,L_SCR1(%a6) | ...save N temporarily
660 andil #0x3F,%d0 | ...D0 is J = N mod 64
661 lsll #4,%d0
662 addal %d0,%a1 | ...address of 2^(J/64)
663 movel L_SCR1(%a6),%d0
664 asrl #6,%d0 | ...D0 is M
665 movel %d0,L_SCR1(%a6) | ...save a copy of M
666 | MOVE.W #$3FDC,L2 ...prefetch L2 in CB mode
667
668 |--Step 3.
669 |--fp1,fp2 saved on the stack. fp0 is N, fp1 is X,
670 |--a0 points to 2^(J/64), D0 and a1 both contain M
671 fmovex %fp0,%fp2
672 fmuls #0xBC317218,%fp0 | ...N * L1, L1 = lead(-log2/64)
673 fmulx L2,%fp2 | ...N * L2, L1+L2 = -log2/64
674 faddx %fp1,%fp0 | ...X + N*L1
675 faddx %fp2,%fp0 | ...fp0 is R, reduced arg.
676 | MOVE.W #$3FC5,EM1A2 ...load EM1A2 in cache
677 addiw #0x3FFF,%d0 | ...D0 is biased expo. of 2^M
678
679 |--Step 4.
680 |--WE NOW COMPUTE EXP(R)-1 BY A POLYNOMIAL
681 |-- R + R*R*(A1 + R*(A2 + R*(A3 + R*(A4 + R*(A5 + R*A6)))))
682 |--TO FULLY UTILIZE THE PIPELINE, WE COMPUTE S = R*R
683 |--[R*S*(A2+S*(A4+S*A6))] + [R+S*(A1+S*(A3+S*A5))]
684
685 fmovex %fp0,%fp1
686 fmulx %fp1,%fp1 | ...fp1 IS S = R*R
687
688 fmoves #0x3950097B,%fp2 | ...fp2 IS a6
689 | MOVE.W #0,2(%a1) ...load 2^(J/64) in cache
690
691 fmulx %fp1,%fp2 | ...fp2 IS S*A6
692 fmovex %fp1,%fp3
693 fmuls #0x3AB60B6A,%fp3 | ...fp3 IS S*A5
694
695 faddd EM1A4,%fp2 | ...fp2 IS A4+S*A6
696 faddd EM1A3,%fp3 | ...fp3 IS A3+S*A5
697 movew %d0,SC(%a6) | ...SC is 2^(M) in extended
698 clrw SC+2(%a6)
699 movel #0x80000000,SC+4(%a6)
700 clrl SC+8(%a6)
701
702 fmulx %fp1,%fp2 | ...fp2 IS S*(A4+S*A6)
703 movel L_SCR1(%a6),%d0 | ...D0 is M
704 negw %d0 | ...D0 is -M
705 fmulx %fp1,%fp3 | ...fp3 IS S*(A3+S*A5)
706 addiw #0x3FFF,%d0 | ...biased expo. of 2^(-M)
707 faddd EM1A2,%fp2 | ...fp2 IS A2+S*(A4+S*A6)
708 fadds #0x3F000000,%fp3 | ...fp3 IS A1+S*(A3+S*A5)
709
710 fmulx %fp1,%fp2 | ...fp2 IS S*(A2+S*(A4+S*A6))
711 oriw #0x8000,%d0 | ...signed/expo. of -2^(-M)
712 movew %d0,ONEBYSC(%a6) | ...OnebySc is -2^(-M)
713 clrw ONEBYSC+2(%a6)
714 movel #0x80000000,ONEBYSC+4(%a6)
715 clrl ONEBYSC+8(%a6)
716 fmulx %fp3,%fp1 | ...fp1 IS S*(A1+S*(A3+S*A5))
717 | ...fp3 released
718
719 fmulx %fp0,%fp2 | ...fp2 IS R*S*(A2+S*(A4+S*A6))
720 faddx %fp1,%fp0 | ...fp0 IS R+S*(A1+S*(A3+S*A5))
721 | ...fp1 released
722
723 faddx %fp2,%fp0 | ...fp0 IS EXP(R)-1
724 | ...fp2 released
725 fmovemx (%a7)+,%fp2-%fp2/%fp3 | ...fp2 restored
726
727 |--Step 5
728 |--Compute 2^(J/64)*p
729
730 fmulx (%a1),%fp0 | ...2^(J/64)*(Exp(R)-1)
731
732 |--Step 6
733 |--Step 6.1
734 movel L_SCR1(%a6),%d0 | ...retrieve M
735 cmpil #63,%d0
736 bles MLE63
737 |--Step 6.2 M >= 64
738 fmoves 12(%a1),%fp1 | ...fp1 is t
739 faddx ONEBYSC(%a6),%fp1 | ...fp1 is t+OnebySc
740 faddx %fp1,%fp0 | ...p+(t+OnebySc), fp1 released
741 faddx (%a1),%fp0 | ...T+(p+(t+OnebySc))
742 bras EM1SCALE
743 MLE63:
744 |--Step 6.3 M <= 63
745 cmpil #-3,%d0
746 bges MGEN3
747 MLTN3:
748 |--Step 6.4 M <= -4
749 fadds 12(%a1),%fp0 | ...p+t
750 faddx (%a1),%fp0 | ...T+(p+t)
751 faddx ONEBYSC(%a6),%fp0 | ...OnebySc + (T+(p+t))
752 bras EM1SCALE
753 MGEN3:
754 |--Step 6.5 -3 <= M <= 63
755 fmovex (%a1)+,%fp1 | ...fp1 is T
756 fadds (%a1),%fp0 | ...fp0 is p+t
757 faddx ONEBYSC(%a6),%fp1 | ...fp1 is T+OnebySc
758 faddx %fp1,%fp0 | ...(T+OnebySc)+(p+t)
759
760 EM1SCALE:
761 |--Step 6.6
762 fmovel %d1,%FPCR
763 fmulx SC(%a6),%fp0
764
765 bra t_frcinx
766
767 EM1SM:
768 |--Step 7 |X| < 1/4.
769 cmpil #0x3FBE0000,%d0 | ...2^(-65)
770 bges EM1POLY
771
772 EM1TINY:
773 |--Step 8 |X| < 2^(-65)
774 cmpil #0x00330000,%d0 | ...2^(-16312)
775 blts EM12TINY
776 |--Step 8.2
777 movel #0x80010000,SC(%a6) | ...SC is -2^(-16382)
778 movel #0x80000000,SC+4(%a6)
779 clrl SC+8(%a6)
780 fmovex (%a0),%fp0
781 fmovel %d1,%FPCR
782 faddx SC(%a6),%fp0
783
784 bra t_frcinx
785
786 EM12TINY:
787 |--Step 8.3
788 fmovex (%a0),%fp0
789 fmuld TWO140,%fp0
790 movel #0x80010000,SC(%a6)
791 movel #0x80000000,SC+4(%a6)
792 clrl SC+8(%a6)
793 faddx SC(%a6),%fp0
794 fmovel %d1,%FPCR
795 fmuld TWON140,%fp0
796
797 bra t_frcinx
798
799 EM1POLY:
800 |--Step 9 exp(X)-1 by a simple polynomial
801 fmovex (%a0),%fp0 | ...fp0 is X
802 fmulx %fp0,%fp0 | ...fp0 is S := X*X
803 fmovemx %fp2-%fp2/%fp3,-(%a7) | ...save fp2
804 fmoves #0x2F30CAA8,%fp1 | ...fp1 is B12
805 fmulx %fp0,%fp1 | ...fp1 is S*B12
806 fmoves #0x310F8290,%fp2 | ...fp2 is B11
807 fadds #0x32D73220,%fp1 | ...fp1 is B10+S*B12
808
809 fmulx %fp0,%fp2 | ...fp2 is S*B11
810 fmulx %fp0,%fp1 | ...fp1 is S*(B10 + ...
811
812 fadds #0x3493F281,%fp2 | ...fp2 is B9+S*...
813 faddd EM1B8,%fp1 | ...fp1 is B8+S*...
814
815 fmulx %fp0,%fp2 | ...fp2 is S*(B9+...
816 fmulx %fp0,%fp1 | ...fp1 is S*(B8+...
817
818 faddd EM1B7,%fp2 | ...fp2 is B7+S*...
819 faddd EM1B6,%fp1 | ...fp1 is B6+S*...
820
821 fmulx %fp0,%fp2 | ...fp2 is S*(B7+...
822 fmulx %fp0,%fp1 | ...fp1 is S*(B6+...
823
824 faddd EM1B5,%fp2 | ...fp2 is B5+S*...
825 faddd EM1B4,%fp1 | ...fp1 is B4+S*...
826
827 fmulx %fp0,%fp2 | ...fp2 is S*(B5+...
828 fmulx %fp0,%fp1 | ...fp1 is S*(B4+...
829
830 faddd EM1B3,%fp2 | ...fp2 is B3+S*...
831 faddx EM1B2,%fp1 | ...fp1 is B2+S*...
832
833 fmulx %fp0,%fp2 | ...fp2 is S*(B3+...
834 fmulx %fp0,%fp1 | ...fp1 is S*(B2+...
835
836 fmulx %fp0,%fp2 | ...fp2 is S*S*(B3+...)
837 fmulx (%a0),%fp1 | ...fp1 is X*S*(B2...
838
839 fmuls #0x3F000000,%fp0 | ...fp0 is S*B1
840 faddx %fp2,%fp1 | ...fp1 is Q
841 | ...fp2 released
842
843 fmovemx (%a7)+,%fp2-%fp2/%fp3 | ...fp2 restored
844
845 faddx %fp1,%fp0 | ...fp0 is S*B1+Q
846 | ...fp1 released
847
848 fmovel %d1,%FPCR
849 faddx (%a0),%fp0
850
851 bra t_frcinx
852
853 EM1BIG:
854 |--Step 10 |X| > 70 log2
855 movel (%a0),%d0
856 cmpil #0,%d0
857 bgt EXPC1
858 |--Step 10.2
859 fmoves #0xBF800000,%fp0 | ...fp0 is -1
860 fmovel %d1,%FPCR
861 fadds #0x00800000,%fp0 | ...-1 + 2^(-126)
862
863 bra t_frcinx
864
865 |end
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