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 |
331 |               Copyright (C) Motorola, Inc. 1990
332 |                       All Rights Reserved
333 |
334 |       For details on the license for this file, please see the
335 |       file, README, in this same directory.
337 |setox  idnt    2,1 | Motorola 040 Floating Point Software Package
339         |section        8
341 #include "fpsp.h"
343 L2:     .long   0x3FDC0000,0x82E30865,0x4361C4C6,0x00000000
345 EXPA3:  .long   0x3FA55555,0x55554431
346 EXPA2:  .long   0x3FC55555,0x55554018
348 HUGE:   .long   0x7FFE0000,0xFFFFFFFF,0xFFFFFFFF,0x00000000
349 TINY:   .long   0x00010000,0xFFFFFFFF,0xFFFFFFFF,0x00000000
351 EM1A4:  .long   0x3F811111,0x11174385
352 EM1A3:  .long   0x3FA55555,0x55554F5A
354 EM1A2:  .long   0x3FC55555,0x55555555,0x00000000,0x00000000
356 EM1B8:  .long   0x3EC71DE3,0xA5774682
357 EM1B7:  .long   0x3EFA01A0,0x19D7CB68
359 EM1B6:  .long   0x3F2A01A0,0x1A019DF3
360 EM1B5:  .long   0x3F56C16C,0x16C170E2
362 EM1B4:  .long   0x3F811111,0x11111111
363 EM1B3:  .long   0x3FA55555,0x55555555
365 EM1B2:  .long   0x3FFC0000,0xAAAAAAAA,0xAAAAAAAB
366         .long   0x00000000
368 TWO140: .long   0x48B00000,0x00000000
369 TWON140:        .long   0x37300000,0x00000000
371 EXPTBL:
372         .long   0x3FFF0000,0x80000000,0x00000000,0x00000000
373         .long   0x3FFF0000,0x8164D1F3,0xBC030774,0x9F841A9B
374         .long   0x3FFF0000,0x82CD8698,0xAC2BA1D8,0x9FC1D5B9
375         .long   0x3FFF0000,0x843A28C3,0xACDE4048,0xA0728369
376         .long   0x3FFF0000,0x85AAC367,0xCC487B14,0x1FC5C95C
377         .long   0x3FFF0000,0x871F6196,0x9E8D1010,0x1EE85C9F
378         .long   0x3FFF0000,0x88980E80,0x92DA8528,0x9FA20729
379         .long   0x3FFF0000,0x8A14D575,0x496EFD9C,0xA07BF9AF
380         .long   0x3FFF0000,0x8B95C1E3,0xEA8BD6E8,0xA0020DCF
381         .long   0x3FFF0000,0x8D1ADF5B,0x7E5BA9E4,0x205A63DA
382         .long   0x3FFF0000,0x8EA4398B,0x45CD53C0,0x1EB70051
383         .long   0x3FFF0000,0x9031DC43,0x1466B1DC,0x1F6EB029
384         .long   0x3FFF0000,0x91C3D373,0xAB11C338,0xA0781494
385         .long   0x3FFF0000,0x935A2B2F,0x13E6E92C,0x9EB319B0
386         .long   0x3FFF0000,0x94F4EFA8,0xFEF70960,0x2017457D
387         .long   0x3FFF0000,0x96942D37,0x20185A00,0x1F11D537
388         .long   0x3FFF0000,0x9837F051,0x8DB8A970,0x9FB952DD
389         .long   0x3FFF0000,0x99E04593,0x20B7FA64,0x1FE43087
390         .long   0x3FFF0000,0x9B8D39B9,0xD54E5538,0x1FA2A818
391         .long   0x3FFF0000,0x9D3ED9A7,0x2CFFB750,0x1FDE494D
392         .long   0x3FFF0000,0x9EF53260,0x91A111AC,0x20504890
393         .long   0x3FFF0000,0xA0B0510F,0xB9714FC4,0xA073691C
394         .long   0x3FFF0000,0xA2704303,0x0C496818,0x1F9B7A05
395         .long   0x3FFF0000,0xA43515AE,0x09E680A0,0xA0797126
396         .long   0x3FFF0000,0xA5FED6A9,0xB15138EC,0xA071A140
397         .long   0x3FFF0000,0xA7CD93B4,0xE9653568,0x204F62DA
398         .long   0x3FFF0000,0xA9A15AB4,0xEA7C0EF8,0x1F283C4A
399         .long   0x3FFF0000,0xAB7A39B5,0xA93ED338,0x9F9A7FDC
400         .long   0x3FFF0000,0xAD583EEA,0x42A14AC8,0xA05B3FAC
401         .long   0x3FFF0000,0xAF3B78AD,0x690A4374,0x1FDF2610
402         .long   0x3FFF0000,0xB123F581,0xD2AC2590,0x9F705F90
403         .long   0x3FFF0000,0xB311C412,0xA9112488,0x201F678A
404         .long   0x3FFF0000,0xB504F333,0xF9DE6484,0x1F32FB13
405         .long   0x3FFF0000,0xB6FD91E3,0x28D17790,0x20038B30
406         .long   0x3FFF0000,0xB8FBAF47,0x62FB9EE8,0x200DC3CC
407         .long   0x3FFF0000,0xBAFF5AB2,0x133E45FC,0x9F8B2AE6
408         .long   0x3FFF0000,0xBD08A39F,0x580C36C0,0xA02BBF70
409         .long   0x3FFF0000,0xBF1799B6,0x7A731084,0xA00BF518
410         .long   0x3FFF0000,0xC12C4CCA,0x66709458,0xA041DD41
411         .long   0x3FFF0000,0xC346CCDA,0x24976408,0x9FDF137B
412         .long   0x3FFF0000,0xC5672A11,0x5506DADC,0x201F1568
413         .long   0x3FFF0000,0xC78D74C8,0xABB9B15C,0x1FC13A2E
414         .long   0x3FFF0000,0xC9B9BD86,0x6E2F27A4,0xA03F8F03
415         .long   0x3FFF0000,0xCBEC14FE,0xF2727C5C,0x1FF4907D
416         .long   0x3FFF0000,0xCE248C15,0x1F8480E4,0x9E6E53E4
417         .long   0x3FFF0000,0xD06333DA,0xEF2B2594,0x1FD6D45C
418         .long   0x3FFF0000,0xD2A81D91,0xF12AE45C,0xA076EDB9
419         .long   0x3FFF0000,0xD4F35AAB,0xCFEDFA20,0x9FA6DE21
420         .long   0x3FFF0000,0xD744FCCA,0xD69D6AF4,0x1EE69A2F
421         .long   0x3FFF0000,0xD99D15C2,0x78AFD7B4,0x207F439F
422         .long   0x3FFF0000,0xDBFBB797,0xDAF23754,0x201EC207
423         .long   0x3FFF0000,0xDE60F482,0x5E0E9124,0x9E8BE175
424         .long   0x3FFF0000,0xE0CCDEEC,0x2A94E110,0x20032C4B
425         .long   0x3FFF0000,0xE33F8972,0xBE8A5A50,0x2004DFF5
426         .long   0x3FFF0000,0xE5B906E7,0x7C8348A8,0x1E72F47A
427         .long   0x3FFF0000,0xE8396A50,0x3C4BDC68,0x1F722F22
428         .long   0x3FFF0000,0xEAC0C6E7,0xDD243930,0xA017E945
429         .long   0x3FFF0000,0xED4F301E,0xD9942B84,0x1F401A5B
430         .long   0x3FFF0000,0xEFE4B99B,0xDCDAF5CC,0x9FB9A9E3
431         .long   0x3FFF0000,0xF281773C,0x59FFB138,0x20744C05
432         .long   0x3FFF0000,0xF5257D15,0x2486CC2C,0x1F773A19
433         .long   0x3FFF0000,0xF7D0DF73,0x0AD13BB8,0x1FFE90D5
434         .long   0x3FFF0000,0xFA83B2DB,0x722A033C,0xA041ED22
435         .long   0x3FFF0000,0xFD3E0C0C,0xF486C174,0x1F853F3A
437         .set    ADJFLAG,L_SCR2
438         .set    SCALE,FP_SCR1
439         .set    ADJSCALE,FP_SCR2
440         .set    SC,FP_SCR3
441         .set    ONEBYSC,FP_SCR4
443         | xref  t_frcinx
444         |xref   t_extdnrm
445         |xref   t_unfl
446         |xref   t_ovfl
448         .global setoxd
449 setoxd:
450 |--entry point for EXP(X), X is denormalized
451         movel           (%a0),%d0
452         andil           #0x80000000,%d0
453         oril            #0x00800000,%d0         | ...sign(X)*2^(-126)
454         movel           %d0,-(%sp)
455         fmoves          #0x3F800000,%fp0
456         fmovel          %d1,%fpcr
457         fadds           (%sp)+,%fp0
458         bra             t_frcinx
460         .global setox
461 setox:
462 |--entry point for EXP(X), here X is finite, non-zero, and not NaN's
464 |--Step 1.
465         movel           (%a0),%d0        | ...load part of input X
466         andil           #0x7FFF0000,%d0 | ...biased expo. of X
467         cmpil           #0x3FBE0000,%d0 | ...2^(-65)
468         bges            EXPC1           | ...normal case
469         bra             EXPSM
471 EXPC1:
472 |--The case |X| >= 2^(-65)
473         movew           4(%a0),%d0      | ...expo. and partial sig. of |X|
474         cmpil           #0x400CB167,%d0 | ...16380 log2 trunc. 16 bits
475         blts            EXPMAIN  | ...normal case
476         bra             EXPBIG
478 EXPMAIN:
479 |--Step 2.
480 |--This is the normal branch:   2^(-65) <= |X| < 16380 log2.
481         fmovex          (%a0),%fp0      | ...load input from (a0)
483         fmovex          %fp0,%fp1
484         fmuls           #0x42B8AA3B,%fp0        | ...64/log2 * X
485         fmovemx %fp2-%fp2/%fp3,-(%a7)           | ...save fp2
486         movel           #0,ADJFLAG(%a6)
487         fmovel          %fp0,%d0                | ...N = int( X * 64/log2 )
488         lea             EXPTBL,%a1
489         fmovel          %d0,%fp0                | ...convert to floating-format
491         movel           %d0,L_SCR1(%a6) | ...save N temporarily
492         andil           #0x3F,%d0               | ...D0 is J = N mod 64
493         lsll            #4,%d0
494         addal           %d0,%a1         | ...address of 2^(J/64)
495         movel           L_SCR1(%a6),%d0
496         asrl            #6,%d0          | ...D0 is M
497         addiw           #0x3FFF,%d0     | ...biased expo. of 2^(M)
498         movew           L2,L_SCR1(%a6)  | ...prefetch L2, no need in CB
500 EXPCONT1:
501 |--Step 3.
502 |--fp1,fp2 saved on the stack. fp0 is N, fp1 is X,
503 |--a0 points to 2^(J/64), D0 is biased expo. of 2^(M)
504         fmovex          %fp0,%fp2
505         fmuls           #0xBC317218,%fp0        | ...N * L1, L1 = lead(-log2/64)
506         fmulx           L2,%fp2         | ...N * L2, L1+L2 = -log2/64
507         faddx           %fp1,%fp0               | ...X + N*L1
508         faddx           %fp2,%fp0               | ...fp0 is R, reduced arg.
509 |       MOVE.W          #\$3FA5,EXPA3    ...load EXPA3 in cache
511 |--Step 4.
512 |--WE NOW COMPUTE EXP(R)-1 BY A POLYNOMIAL
513 |-- R + R*R*(A1 + R*(A2 + R*(A3 + R*(A4 + R*A5))))
514 |--TO FULLY UTILIZE THE PIPELINE, WE COMPUTE S = R*R
515 |--[R+R*S*(A2+S*A4)] + [S*(A1+S*(A3+S*A5))]
517         fmovex          %fp0,%fp1
518         fmulx           %fp1,%fp1               | ...fp1 IS S = R*R
520         fmoves          #0x3AB60B70,%fp2        | ...fp2 IS A5
521 |       MOVE.W          #0,2(%a1)       ...load 2^(J/64) in cache
523         fmulx           %fp1,%fp2               | ...fp2 IS S*A5
524         fmovex          %fp1,%fp3
525         fmuls           #0x3C088895,%fp3        | ...fp3 IS S*A4
527         faddd           EXPA3,%fp2      | ...fp2 IS A3+S*A5
528         faddd           EXPA2,%fp3      | ...fp3 IS A2+S*A4
530         fmulx           %fp1,%fp2               | ...fp2 IS S*(A3+S*A5)
531         movew           %d0,SCALE(%a6)  | ...SCALE is 2^(M) in extended
532         clrw            SCALE+2(%a6)
533         movel           #0x80000000,SCALE+4(%a6)
534         clrl            SCALE+8(%a6)
536         fmulx           %fp1,%fp3               | ...fp3 IS S*(A2+S*A4)
538         fadds           #0x3F000000,%fp2        | ...fp2 IS A1+S*(A3+S*A5)
539         fmulx           %fp0,%fp3               | ...fp3 IS R*S*(A2+S*A4)
541         fmulx           %fp1,%fp2               | ...fp2 IS S*(A1+S*(A3+S*A5))
542         faddx           %fp3,%fp0               | ...fp0 IS R+R*S*(A2+S*A4),
543 |                                       ...fp3 released
545         fmovex          (%a1)+,%fp1     | ...fp1 is lead. pt. of 2^(J/64)
546         faddx           %fp2,%fp0               | ...fp0 is EXP(R) - 1
547 |                                       ...fp2 released
549 |--Step 5
550 |--final reconstruction process
551 |--EXP(X) = 2^M * ( 2^(J/64) + 2^(J/64)*(EXP(R)-1) )
553         fmulx           %fp1,%fp0               | ...2^(J/64)*(Exp(R)-1)
554         fmovemx (%a7)+,%fp2-%fp2/%fp3   | ...fp2 restored
555         fadds           (%a1),%fp0      | ...accurate 2^(J/64)
557         faddx           %fp1,%fp0               | ...2^(J/64) + 2^(J/64)*...
558         movel           ADJFLAG(%a6),%d0
560 |--Step 6
561         tstl            %d0
562         beqs            NORMAL
563 ADJUST:
564         fmulx           ADJSCALE(%a6),%fp0
565 NORMAL:
566         fmovel          %d1,%FPCR               | ...restore user FPCR
567         fmulx           SCALE(%a6),%fp0 | ...multiply 2^(M)
568         bra             t_frcinx
570 EXPSM:
571 |--Step 7
572         fmovemx (%a0),%fp0-%fp0 | ...in case X is denormalized
573         fmovel          %d1,%FPCR
574         fadds           #0x3F800000,%fp0        | ...1+X in user mode
575         bra             t_frcinx
577 EXPBIG:
578 |--Step 8
579         cmpil           #0x400CB27C,%d0 | ...16480 log2
580         bgts            EXP2BIG
581 |--Steps 8.2 -- 8.6
582         fmovex          (%a0),%fp0      | ...load input from (a0)
584         fmovex          %fp0,%fp1
585         fmuls           #0x42B8AA3B,%fp0        | ...64/log2 * X
586         fmovemx  %fp2-%fp2/%fp3,-(%a7)          | ...save fp2
587         movel           #1,ADJFLAG(%a6)
588         fmovel          %fp0,%d0                | ...N = int( X * 64/log2 )
589         lea             EXPTBL,%a1
590         fmovel          %d0,%fp0                | ...convert to floating-format
591         movel           %d0,L_SCR1(%a6)                 | ...save N temporarily
592         andil           #0x3F,%d0                | ...D0 is J = N mod 64
593         lsll            #4,%d0
594         addal           %d0,%a1                 | ...address of 2^(J/64)
595         movel           L_SCR1(%a6),%d0
596         asrl            #6,%d0                  | ...D0 is K
597         movel           %d0,L_SCR1(%a6)                 | ...save K temporarily
598         asrl            #1,%d0                  | ...D0 is M1
599         subl            %d0,L_SCR1(%a6)                 | ...a1 is M
600         addiw           #0x3FFF,%d0             | ...biased expo. of 2^(M1)
601         movew           %d0,ADJSCALE(%a6)               | ...ADJSCALE := 2^(M1)
602         clrw            ADJSCALE+2(%a6)
603         movel           #0x80000000,ADJSCALE+4(%a6)
604         clrl            ADJSCALE+8(%a6)
605         movel           L_SCR1(%a6),%d0                 | ...D0 is M
606         addiw           #0x3FFF,%d0             | ...biased expo. of 2^(M)
607         bra             EXPCONT1                | ...go back to Step 3
609 EXP2BIG:
610 |--Step 9
611         fmovel          %d1,%FPCR
612         movel           (%a0),%d0
613         bclrb           #sign_bit,(%a0)         | ...setox always returns positive
614         cmpil           #0,%d0
615         blt             t_unfl
616         bra             t_ovfl
618         .global setoxm1d
619 setoxm1d:
620 |--entry point for EXPM1(X), here X is denormalized
621 |--Step 0.
622         bra             t_extdnrm
625         .global setoxm1
626 setoxm1:
627 |--entry point for EXPM1(X), here X is finite, non-zero, non-NaN
629 |--Step 1.
630 |--Step 1.1
631         movel           (%a0),%d0        | ...load part of input X
632         andil           #0x7FFF0000,%d0 | ...biased expo. of X
633         cmpil           #0x3FFD0000,%d0 | ...1/4
634         bges            EM1CON1  | ...|X| >= 1/4
635         bra             EM1SM
637 EM1CON1:
638 |--Step 1.3
639 |--The case |X| >= 1/4
640         movew           4(%a0),%d0      | ...expo. and partial sig. of |X|
641         cmpil           #0x4004C215,%d0 | ...70log2 rounded up to 16 bits
642         bles            EM1MAIN  | ...1/4 <= |X| <= 70log2
643         bra             EM1BIG
645 EM1MAIN:
646 |--Step 2.
647 |--This is the case:    1/4 <= |X| <= 70 log2.
648         fmovex          (%a0),%fp0      | ...load input from (a0)
650         fmovex          %fp0,%fp1
651         fmuls           #0x42B8AA3B,%fp0        | ...64/log2 * X
652         fmovemx %fp2-%fp2/%fp3,-(%a7)           | ...save fp2
653 |       MOVE.W          #\$3F81,EM1A4            ...prefetch in CB mode
654         fmovel          %fp0,%d0                | ...N = int( X * 64/log2 )
655         lea             EXPTBL,%a1
656         fmovel          %d0,%fp0                | ...convert to floating-format
658         movel           %d0,L_SCR1(%a6)                 | ...save N temporarily
659         andil           #0x3F,%d0                | ...D0 is J = N mod 64
660         lsll            #4,%d0
661         addal           %d0,%a1                 | ...address of 2^(J/64)
662         movel           L_SCR1(%a6),%d0
663         asrl            #6,%d0                  | ...D0 is M
664         movel           %d0,L_SCR1(%a6)                 | ...save a copy of M
665 |       MOVE.W          #\$3FDC,L2               ...prefetch L2 in CB mode
667 |--Step 3.
668 |--fp1,fp2 saved on the stack. fp0 is N, fp1 is X,
669 |--a0 points to 2^(J/64), D0 and a1 both contain M
670         fmovex          %fp0,%fp2
671         fmuls           #0xBC317218,%fp0        | ...N * L1, L1 = lead(-log2/64)
672         fmulx           L2,%fp2         | ...N * L2, L1+L2 = -log2/64
673         faddx           %fp1,%fp0        | ...X + N*L1
674         faddx           %fp2,%fp0        | ...fp0 is R, reduced arg.
675 |       MOVE.W          #\$3FC5,EM1A2            ...load EM1A2 in cache
676         addiw           #0x3FFF,%d0             | ...D0 is biased expo. of 2^M
678 |--Step 4.
679 |--WE NOW COMPUTE EXP(R)-1 BY A POLYNOMIAL
680 |-- R + R*R*(A1 + R*(A2 + R*(A3 + R*(A4 + R*(A5 + R*A6)))))
681 |--TO FULLY UTILIZE THE PIPELINE, WE COMPUTE S = R*R
682 |--[R*S*(A2+S*(A4+S*A6))] + [R+S*(A1+S*(A3+S*A5))]
684         fmovex          %fp0,%fp1
685         fmulx           %fp1,%fp1               | ...fp1 IS S = R*R
687         fmoves          #0x3950097B,%fp2        | ...fp2 IS a6
688 |       MOVE.W          #0,2(%a1)       ...load 2^(J/64) in cache
690         fmulx           %fp1,%fp2               | ...fp2 IS S*A6
691         fmovex          %fp1,%fp3
692         fmuls           #0x3AB60B6A,%fp3        | ...fp3 IS S*A5
694         faddd           EM1A4,%fp2      | ...fp2 IS A4+S*A6
695         faddd           EM1A3,%fp3      | ...fp3 IS A3+S*A5
696         movew           %d0,SC(%a6)             | ...SC is 2^(M) in extended
697         clrw            SC+2(%a6)
698         movel           #0x80000000,SC+4(%a6)
699         clrl            SC+8(%a6)
701         fmulx           %fp1,%fp2               | ...fp2 IS S*(A4+S*A6)
702         movel           L_SCR1(%a6),%d0         | ...D0 is      M
703         negw            %d0             | ...D0 is -M
704         fmulx           %fp1,%fp3               | ...fp3 IS S*(A3+S*A5)
705         addiw           #0x3FFF,%d0     | ...biased expo. of 2^(-M)
706         faddd           EM1A2,%fp2      | ...fp2 IS A2+S*(A4+S*A6)
707         fadds           #0x3F000000,%fp3        | ...fp3 IS A1+S*(A3+S*A5)
709         fmulx           %fp1,%fp2               | ...fp2 IS S*(A2+S*(A4+S*A6))
710         oriw            #0x8000,%d0     | ...signed/expo. of -2^(-M)
711         movew           %d0,ONEBYSC(%a6)        | ...OnebySc is -2^(-M)
712         clrw            ONEBYSC+2(%a6)
713         movel           #0x80000000,ONEBYSC+4(%a6)
714         clrl            ONEBYSC+8(%a6)
715         fmulx           %fp3,%fp1               | ...fp1 IS S*(A1+S*(A3+S*A5))
716 |                                       ...fp3 released
718         fmulx           %fp0,%fp2               | ...fp2 IS R*S*(A2+S*(A4+S*A6))
719         faddx           %fp1,%fp0               | ...fp0 IS R+S*(A1+S*(A3+S*A5))
720 |                                       ...fp1 released
722         faddx           %fp2,%fp0               | ...fp0 IS EXP(R)-1
723 |                                       ...fp2 released
724         fmovemx (%a7)+,%fp2-%fp2/%fp3   | ...fp2 restored
726 |--Step 5
727 |--Compute 2^(J/64)*p
729         fmulx           (%a1),%fp0      | ...2^(J/64)*(Exp(R)-1)
731 |--Step 6
732 |--Step 6.1
733         movel           L_SCR1(%a6),%d0         | ...retrieve M
734         cmpil           #63,%d0
735         bles            MLE63
736 |--Step 6.2     M >= 64
737         fmoves          12(%a1),%fp1    | ...fp1 is t
738         faddx           ONEBYSC(%a6),%fp1       | ...fp1 is t+OnebySc
739         faddx           %fp1,%fp0               | ...p+(t+OnebySc), fp1 released
740         faddx           (%a1),%fp0      | ...T+(p+(t+OnebySc))
741         bras            EM1SCALE
742 MLE63:
743 |--Step 6.3     M <= 63
744         cmpil           #-3,%d0
745         bges            MGEN3
746 MLTN3:
747 |--Step 6.4     M <= -4
748         fadds           12(%a1),%fp0    | ...p+t
749         faddx           (%a1),%fp0      | ...T+(p+t)
750         faddx           ONEBYSC(%a6),%fp0       | ...OnebySc + (T+(p+t))
751         bras            EM1SCALE
752 MGEN3:
753 |--Step 6.5     -3 <= M <= 63
754         fmovex          (%a1)+,%fp1     | ...fp1 is T
755         fadds           (%a1),%fp0      | ...fp0 is p+t
756         faddx           ONEBYSC(%a6),%fp1       | ...fp1 is T+OnebySc
757         faddx           %fp1,%fp0               | ...(T+OnebySc)+(p+t)
759 EM1SCALE:
760 |--Step 6.6
761         fmovel          %d1,%FPCR
762         fmulx           SC(%a6),%fp0
764         bra             t_frcinx
766 EM1SM:
767 |--Step 7       |X| < 1/4.
768         cmpil           #0x3FBE0000,%d0 | ...2^(-65)
769         bges            EM1POLY
771 EM1TINY:
772 |--Step 8       |X| < 2^(-65)
773         cmpil           #0x00330000,%d0 | ...2^(-16312)
774         blts            EM12TINY
775 |--Step 8.2
776         movel           #0x80010000,SC(%a6)     | ...SC is -2^(-16382)
777         movel           #0x80000000,SC+4(%a6)
778         clrl            SC+8(%a6)
779         fmovex          (%a0),%fp0
780         fmovel          %d1,%FPCR
781         faddx           SC(%a6),%fp0
783         bra             t_frcinx
785 EM12TINY:
786 |--Step 8.3
787         fmovex          (%a0),%fp0
788         fmuld           TWO140,%fp0
789         movel           #0x80010000,SC(%a6)
790         movel           #0x80000000,SC+4(%a6)
791         clrl            SC+8(%a6)
792         faddx           SC(%a6),%fp0
793         fmovel          %d1,%FPCR
794         fmuld           TWON140,%fp0
796         bra             t_frcinx
798 EM1POLY:
799 |--Step 9       exp(X)-1 by a simple polynomial
800         fmovex          (%a0),%fp0      | ...fp0 is X
801         fmulx           %fp0,%fp0               | ...fp0 is S := X*X
802         fmovemx %fp2-%fp2/%fp3,-(%a7)   | ...save fp2
803         fmoves          #0x2F30CAA8,%fp1        | ...fp1 is B12
804         fmulx           %fp0,%fp1               | ...fp1 is S*B12
805         fmoves          #0x310F8290,%fp2        | ...fp2 is B11
806         fadds           #0x32D73220,%fp1        | ...fp1 is B10+S*B12
808         fmulx           %fp0,%fp2               | ...fp2 is S*B11
809         fmulx           %fp0,%fp1               | ...fp1 is S*(B10 + ...
811         fadds           #0x3493F281,%fp2        | ...fp2 is B9+S*...
812         faddd           EM1B8,%fp1      | ...fp1 is B8+S*...
814         fmulx           %fp0,%fp2               | ...fp2 is S*(B9+...
815         fmulx           %fp0,%fp1               | ...fp1 is S*(B8+...
817         faddd           EM1B7,%fp2      | ...fp2 is B7+S*...
818         faddd           EM1B6,%fp1      | ...fp1 is B6+S*...
820         fmulx           %fp0,%fp2               | ...fp2 is S*(B7+...
821         fmulx           %fp0,%fp1               | ...fp1 is S*(B6+...
823         faddd           EM1B5,%fp2      | ...fp2 is B5+S*...
824         faddd           EM1B4,%fp1      | ...fp1 is B4+S*...
826         fmulx           %fp0,%fp2               | ...fp2 is S*(B5+...
827         fmulx           %fp0,%fp1               | ...fp1 is S*(B4+...
829         faddd           EM1B3,%fp2      | ...fp2 is B3+S*...
830         faddx           EM1B2,%fp1      | ...fp1 is B2+S*...
832         fmulx           %fp0,%fp2               | ...fp2 is S*(B3+...
833         fmulx           %fp0,%fp1               | ...fp1 is S*(B2+...
835         fmulx           %fp0,%fp2               | ...fp2 is S*S*(B3+...)
836         fmulx           (%a0),%fp1      | ...fp1 is X*S*(B2...
838         fmuls           #0x3F000000,%fp0        | ...fp0 is S*B1
839         faddx           %fp2,%fp1               | ...fp1 is Q
840 |                                       ...fp2 released
842         fmovemx (%a7)+,%fp2-%fp2/%fp3   | ...fp2 restored
844         faddx           %fp1,%fp0               | ...fp0 is S*B1+Q
845 |                                       ...fp1 released
847         fmovel          %d1,%FPCR
848         faddx           (%a0),%fp0
850         bra             t_frcinx
852 EM1BIG:
853 |--Step 10      |X| > 70 log2
854         movel           (%a0),%d0
855         cmpil           #0,%d0
856         bgt             EXPC1
857 |--Step 10.2
858         fmoves          #0xBF800000,%fp0        | ...fp0 is -1
859         fmovel          %d1,%FPCR
860         fadds           #0x00800000,%fp0        | ...-1 + 2^(-126)
862         bra             t_frcinx
864         |end