| blob | 145af5447581f1f3719549dd9cf7494c02dc44be |
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