1 | 1 |
2 | setox.sa 3.1 12/10/90 2 | setox.sa 3.1 12/10/90
3 | 3 |
4 | The entry point setox computes the exp 4 | The entry point setox computes the exponential of a value.
5 | setoxd does the same except the input 5 | setoxd does the same except the input value is a denormalized
6 | number. setoxm1 computes exp(X)-1, and 6 | number. setoxm1 computes exp(X)-1, and setoxm1d computes
7 | exp(X)-1 for denormalized X. 7 | exp(X)-1 for denormalized X.
8 | 8 |
9 | INPUT 9 | INPUT
10 | ----- 10 | -----
11 | Double-extended value in memory locati 11 | Double-extended value in memory location pointed to by address
12 | register a0. 12 | register a0.
13 | 13 |
14 | OUTPUT 14 | OUTPUT
15 | ------ 15 | ------
16 | exp(X) or exp(X)-1 returned in floatin 16 | exp(X) or exp(X)-1 returned in floating-point register fp0.
17 | 17 |
18 | ACCURACY and MONOTONICITY 18 | ACCURACY and MONOTONICITY
19 | ------------------------- 19 | -------------------------
20 | The returned result is within 0.85 ulp 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 re 21 | within 0.5001 ulp to 53 bits if the result is subsequently rounded
22 | to double precision. The result is pro 22 | to double precision. The result is provably monotonic in double
23 | precision. 23 | precision.
24 | 24 |
25 | SPEED 25 | SPEED
26 | ----- 26 | -----
27 | Two timings are measured, both in the 27 | Two timings are measured, both in the copy-back mode. The
28 | first one is measured when the functio 28 | first one is measured when the function is invoked the first time
29 | (so the instructions and data are not 29 | (so the instructions and data are not in cache), and the
30 | second one is measured when the functi 30 | second one is measured when the function is reinvoked at the same
31 | input argument. 31 | input argument.
32 | 32 |
33 | The program setox takes approximately 33 | The program setox takes approximately 210/190 cycles for input
34 | argument X whose magnitude is less tha 34 | argument X whose magnitude is less than 16380 log2, which
35 | is the usual situation. For the less c 35 | is the usual situation. For the less common arguments,
36 | depending on their values, the program 36 | depending on their values, the program may run faster or slower --
37 | but no worse than 10% slower even in t 37 | but no worse than 10% slower even in the extreme cases.
38 | 38 |
39 | The program setoxm1 takes approximatel 39 | The program setoxm1 takes approximately ??? / ??? cycles for input
40 | argument X, 0.25 <= |X| < 70log2. For 40 | argument X, 0.25 <= |X| < 70log2. For |X| < 0.25, it takes
41 | approximately ??? / ??? cycles. For th 41 | approximately ??? / ??? cycles. For the less common arguments,
42 | depending on their values, the program 42 | depending on their values, the program may run faster or slower --
43 | but no worse than 10% slower even in t 43 | but no worse than 10% slower even in the extreme cases.
44 | 44 |
45 | ALGORITHM and IMPLEMENTATION NOTES 45 | ALGORITHM and IMPLEMENTATION NOTES
46 | ---------------------------------- 46 | ----------------------------------
47 | 47 |
48 | setoxd 48 | setoxd
49 | ------ 49 | ------
50 | Step 1. Set ans := 1.0 50 | Step 1. Set ans := 1.0
51 | 51 |
52 | Step 2. Return ans := ans + sign(X)*2 52 | Step 2. Return ans := ans + sign(X)*2^(-126). Exit.
53 | Notes: This will always generate one 53 | Notes: This will always generate one exception -- inexact.
54 | 54 |
55 | 55 |
56 | setox 56 | setox
57 | ----- 57 | -----
58 | 58 |
59 | Step 1. Filter out extreme cases of in 59 | Step 1. Filter out extreme cases of input argument.
60 | 1.1 If |X| >= 2^(-65), go 60 | 1.1 If |X| >= 2^(-65), go to Step 1.3.
61 | 1.2 Go to Step 7. 61 | 1.2 Go to Step 7.
62 | 1.3 If |X| < 16380 log(2), 62 | 1.3 If |X| < 16380 log(2), go to Step 2.
63 | 1.4 Go to Step 8. 63 | 1.4 Go to Step 8.
64 | Notes: The usual case should take the 64 | Notes: The usual case should take the branches 1.1 -> 1.3 -> 2.
65 | To avoid the use of floating- 65 | To avoid the use of floating-point comparisons, a
66 | compact representation of |X| 66 | compact representation of |X| is used. This format is a
67 | 32-bit integer, the upper (mo 67 | 32-bit integer, the upper (more significant) 16 bits are
68 | the sign and biased exponent 68 | the sign and biased exponent field of |X|; the lower 16
69 | bits are the 16 most signific 69 | bits are the 16 most significant fraction (including the
70 | explicit bit) bits of |X|. Co 70 | explicit bit) bits of |X|. Consequently, the comparisons
71 | in Steps 1.1 and 1.3 can be p 71 | in Steps 1.1 and 1.3 can be performed by integer comparison.
72 | Note also that the constant 1 72 | Note also that the constant 16380 log(2) used in Step 1.3
73 | is also in the compact form. 73 | is also in the compact form. Thus taking the branch
74 | to Step 2 guarantees |X| < 16 74 | to Step 2 guarantees |X| < 16380 log(2). There is no harm
75 | to have a small number of cas 75 | to have a small number of cases where |X| is less than,
76 | but close to, 16380 log(2) an 76 | but close to, 16380 log(2) and the branch to Step 9 is
77 | taken. 77 | taken.
78 | 78 |
79 | Step 2. Calculate N = round-to-nearest 79 | Step 2. Calculate N = round-to-nearest-int( X * 64/log2 ).
80 | 2.1 Set AdjFlag := 0 (indi 80 | 2.1 Set AdjFlag := 0 (indicates the branch 1.3 -> 2 was taken)
81 | 2.2 N := round-to-nearest- 81 | 2.2 N := round-to-nearest-integer( X * 64/log2 ).
82 | 2.3 Calculate J = N 82 | 2.3 Calculate J = N mod 64; so J = 0,1,2,..., or 63.
83 | 2.4 Calculate M = (N 83 | 2.4 Calculate M = (N - J)/64; so N = 64M + J.
84 | 2.5 Calculate the address 84 | 2.5 Calculate the address of the stored value of 2^(J/64).
85 | 2.6 Create the value Scale 85 | 2.6 Create the value Scale = 2^M.
86 | Notes: The calculation in 2.2 is real 86 | Notes: The calculation in 2.2 is really performed by
87 | 87 |
88 | Z := X * constant 88 | Z := X * constant
89 | N := round-to-nearest- 89 | N := round-to-nearest-integer(Z)
90 | 90 |
91 | where 91 | where
92 | 92 |
93 | constant := single-pre 93 | constant := single-precision( 64/log 2 ).
94 | 94 |
95 | Using a single-precision cons 95 | Using a single-precision constant avoids memory access.
96 | Another effect of using a sin 96 | Another effect of using a single-precision "constant" is
97 | that the calculated value Z i 97 | that the calculated value Z is
98 | 98 |
99 | Z = X*(64/log2)*(1+eps 99 | Z = X*(64/log2)*(1+eps), |eps| <= 2^(-24).
100 | 100 |
101 | This error has to be consider 101 | This error has to be considered later in Steps 3 and 4.
102 | 102 |
103 | Step 3. Calculate X - N*log2/64. 103 | Step 3. Calculate X - N*log2/64.
104 | 3.1 R := X + N*L1, where L 104 | 3.1 R := X + N*L1, where L1 := single-precision(-log2/64).
105 | 3.2 R := R + N*L2, L2 := e 105 | 3.2 R := R + N*L2, L2 := extended-precision(-log2/64 - L1).
106 | Notes: a) The way L1 and L2 are chose 106 | Notes: a) The way L1 and L2 are chosen ensures L1+L2 approximate
107 | the value -log2/64 107 | the value -log2/64 to 88 bits of accuracy.
108 | b) N*L1 is exact because N is 108 | b) N*L1 is exact because N is no longer than 22 bits and
109 | L1 is no longer than 24 bits. 109 | L1 is no longer than 24 bits.
110 | c) The calculation X+N*L1 is 110 | c) The calculation X+N*L1 is also exact due to cancellation.
111 | Thus, R is practically X+N(L1 111 | Thus, R is practically X+N(L1+L2) to full 64 bits.
112 | d) It is important to estimat 112 | d) It is important to estimate how large can |R| be after
113 | Step 3.2. 113 | Step 3.2.
114 | 114 |
115 | N = rnd-to-int( X*64/l 115 | N = rnd-to-int( X*64/log2 (1+eps) ), |eps|<=2^(-24)
116 | X*64/log2 (1+eps) 116 | X*64/log2 (1+eps) = N + f, |f| <= 0.5
117 | X*64/log2 - N = 117 | X*64/log2 - N = f - eps*X 64/log2
118 | X - N*log2/64 = 118 | X - N*log2/64 = f*log2/64 - eps*X
119 | 119 |
120 | 120 |
121 | Now |X| <= 16446 log2, thus 121 | Now |X| <= 16446 log2, thus
122 | 122 |
123 | |X - N*log2/64| <= (0. 123 | |X - N*log2/64| <= (0.5 + 16446/2^(18))*log2/64
124 | <= 0.5 124 | <= 0.57 log2/64.
125 | This bound will be used in St 125 | This bound will be used in Step 4.
126 | 126 |
127 | Step 4. Approximate exp(R)-1 by a poly 127 | Step 4. Approximate exp(R)-1 by a polynomial
128 | p = R + R*R*(A1 + R*(A 128 | p = R + R*R*(A1 + R*(A2 + R*(A3 + R*(A4 + R*A5))))
129 | Notes: a) In order to reduce memory a 129 | Notes: a) In order to reduce memory access, the coefficients are
130 | made as "short" as possible: 130 | made as "short" as possible: A1 (which is 1/2), A4 and A5
131 | are single precision; A2 and 131 | are single precision; A2 and A3 are double precision.
132 | b) Even with the restrictions 132 | b) Even with the restrictions above,
133 | |p - (exp(R)-1)| < 2^( 133 | |p - (exp(R)-1)| < 2^(-68.8) for all |R| <= 0.0062.
134 | Note that 0.0062 is slightly 134 | Note that 0.0062 is slightly bigger than 0.57 log2/64.
135 | c) To fully utilize the pipel 135 | c) To fully utilize the pipeline, p is separated into
136 | two independent pieces of rou 136 | two independent pieces of roughly equal complexities
137 | p = [ R + R*S*(A2 + S* 137 | p = [ R + R*S*(A2 + S*A4) ] +
138 | [ S*(A1 + S*(A 138 | [ S*(A1 + S*(A3 + S*A5)) ]
139 | where S = R*R. 139 | where S = R*R.
140 | 140 |
141 | Step 5. Compute 2^(J/64)*exp(R) = 2^(J 141 | Step 5. Compute 2^(J/64)*exp(R) = 2^(J/64)*(1+p) by
142 | ans := T + ( T 142 | ans := T + ( T*p + t)
143 | where T and t are the stored 143 | where T and t are the stored values for 2^(J/64).
144 | Notes: 2^(J/64) is stored as T and t 144 | Notes: 2^(J/64) is stored as T and t where T+t approximates
145 | 2^(J/64) to roughly 85 bits; 145 | 2^(J/64) to roughly 85 bits; T is in extended precision
146 | and t is in single precision. 146 | and t is in single precision. Note also that T is rounded
147 | to 62 bits so that the last t 147 | to 62 bits so that the last two bits of T are zero. The
148 | reason for such a special for 148 | reason for such a special form is that T-1, T-2, and T-8
149 | will all be exact --- a prope 149 | will all be exact --- a property that will give much
150 | more accurate computation of 150 | more accurate computation of the function EXPM1.
151 | 151 |
152 | Step 6. Reconstruction of exp(X) 152 | Step 6. Reconstruction of exp(X)
153 | exp(X) = 2^M * 2^(J/64 153 | exp(X) = 2^M * 2^(J/64) * exp(R).
154 | 6.1 If AdjFlag = 0, go to 154 | 6.1 If AdjFlag = 0, go to 6.3
155 | 6.2 ans := ans * AdjScale 155 | 6.2 ans := ans * AdjScale
156 | 6.3 Restore the user FPCR 156 | 6.3 Restore the user FPCR
157 | 6.4 Return ans := ans * Sc 157 | 6.4 Return ans := ans * Scale. Exit.
158 | Notes: If AdjFlag = 0, we have X = Ml 158 | Notes: If AdjFlag = 0, we have X = Mlog2 + Jlog2/64 + R,
159 | |M| <= 16380, and Scale = 2^M 159 | |M| <= 16380, and Scale = 2^M. Moreover, exp(X) will
160 | neither overflow nor underflo 160 | neither overflow nor underflow. If AdjFlag = 1, that
161 | means that 161 | means that
162 | X = (M1+M)log2 + Jlog2 162 | X = (M1+M)log2 + Jlog2/64 + R, |M1+M| >= 16380.
163 | Hence, exp(X) may overflow or 163 | Hence, exp(X) may overflow or underflow or neither.
164 | When that is the case, AdjSca 164 | When that is the case, AdjScale = 2^(M1) where M1 is
165 | approximately M. Thus 6.2 wil 165 | approximately M. Thus 6.2 will never cause over/underflow.
166 | Possible exception in 6.4 is 166 | Possible exception in 6.4 is overflow or underflow.
167 | The inexact exception is not 167 | The inexact exception is not generated in 6.4. Although
168 | one can argue that the inexac 168 | one can argue that the inexact flag should always be
169 | raised, to simulate that exce 169 | raised, to simulate that exception cost to much than the
170 | flag is worth in practical us 170 | flag is worth in practical uses.
171 | 171 |
172 | Step 7. Return 1 + X. 172 | Step 7. Return 1 + X.
173 | 7.1 ans := X 173 | 7.1 ans := X
174 | 7.2 Restore user FPCR. 174 | 7.2 Restore user FPCR.
175 | 7.3 Return ans := 1 + ans. 175 | 7.3 Return ans := 1 + ans. Exit
176 | Notes: For non-zero X, the inexact ex 176 | Notes: For non-zero X, the inexact exception will always be
177 | raised by 7.3. That is the on 177 | raised by 7.3. That is the only exception raised by 7.3.
178 | Note also that we use the FMO 178 | Note also that we use the FMOVEM instruction to move X
179 | in Step 7.1 to avoid unnecess 179 | in Step 7.1 to avoid unnecessary trapping. (Although
180 | the FMOVEM may not seem relev 180 | the FMOVEM may not seem relevant since X is normalized,
181 | the precaution will be useful 181 | the precaution will be useful in the library version of
182 | this code where the separate 182 | this code where the separate entry for denormalized inputs
183 | will be done away with.) 183 | will be done away with.)
184 | 184 |
185 | Step 8. Handle exp(X) where |X| >= 163 185 | Step 8. Handle exp(X) where |X| >= 16380log2.
186 | 8.1 If |X| > 16480 log2, g 186 | 8.1 If |X| > 16480 log2, go to Step 9.
187 | (mimic 2.2 - 2.6) 187 | (mimic 2.2 - 2.6)
188 | 8.2 N := round-to-integer( 188 | 8.2 N := round-to-integer( X * 64/log2 )
189 | 8.3 Calculate J = N mod 64 189 | 8.3 Calculate J = N mod 64, J = 0,1,...,63
190 | 8.4 K := (N-J)/64, M1 := t 190 | 8.4 K := (N-J)/64, M1 := truncate(K/2), M = K-M1, AdjFlag := 1.
191 | 8.5 Calculate the address 191 | 8.5 Calculate the address of the stored value 2^(J/64).
192 | 8.6 Create the values Scal 192 | 8.6 Create the values Scale = 2^M, AdjScale = 2^M1.
193 | 8.7 Go to Step 3. 193 | 8.7 Go to Step 3.
194 | Notes: Refer to notes for 2.2 - 2.6. 194 | Notes: Refer to notes for 2.2 - 2.6.
195 | 195 |
196 | Step 9. Handle exp(X), |X| > 16480 log 196 | Step 9. Handle exp(X), |X| > 16480 log2.
197 | 9.1 If X < 0, go to 9.3 197 | 9.1 If X < 0, go to 9.3
198 | 9.2 ans := Huge, go to 9.4 198 | 9.2 ans := Huge, go to 9.4
199 | 9.3 ans := Tiny. 199 | 9.3 ans := Tiny.
200 | 9.4 Restore user FPCR. 200 | 9.4 Restore user FPCR.
201 | 9.5 Return ans := ans * an 201 | 9.5 Return ans := ans * ans. Exit.
202 | Notes: Exp(X) will surely overflow or 202 | Notes: Exp(X) will surely overflow or underflow, depending on
203 | X's sign. "Huge" and "Tiny" a 203 | X's sign. "Huge" and "Tiny" are respectively large/tiny
204 | extended-precision numbers wh 204 | extended-precision numbers whose square over/underflow
205 | with an inexact result. Thus, 205 | with an inexact result. Thus, 9.5 always raises the
206 | inexact together with either 206 | inexact together with either overflow or underflow.
207 | 207 |
208 | 208 |
209 | setoxm1d 209 | setoxm1d
210 | -------- 210 | --------
211 | 211 |
212 | Step 1. Set ans := 0 212 | Step 1. Set ans := 0
213 | 213 |
214 | Step 2. Return ans := X + ans. Exit. 214 | Step 2. Return ans := X + ans. Exit.
215 | Notes: This will return X with the ap 215 | Notes: This will return X with the appropriate rounding
216 | precision prescribed by the u 216 | precision prescribed by the user FPCR.
217 | 217 |
218 | setoxm1 218 | setoxm1
219 | ------- 219 | -------
220 | 220 |
221 | Step 1. Check |X| 221 | Step 1. Check |X|
222 | 1.1 If |X| >= 1/4, go to S 222 | 1.1 If |X| >= 1/4, go to Step 1.3.
223 | 1.2 Go to Step 7. 223 | 1.2 Go to Step 7.
224 | 1.3 If |X| < 70 log(2), go 224 | 1.3 If |X| < 70 log(2), go to Step 2.
225 | 1.4 Go to Step 10. 225 | 1.4 Go to Step 10.
226 | Notes: The usual case should take the 226 | Notes: The usual case should take the branches 1.1 -> 1.3 -> 2.
227 | However, it is conceivable |X 227 | However, it is conceivable |X| can be small very often
228 | because EXPM1 is intended to 228 | because EXPM1 is intended to evaluate exp(X)-1 accurately
229 | when |X| is small. For furthe 229 | when |X| is small. For further details on the comparisons,
230 | see the notes on Step 1 of se 230 | see the notes on Step 1 of setox.
231 | 231 |
232 | Step 2. Calculate N = round-to-nearest 232 | Step 2. Calculate N = round-to-nearest-int( X * 64/log2 ).
233 | 2.1 N := round-to-nearest- 233 | 2.1 N := round-to-nearest-integer( X * 64/log2 ).
234 | 2.2 Calculate J = N 234 | 2.2 Calculate J = N mod 64; so J = 0,1,2,..., or 63.
235 | 2.3 Calculate M = (N 235 | 2.3 Calculate M = (N - J)/64; so N = 64M + J.
236 | 2.4 Calculate the address 236 | 2.4 Calculate the address of the stored value of 2^(J/64).
237 | 2.5 Create the values Sc = 237 | 2.5 Create the values Sc = 2^M and OnebySc := -2^(-M).
238 | Notes: See the notes on Step 2 of set 238 | Notes: See the notes on Step 2 of setox.
239 | 239 |
240 | Step 3. Calculate X - N*log2/64. 240 | Step 3. Calculate X - N*log2/64.
241 | 3.1 R := X + N*L1, where L 241 | 3.1 R := X + N*L1, where L1 := single-precision(-log2/64).
242 | 3.2 R := R + N*L2, L2 := e 242 | 3.2 R := R + N*L2, L2 := extended-precision(-log2/64 - L1).
243 | Notes: Applying the analysis of Step 243 | Notes: Applying the analysis of Step 3 of setox in this case
244 | shows that |R| <= 0.0055 (not 244 | shows that |R| <= 0.0055 (note that |X| <= 70 log2 in
245 | this case). 245 | this case).
246 | 246 |
247 | Step 4. Approximate exp(R)-1 by a poly 247 | Step 4. Approximate exp(R)-1 by a polynomial
248 | p = R+R*R*(A1+R*(A2+R* 248 | p = R+R*R*(A1+R*(A2+R*(A3+R*(A4+R*(A5+R*A6)))))
249 | Notes: a) In order to reduce memory a 249 | Notes: a) In order to reduce memory access, the coefficients are
250 | made as "short" as possible: 250 | made as "short" as possible: A1 (which is 1/2), A5 and A6
251 | are single precision; A2, A3 251 | are single precision; A2, A3 and A4 are double precision.
252 | b) Even with the restriction 252 | b) Even with the restriction above,
253 | |p - (exp(R)-1)| < 253 | |p - (exp(R)-1)| < |R| * 2^(-72.7)
254 | for all |R| <= 0.0055. 254 | for all |R| <= 0.0055.
255 | c) To fully utilize the pipel 255 | c) To fully utilize the pipeline, p is separated into
256 | two independent pieces of rou 256 | two independent pieces of roughly equal complexity
257 | p = [ R*S*(A2 + S*(A4 257 | p = [ R*S*(A2 + S*(A4 + S*A6)) ] +
258 | [ R + S*(A1 + 258 | [ R + S*(A1 + S*(A3 + S*A5)) ]
259 | where S = R*R. 259 | where S = R*R.
260 | 260 |
261 | Step 5. Compute 2^(J/64)*p by 261 | Step 5. Compute 2^(J/64)*p by
262 | p := T*p 262 | p := T*p
263 | where T and t are the stored 263 | where T and t are the stored values for 2^(J/64).
264 | Notes: 2^(J/64) is stored as T and t 264 | Notes: 2^(J/64) is stored as T and t where T+t approximates
265 | 2^(J/64) to roughly 85 bits; 265 | 2^(J/64) to roughly 85 bits; T is in extended precision
266 | and t is in single precision. 266 | and t is in single precision. Note also that T is rounded
267 | to 62 bits so that the last t 267 | to 62 bits so that the last two bits of T are zero. The
268 | reason for such a special for 268 | reason for such a special form is that T-1, T-2, and T-8
269 | will all be exact --- a prope 269 | will all be exact --- a property that will be exploited
270 | in Step 6 below. The total re 270 | in Step 6 below. The total relative error in p is no
271 | bigger than 2^(-67.7) compare 271 | bigger than 2^(-67.7) compared to the final result.
272 | 272 |
273 | Step 6. Reconstruction of exp(X)-1 273 | Step 6. Reconstruction of exp(X)-1
274 | exp(X)-1 = 2^M * ( 2^( 274 | exp(X)-1 = 2^M * ( 2^(J/64) + p - 2^(-M) ).
275 | 6.1 If M <= 63, go to Step 275 | 6.1 If M <= 63, go to Step 6.3.
276 | 6.2 ans := T + (p + (t + O 276 | 6.2 ans := T + (p + (t + OnebySc)). Go to 6.6
277 | 6.3 If M >= -3, go to 6.5. 277 | 6.3 If M >= -3, go to 6.5.
278 | 6.4 ans := (T + (p + t)) + 278 | 6.4 ans := (T + (p + t)) + OnebySc. Go to 6.6
279 | 6.5 ans := (T + OnebySc) + 279 | 6.5 ans := (T + OnebySc) + (p + t).
280 | 6.6 Restore user FPCR. 280 | 6.6 Restore user FPCR.
281 | 6.7 Return ans := Sc * ans 281 | 6.7 Return ans := Sc * ans. Exit.
282 | Notes: The various arrangements of th 282 | Notes: The various arrangements of the expressions give accurate
283 | evaluations. 283 | evaluations.
284 | 284 |
285 | Step 7. exp(X)-1 for |X| < 1/4. 285 | Step 7. exp(X)-1 for |X| < 1/4.
286 | 7.1 If |X| >= 2^(-65), go 286 | 7.1 If |X| >= 2^(-65), go to Step 9.
287 | 7.2 Go to Step 8. 287 | 7.2 Go to Step 8.
288 | 288 |
289 | Step 8. Calculate exp(X)-1, |X| < 2^(- 289 | Step 8. Calculate exp(X)-1, |X| < 2^(-65).
290 | 8.1 If |X| < 2^(-16312), g 290 | 8.1 If |X| < 2^(-16312), goto 8.3
291 | 8.2 Restore FPCR; return a 291 | 8.2 Restore FPCR; return ans := X - 2^(-16382). Exit.
292 | 8.3 X := X * 2^(140). 292 | 8.3 X := X * 2^(140).
293 | 8.4 Restore FPCR; ans := a 293 | 8.4 Restore FPCR; ans := ans - 2^(-16382).
294 | Return ans := ans*2^(140). Ex 294 | Return ans := ans*2^(140). Exit
295 | Notes: The idea is to return "X - tin 295 | Notes: The idea is to return "X - tiny" under the user
296 | precision and rounding modes. 296 | precision and rounding modes. To avoid unnecessary
297 | inefficiency, we stay away fr 297 | inefficiency, we stay away from denormalized numbers the
298 | best we can. For |X| >= 2^(-1 298 | best we can. For |X| >= 2^(-16312), the straightforward
299 | 8.2 generates the inexact exc 299 | 8.2 generates the inexact exception as the case warrants.
300 | 300 |
301 | Step 9. Calculate exp(X)-1, |X| < 1/4, 301 | Step 9. Calculate exp(X)-1, |X| < 1/4, by a polynomial
302 | p = X + X*X*(B1 + X*(B 302 | p = X + X*X*(B1 + X*(B2 + ... + X*B12))
303 | Notes: a) In order to reduce memory a 303 | Notes: a) In order to reduce memory access, the coefficients are
304 | made as "short" as possible: 304 | made as "short" as possible: B1 (which is 1/2), B9 to B12
305 | are single precision; B3 to B 305 | are single precision; B3 to B8 are double precision; and
306 | B2 is double extended. 306 | B2 is double extended.
307 | b) Even with the restriction 307 | b) Even with the restriction above,
308 | |p - (exp(X)-1)| < |X| 308 | |p - (exp(X)-1)| < |X| 2^(-70.6)
309 | for all |X| <= 0.251. 309 | for all |X| <= 0.251.
310 | Note that 0.251 is slightly b 310 | Note that 0.251 is slightly bigger than 1/4.
311 | c) To fully preserve accuracy 311 | c) To fully preserve accuracy, the polynomial is computed
312 | as X + ( S*B1 + Q ) wh 312 | as X + ( S*B1 + Q ) where S = X*X and
313 | Q = X*S*(B 313 | Q = X*S*(B2 + X*(B3 + ... + X*B12))
314 | d) To fully utilize the pipel 314 | d) To fully utilize the pipeline, Q is separated into
315 | two independent pieces of rou 315 | two independent pieces of roughly equal complexity
316 | Q = [ X*S*(B2 + S*(B4 316 | Q = [ X*S*(B2 + S*(B4 + ... + S*B12)) ] +
317 | [ S*S*(B3 + S* 317 | [ S*S*(B3 + S*(B5 + ... + S*B11)) ]
318 | 318 |
319 | Step 10. Calculate exp(X)-1 for 319 | Step 10. Calculate exp(X)-1 for |X| >= 70 log 2.
320 | 10.1 If X >= 70log2 , exp(X) - 320 | 10.1 If X >= 70log2 , exp(X) - 1 = exp(X) for all practical
321 | purposes. Therefore, go to St 321 | purposes. Therefore, go to Step 1 of setox.
322 | 10.2 If X <= -70log2, exp(X) - 322 | 10.2 If X <= -70log2, exp(X) - 1 = -1 for all practical purposes.
323 | ans := -1 323 | ans := -1
324 | Restore user FPCR 324 | Restore user FPCR
325 | Return ans := ans + 2^(-126). 325 | Return ans := ans + 2^(-126). Exit.
326 | Notes: 10.2 will always create an ine 326 | Notes: 10.2 will always create an inexact and return -1 + tiny
327 | in the user rounding precisio 327 | in the user rounding precision and mode.
328 | 328 |
329 | 329 |
330 330
331 | Copyright (C) Motorola, Inc. 1 331 | Copyright (C) Motorola, Inc. 1990
332 | All Rights Reserved 332 | All Rights Reserved
333 | 333 |
334 | For details on the license for this fi 334 | For details on the license for this file, please see the
335 | file, README, in this same directory. 335 | file, README, in this same directory.
336 336
337 |setox idnt 2,1 | Motorola 040 Floating Po 337 |setox idnt 2,1 | Motorola 040 Floating Point Software Package
338 338
339 |section 8 339 |section 8
340 340
341 #include "fpsp.h" 341 #include "fpsp.h"
342 342
343 L2: .long 0x3FDC0000,0x82E30865,0x4361C4 343 L2: .long 0x3FDC0000,0x82E30865,0x4361C4C6,0x00000000
344 344
345 EXPA3: .long 0x3FA55555,0x55554431 345 EXPA3: .long 0x3FA55555,0x55554431
346 EXPA2: .long 0x3FC55555,0x55554018 346 EXPA2: .long 0x3FC55555,0x55554018
347 347
348 HUGE: .long 0x7FFE0000,0xFFFFFFFF,0xFFFFFF 348 HUGE: .long 0x7FFE0000,0xFFFFFFFF,0xFFFFFFFF,0x00000000
349 TINY: .long 0x00010000,0xFFFFFFFF,0xFFFFFF 349 TINY: .long 0x00010000,0xFFFFFFFF,0xFFFFFFFF,0x00000000
350 350
351 EM1A4: .long 0x3F811111,0x11174385 351 EM1A4: .long 0x3F811111,0x11174385
352 EM1A3: .long 0x3FA55555,0x55554F5A 352 EM1A3: .long 0x3FA55555,0x55554F5A
353 353
354 EM1A2: .long 0x3FC55555,0x55555555,0x000000 354 EM1A2: .long 0x3FC55555,0x55555555,0x00000000,0x00000000
355 355
356 EM1B8: .long 0x3EC71DE3,0xA5774682 356 EM1B8: .long 0x3EC71DE3,0xA5774682
357 EM1B7: .long 0x3EFA01A0,0x19D7CB68 357 EM1B7: .long 0x3EFA01A0,0x19D7CB68
358 358
359 EM1B6: .long 0x3F2A01A0,0x1A019DF3 359 EM1B6: .long 0x3F2A01A0,0x1A019DF3
360 EM1B5: .long 0x3F56C16C,0x16C170E2 360 EM1B5: .long 0x3F56C16C,0x16C170E2
361 361
362 EM1B4: .long 0x3F811111,0x11111111 362 EM1B4: .long 0x3F811111,0x11111111
363 EM1B3: .long 0x3FA55555,0x55555555 363 EM1B3: .long 0x3FA55555,0x55555555
364 364
365 EM1B2: .long 0x3FFC0000,0xAAAAAAAA,0xAAAAAA 365 EM1B2: .long 0x3FFC0000,0xAAAAAAAA,0xAAAAAAAB
366 .long 0x00000000 366 .long 0x00000000
367 367
368 TWO140: .long 0x48B00000,0x00000000 368 TWO140: .long 0x48B00000,0x00000000
369 TWON140: .long 0x37300000,0x00000000 369 TWON140: .long 0x37300000,0x00000000
370 370
371 EXPTBL: 371 EXPTBL:
372 .long 0x3FFF0000,0x80000000,0x000000 372 .long 0x3FFF0000,0x80000000,0x00000000,0x00000000
373 .long 0x3FFF0000,0x8164D1F3,0xBC0307 373 .long 0x3FFF0000,0x8164D1F3,0xBC030774,0x9F841A9B
374 .long 0x3FFF0000,0x82CD8698,0xAC2BA1 374 .long 0x3FFF0000,0x82CD8698,0xAC2BA1D8,0x9FC1D5B9
375 .long 0x3FFF0000,0x843A28C3,0xACDE40 375 .long 0x3FFF0000,0x843A28C3,0xACDE4048,0xA0728369
376 .long 0x3FFF0000,0x85AAC367,0xCC487B 376 .long 0x3FFF0000,0x85AAC367,0xCC487B14,0x1FC5C95C
377 .long 0x3FFF0000,0x871F6196,0x9E8D10 377 .long 0x3FFF0000,0x871F6196,0x9E8D1010,0x1EE85C9F
378 .long 0x3FFF0000,0x88980E80,0x92DA85 378 .long 0x3FFF0000,0x88980E80,0x92DA8528,0x9FA20729
379 .long 0x3FFF0000,0x8A14D575,0x496EFD 379 .long 0x3FFF0000,0x8A14D575,0x496EFD9C,0xA07BF9AF
380 .long 0x3FFF0000,0x8B95C1E3,0xEA8BD6 380 .long 0x3FFF0000,0x8B95C1E3,0xEA8BD6E8,0xA0020DCF
381 .long 0x3FFF0000,0x8D1ADF5B,0x7E5BA9 381 .long 0x3FFF0000,0x8D1ADF5B,0x7E5BA9E4,0x205A63DA
382 .long 0x3FFF0000,0x8EA4398B,0x45CD53 382 .long 0x3FFF0000,0x8EA4398B,0x45CD53C0,0x1EB70051
383 .long 0x3FFF0000,0x9031DC43,0x1466B1 383 .long 0x3FFF0000,0x9031DC43,0x1466B1DC,0x1F6EB029
384 .long 0x3FFF0000,0x91C3D373,0xAB11C3 384 .long 0x3FFF0000,0x91C3D373,0xAB11C338,0xA0781494
385 .long 0x3FFF0000,0x935A2B2F,0x13E6E9 385 .long 0x3FFF0000,0x935A2B2F,0x13E6E92C,0x9EB319B0
386 .long 0x3FFF0000,0x94F4EFA8,0xFEF709 386 .long 0x3FFF0000,0x94F4EFA8,0xFEF70960,0x2017457D
387 .long 0x3FFF0000,0x96942D37,0x20185A 387 .long 0x3FFF0000,0x96942D37,0x20185A00,0x1F11D537
388 .long 0x3FFF0000,0x9837F051,0x8DB8A9 388 .long 0x3FFF0000,0x9837F051,0x8DB8A970,0x9FB952DD
389 .long 0x3FFF0000,0x99E04593,0x20B7FA 389 .long 0x3FFF0000,0x99E04593,0x20B7FA64,0x1FE43087
390 .long 0x3FFF0000,0x9B8D39B9,0xD54E55 390 .long 0x3FFF0000,0x9B8D39B9,0xD54E5538,0x1FA2A818
391 .long 0x3FFF0000,0x9D3ED9A7,0x2CFFB7 391 .long 0x3FFF0000,0x9D3ED9A7,0x2CFFB750,0x1FDE494D
392 .long 0x3FFF0000,0x9EF53260,0x91A111 392 .long 0x3FFF0000,0x9EF53260,0x91A111AC,0x20504890
393 .long 0x3FFF0000,0xA0B0510F,0xB9714F 393 .long 0x3FFF0000,0xA0B0510F,0xB9714FC4,0xA073691C
394 .long 0x3FFF0000,0xA2704303,0x0C4968 394 .long 0x3FFF0000,0xA2704303,0x0C496818,0x1F9B7A05
395 .long 0x3FFF0000,0xA43515AE,0x09E680 395 .long 0x3FFF0000,0xA43515AE,0x09E680A0,0xA0797126
396 .long 0x3FFF0000,0xA5FED6A9,0xB15138 396 .long 0x3FFF0000,0xA5FED6A9,0xB15138EC,0xA071A140
397 .long 0x3FFF0000,0xA7CD93B4,0xE96535 397 .long 0x3FFF0000,0xA7CD93B4,0xE9653568,0x204F62DA
398 .long 0x3FFF0000,0xA9A15AB4,0xEA7C0E 398 .long 0x3FFF0000,0xA9A15AB4,0xEA7C0EF8,0x1F283C4A
399 .long 0x3FFF0000,0xAB7A39B5,0xA93ED3 399 .long 0x3FFF0000,0xAB7A39B5,0xA93ED338,0x9F9A7FDC
400 .long 0x3FFF0000,0xAD583EEA,0x42A14A 400 .long 0x3FFF0000,0xAD583EEA,0x42A14AC8,0xA05B3FAC
401 .long 0x3FFF0000,0xAF3B78AD,0x690A43 401 .long 0x3FFF0000,0xAF3B78AD,0x690A4374,0x1FDF2610
402 .long 0x3FFF0000,0xB123F581,0xD2AC25 402 .long 0x3FFF0000,0xB123F581,0xD2AC2590,0x9F705F90
403 .long 0x3FFF0000,0xB311C412,0xA91124 403 .long 0x3FFF0000,0xB311C412,0xA9112488,0x201F678A
404 .long 0x3FFF0000,0xB504F333,0xF9DE64 404 .long 0x3FFF0000,0xB504F333,0xF9DE6484,0x1F32FB13
405 .long 0x3FFF0000,0xB6FD91E3,0x28D177 405 .long 0x3FFF0000,0xB6FD91E3,0x28D17790,0x20038B30
406 .long 0x3FFF0000,0xB8FBAF47,0x62FB9E 406 .long 0x3FFF0000,0xB8FBAF47,0x62FB9EE8,0x200DC3CC
407 .long 0x3FFF0000,0xBAFF5AB2,0x133E45 407 .long 0x3FFF0000,0xBAFF5AB2,0x133E45FC,0x9F8B2AE6
408 .long 0x3FFF0000,0xBD08A39F,0x580C36 408 .long 0x3FFF0000,0xBD08A39F,0x580C36C0,0xA02BBF70
409 .long 0x3FFF0000,0xBF1799B6,0x7A7310 409 .long 0x3FFF0000,0xBF1799B6,0x7A731084,0xA00BF518
410 .long 0x3FFF0000,0xC12C4CCA,0x667094 410 .long 0x3FFF0000,0xC12C4CCA,0x66709458,0xA041DD41
411 .long 0x3FFF0000,0xC346CCDA,0x249764 411 .long 0x3FFF0000,0xC346CCDA,0x24976408,0x9FDF137B
412 .long 0x3FFF0000,0xC5672A11,0x5506DA 412 .long 0x3FFF0000,0xC5672A11,0x5506DADC,0x201F1568
413 .long 0x3FFF0000,0xC78D74C8,0xABB9B1 413 .long 0x3FFF0000,0xC78D74C8,0xABB9B15C,0x1FC13A2E
414 .long 0x3FFF0000,0xC9B9BD86,0x6E2F27 414 .long 0x3FFF0000,0xC9B9BD86,0x6E2F27A4,0xA03F8F03
415 .long 0x3FFF0000,0xCBEC14FE,0xF2727C 415 .long 0x3FFF0000,0xCBEC14FE,0xF2727C5C,0x1FF4907D
416 .long 0x3FFF0000,0xCE248C15,0x1F8480 416 .long 0x3FFF0000,0xCE248C15,0x1F8480E4,0x9E6E53E4
417 .long 0x3FFF0000,0xD06333DA,0xEF2B25 417 .long 0x3FFF0000,0xD06333DA,0xEF2B2594,0x1FD6D45C
418 .long 0x3FFF0000,0xD2A81D91,0xF12AE4 418 .long 0x3FFF0000,0xD2A81D91,0xF12AE45C,0xA076EDB9
419 .long 0x3FFF0000,0xD4F35AAB,0xCFEDFA 419 .long 0x3FFF0000,0xD4F35AAB,0xCFEDFA20,0x9FA6DE21
420 .long 0x3FFF0000,0xD744FCCA,0xD69D6A 420 .long 0x3FFF0000,0xD744FCCA,0xD69D6AF4,0x1EE69A2F
421 .long 0x3FFF0000,0xD99D15C2,0x78AFD7 421 .long 0x3FFF0000,0xD99D15C2,0x78AFD7B4,0x207F439F
422 .long 0x3FFF0000,0xDBFBB797,0xDAF237 422 .long 0x3FFF0000,0xDBFBB797,0xDAF23754,0x201EC207
423 .long 0x3FFF0000,0xDE60F482,0x5E0E91 423 .long 0x3FFF0000,0xDE60F482,0x5E0E9124,0x9E8BE175
424 .long 0x3FFF0000,0xE0CCDEEC,0x2A94E1 424 .long 0x3FFF0000,0xE0CCDEEC,0x2A94E110,0x20032C4B
425 .long 0x3FFF0000,0xE33F8972,0xBE8A5A 425 .long 0x3FFF0000,0xE33F8972,0xBE8A5A50,0x2004DFF5
426 .long 0x3FFF0000,0xE5B906E7,0x7C8348 426 .long 0x3FFF0000,0xE5B906E7,0x7C8348A8,0x1E72F47A
427 .long 0x3FFF0000,0xE8396A50,0x3C4BDC 427 .long 0x3FFF0000,0xE8396A50,0x3C4BDC68,0x1F722F22
428 .long 0x3FFF0000,0xEAC0C6E7,0xDD2439 428 .long 0x3FFF0000,0xEAC0C6E7,0xDD243930,0xA017E945
429 .long 0x3FFF0000,0xED4F301E,0xD9942B 429 .long 0x3FFF0000,0xED4F301E,0xD9942B84,0x1F401A5B
430 .long 0x3FFF0000,0xEFE4B99B,0xDCDAF5 430 .long 0x3FFF0000,0xEFE4B99B,0xDCDAF5CC,0x9FB9A9E3
431 .long 0x3FFF0000,0xF281773C,0x59FFB1 431 .long 0x3FFF0000,0xF281773C,0x59FFB138,0x20744C05
432 .long 0x3FFF0000,0xF5257D15,0x2486CC 432 .long 0x3FFF0000,0xF5257D15,0x2486CC2C,0x1F773A19
433 .long 0x3FFF0000,0xF7D0DF73,0x0AD13B 433 .long 0x3FFF0000,0xF7D0DF73,0x0AD13BB8,0x1FFE90D5
434 .long 0x3FFF0000,0xFA83B2DB,0x722A03 434 .long 0x3FFF0000,0xFA83B2DB,0x722A033C,0xA041ED22
435 .long 0x3FFF0000,0xFD3E0C0C,0xF486C1 435 .long 0x3FFF0000,0xFD3E0C0C,0xF486C174,0x1F853F3A
436 436
437 .set ADJFLAG,L_SCR2 437 .set ADJFLAG,L_SCR2
438 .set SCALE,FP_SCR1 438 .set SCALE,FP_SCR1
439 .set ADJSCALE,FP_SCR2 439 .set ADJSCALE,FP_SCR2
440 .set SC,FP_SCR3 440 .set SC,FP_SCR3
441 .set ONEBYSC,FP_SCR4 441 .set ONEBYSC,FP_SCR4
442 442
443 | xref t_frcinx 443 | xref t_frcinx
444 |xref t_extdnrm 444 |xref t_extdnrm
445 |xref t_unfl 445 |xref t_unfl
446 |xref t_ovfl 446 |xref t_ovfl
447 447
448 .global setoxd 448 .global setoxd
449 setoxd: 449 setoxd:
450 |--entry point for EXP(X), X is denormalized 450 |--entry point for EXP(X), X is denormalized
451 movel (%a0),%d0 451 movel (%a0),%d0
452 andil #0x80000000,%d0 452 andil #0x80000000,%d0
453 oril #0x00800000,%d0 453 oril #0x00800000,%d0 | ...sign(X)*2^(-126)
454 movel %d0,-(%sp) 454 movel %d0,-(%sp)
455 fmoves #0x3F800000,%fp0 455 fmoves #0x3F800000,%fp0
456 fmovel %d1,%fpcr 456 fmovel %d1,%fpcr
457 fadds (%sp)+,%fp0 457 fadds (%sp)+,%fp0
458 bra t_frcinx 458 bra t_frcinx
459 459
460 .global setox 460 .global setox
461 setox: 461 setox:
462 |--entry point for EXP(X), here X is finite, n 462 |--entry point for EXP(X), here X is finite, non-zero, and not NaN's
463 463
464 |--Step 1. 464 |--Step 1.
465 movel (%a0),%d0 | ... 465 movel (%a0),%d0 | ...load part of input X
466 andil #0x7FFF0000,%d0 | ...b 466 andil #0x7FFF0000,%d0 | ...biased expo. of X
467 cmpil #0x3FBE0000,%d0 | ...2 467 cmpil #0x3FBE0000,%d0 | ...2^(-65)
468 bges EXPC1 | ...n 468 bges EXPC1 | ...normal case
469 bra EXPSM 469 bra EXPSM
470 470
471 EXPC1: 471 EXPC1:
472 |--The case |X| >= 2^(-65) 472 |--The case |X| >= 2^(-65)
473 movew 4(%a0),%d0 | ...e 473 movew 4(%a0),%d0 | ...expo. and partial sig. of |X|
474 cmpil #0x400CB167,%d0 | ...1 474 cmpil #0x400CB167,%d0 | ...16380 log2 trunc. 16 bits
475 blts EXPMAIN | ...normal c 475 blts EXPMAIN | ...normal case
476 bra EXPBIG 476 bra EXPBIG
477 477
478 EXPMAIN: 478 EXPMAIN:
479 |--Step 2. 479 |--Step 2.
480 |--This is the normal branch: 2^(-65) <= |X| 480 |--This is the normal branch: 2^(-65) <= |X| < 16380 log2.
481 fmovex (%a0),%fp0 | ...l 481 fmovex (%a0),%fp0 | ...load input from (a0)
482 482
483 fmovex %fp0,%fp1 483 fmovex %fp0,%fp1
484 fmuls #0x42B8AA3B,%fp0 484 fmuls #0x42B8AA3B,%fp0 | ...64/log2 * X
485 fmovemx %fp2-%fp2/%fp3,-(%a7) 485 fmovemx %fp2-%fp2/%fp3,-(%a7) | ...save fp2
486 movel #0,ADJFLAG(%a6) 486 movel #0,ADJFLAG(%a6)
487 fmovel %fp0,%d0 487 fmovel %fp0,%d0 | ...N = int( X * 64/log2 )
488 lea EXPTBL,%a1 488 lea EXPTBL,%a1
489 fmovel %d0,%fp0 489 fmovel %d0,%fp0 | ...convert to floating-format
490 490
491 movel %d0,L_SCR1(%a6) | ...s 491 movel %d0,L_SCR1(%a6) | ...save N temporarily
492 andil #0x3F,%d0 492 andil #0x3F,%d0 | ...D0 is J = N mod 64
493 lsll #4,%d0 493 lsll #4,%d0
494 addal %d0,%a1 | ...a 494 addal %d0,%a1 | ...address of 2^(J/64)
495 movel L_SCR1(%a6),%d0 495 movel L_SCR1(%a6),%d0
496 asrl #6,%d0 | ...D 496 asrl #6,%d0 | ...D0 is M
497 addiw #0x3FFF,%d0 | ...b 497 addiw #0x3FFF,%d0 | ...biased expo. of 2^(M)
498 movew L2,L_SCR1(%a6) | ...p 498 movew L2,L_SCR1(%a6) | ...prefetch L2, no need in CB
499 499
500 EXPCONT1: 500 EXPCONT1:
501 |--Step 3. 501 |--Step 3.
502 |--fp1,fp2 saved on the stack. fp0 is N, fp1 i 502 |--fp1,fp2 saved on the stack. fp0 is N, fp1 is X,
503 |--a0 points to 2^(J/64), D0 is biased expo. o 503 |--a0 points to 2^(J/64), D0 is biased expo. of 2^(M)
504 fmovex %fp0,%fp2 504 fmovex %fp0,%fp2
505 fmuls #0xBC317218,%fp0 505 fmuls #0xBC317218,%fp0 | ...N * L1, L1 = lead(-log2/64)
506 fmulx L2,%fp2 | ...N 506 fmulx L2,%fp2 | ...N * L2, L1+L2 = -log2/64
507 faddx %fp1,%fp0 507 faddx %fp1,%fp0 | ...X + N*L1
508 faddx %fp2,%fp0 508 faddx %fp2,%fp0 | ...fp0 is R, reduced arg.
509 | MOVE.W #$3FA5,EXPA3 ...loa 509 | MOVE.W #$3FA5,EXPA3 ...load EXPA3 in cache
510 510
511 |--Step 4. 511 |--Step 4.
512 |--WE NOW COMPUTE EXP(R)-1 BY A POLYNOMIAL 512 |--WE NOW COMPUTE EXP(R)-1 BY A POLYNOMIAL
513 |-- R + R*R*(A1 + R*(A2 + R*(A3 + R*(A4 + R*A5 513 |-- R + R*R*(A1 + R*(A2 + R*(A3 + R*(A4 + R*A5))))
514 |--TO FULLY UTILIZE THE PIPELINE, WE COMPUTE S 514 |--TO FULLY UTILIZE THE PIPELINE, WE COMPUTE S = R*R
515 |--[R+R*S*(A2+S*A4)] + [S*(A1+S*(A3+S*A5))] 515 |--[R+R*S*(A2+S*A4)] + [S*(A1+S*(A3+S*A5))]
516 516
517 fmovex %fp0,%fp1 517 fmovex %fp0,%fp1
518 fmulx %fp1,%fp1 518 fmulx %fp1,%fp1 | ...fp1 IS S = R*R
519 519
520 fmoves #0x3AB60B70,%fp2 520 fmoves #0x3AB60B70,%fp2 | ...fp2 IS A5
521 | MOVE.W #0,2(%a1) ...loa 521 | MOVE.W #0,2(%a1) ...load 2^(J/64) in cache
522 522
523 fmulx %fp1,%fp2 523 fmulx %fp1,%fp2 | ...fp2 IS S*A5
524 fmovex %fp1,%fp3 524 fmovex %fp1,%fp3
525 fmuls #0x3C088895,%fp3 525 fmuls #0x3C088895,%fp3 | ...fp3 IS S*A4
526 526
527 faddd EXPA3,%fp2 | ...f 527 faddd EXPA3,%fp2 | ...fp2 IS A3+S*A5
528 faddd EXPA2,%fp3 | ...f 528 faddd EXPA2,%fp3 | ...fp3 IS A2+S*A4
529 529
530 fmulx %fp1,%fp2 530 fmulx %fp1,%fp2 | ...fp2 IS S*(A3+S*A5)
531 movew %d0,SCALE(%a6) | ...S 531 movew %d0,SCALE(%a6) | ...SCALE is 2^(M) in extended
532 clrw SCALE+2(%a6) 532 clrw SCALE+2(%a6)
533 movel #0x80000000,SCALE+4(%a 533 movel #0x80000000,SCALE+4(%a6)
534 clrl SCALE+8(%a6) 534 clrl SCALE+8(%a6)
535 535
536 fmulx %fp1,%fp3 536 fmulx %fp1,%fp3 | ...fp3 IS S*(A2+S*A4)
537 537
538 fadds #0x3F000000,%fp2 538 fadds #0x3F000000,%fp2 | ...fp2 IS A1+S*(A3+S*A5)
539 fmulx %fp0,%fp3 539 fmulx %fp0,%fp3 | ...fp3 IS R*S*(A2+S*A4)
540 540
541 fmulx %fp1,%fp2 541 fmulx %fp1,%fp2 | ...fp2 IS S*(A1+S*(A3+S*A5))
542 faddx %fp3,%fp0 542 faddx %fp3,%fp0 | ...fp0 IS R+R*S*(A2+S*A4),
543 | ...fp3 543 | ...fp3 released
544 544
545 fmovex (%a1)+,%fp1 | ...f 545 fmovex (%a1)+,%fp1 | ...fp1 is lead. pt. of 2^(J/64)
546 faddx %fp2,%fp0 546 faddx %fp2,%fp0 | ...fp0 is EXP(R) - 1
547 | ...fp2 547 | ...fp2 released
548 548
549 |--Step 5 549 |--Step 5
550 |--final reconstruction process 550 |--final reconstruction process
551 |--EXP(X) = 2^M * ( 2^(J/64) + 2^(J/64)*(EXP(R 551 |--EXP(X) = 2^M * ( 2^(J/64) + 2^(J/64)*(EXP(R)-1) )
552 552
553 fmulx %fp1,%fp0 553 fmulx %fp1,%fp0 | ...2^(J/64)*(Exp(R)-1)
554 fmovemx (%a7)+,%fp2-%fp2/%fp3 | ...f 554 fmovemx (%a7)+,%fp2-%fp2/%fp3 | ...fp2 restored
555 fadds (%a1),%fp0 | ...a 555 fadds (%a1),%fp0 | ...accurate 2^(J/64)
556 556
557 faddx %fp1,%fp0 557 faddx %fp1,%fp0 | ...2^(J/64) + 2^(J/64)*...
558 movel ADJFLAG(%a6),%d0 558 movel ADJFLAG(%a6),%d0
559 559
560 |--Step 6 560 |--Step 6
561 tstl %d0 561 tstl %d0
562 beqs NORMAL 562 beqs NORMAL
563 ADJUST: 563 ADJUST:
564 fmulx ADJSCALE(%a6),%fp0 564 fmulx ADJSCALE(%a6),%fp0
565 NORMAL: 565 NORMAL:
566 fmovel %d1,%FPCR 566 fmovel %d1,%FPCR | ...restore user FPCR
567 fmulx SCALE(%a6),%fp0 | ...m 567 fmulx SCALE(%a6),%fp0 | ...multiply 2^(M)
568 bra t_frcinx 568 bra t_frcinx
569 569
570 EXPSM: 570 EXPSM:
571 |--Step 7 571 |--Step 7
572 fmovemx (%a0),%fp0-%fp0 | ...in case X 572 fmovemx (%a0),%fp0-%fp0 | ...in case X is denormalized
573 fmovel %d1,%FPCR 573 fmovel %d1,%FPCR
574 fadds #0x3F800000,%fp0 574 fadds #0x3F800000,%fp0 | ...1+X in user mode
575 bra t_frcinx 575 bra t_frcinx
576 576
577 EXPBIG: 577 EXPBIG:
578 |--Step 8 578 |--Step 8
579 cmpil #0x400CB27C,%d0 | ...1 579 cmpil #0x400CB27C,%d0 | ...16480 log2
580 bgts EXP2BIG 580 bgts EXP2BIG
581 |--Steps 8.2 -- 8.6 581 |--Steps 8.2 -- 8.6
582 fmovex (%a0),%fp0 | ...l 582 fmovex (%a0),%fp0 | ...load input from (a0)
583 583
584 fmovex %fp0,%fp1 584 fmovex %fp0,%fp1
585 fmuls #0x42B8AA3B,%fp0 585 fmuls #0x42B8AA3B,%fp0 | ...64/log2 * X
586 fmovemx %fp2-%fp2/%fp3,-(%a7) 586 fmovemx %fp2-%fp2/%fp3,-(%a7) | ...save fp2
587 movel #1,ADJFLAG(%a6) 587 movel #1,ADJFLAG(%a6)
588 fmovel %fp0,%d0 588 fmovel %fp0,%d0 | ...N = int( X * 64/log2 )
589 lea EXPTBL,%a1 589 lea EXPTBL,%a1
590 fmovel %d0,%fp0 590 fmovel %d0,%fp0 | ...convert to floating-format
591 movel %d0,L_SCR1(%a6) 591 movel %d0,L_SCR1(%a6) | ...save N temporarily
592 andil #0x3F,%d0 592 andil #0x3F,%d0 | ...D0 is J = N mod 64
593 lsll #4,%d0 593 lsll #4,%d0
594 addal %d0,%a1 594 addal %d0,%a1 | ...address of 2^(J/64)
595 movel L_SCR1(%a6),%d0 595 movel L_SCR1(%a6),%d0
596 asrl #6,%d0 596 asrl #6,%d0 | ...D0 is K
597 movel %d0,L_SCR1(%a6) 597 movel %d0,L_SCR1(%a6) | ...save K temporarily
598 asrl #1,%d0 598 asrl #1,%d0 | ...D0 is M1
599 subl %d0,L_SCR1(%a6) 599 subl %d0,L_SCR1(%a6) | ...a1 is M
600 addiw #0x3FFF,%d0 600 addiw #0x3FFF,%d0 | ...biased expo. of 2^(M1)
601 movew %d0,ADJSCALE(%a6) 601 movew %d0,ADJSCALE(%a6) | ...ADJSCALE := 2^(M1)
602 clrw ADJSCALE+2(%a6) 602 clrw ADJSCALE+2(%a6)
603 movel #0x80000000,ADJSCALE+4 603 movel #0x80000000,ADJSCALE+4(%a6)
604 clrl ADJSCALE+8(%a6) 604 clrl ADJSCALE+8(%a6)
605 movel L_SCR1(%a6),%d0 605 movel L_SCR1(%a6),%d0 | ...D0 is M
606 addiw #0x3FFF,%d0 606 addiw #0x3FFF,%d0 | ...biased expo. of 2^(M)
607 bra EXPCONT1 607 bra EXPCONT1 | ...go back to Step 3
608 608
609 EXP2BIG: 609 EXP2BIG:
610 |--Step 9 610 |--Step 9
611 fmovel %d1,%FPCR 611 fmovel %d1,%FPCR
612 movel (%a0),%d0 612 movel (%a0),%d0
613 bclrb #sign_bit,(%a0) 613 bclrb #sign_bit,(%a0) | ...setox always returns positive
614 cmpil #0,%d0 614 cmpil #0,%d0
615 blt t_unfl 615 blt t_unfl
616 bra t_ovfl 616 bra t_ovfl
617 617
618 .global setoxm1d 618 .global setoxm1d
619 setoxm1d: 619 setoxm1d:
620 |--entry point for EXPM1(X), here X is denorma 620 |--entry point for EXPM1(X), here X is denormalized
621 |--Step 0. 621 |--Step 0.
622 bra t_extdnrm 622 bra t_extdnrm
623 623
624 624
625 .global setoxm1 625 .global setoxm1
626 setoxm1: 626 setoxm1:
627 |--entry point for EXPM1(X), here X is finite, 627 |--entry point for EXPM1(X), here X is finite, non-zero, non-NaN
628 628
629 |--Step 1. 629 |--Step 1.
630 |--Step 1.1 630 |--Step 1.1
631 movel (%a0),%d0 | ... 631 movel (%a0),%d0 | ...load part of input X
632 andil #0x7FFF0000,%d0 | ...b 632 andil #0x7FFF0000,%d0 | ...biased expo. of X
633 cmpil #0x3FFD0000,%d0 | ...1 633 cmpil #0x3FFD0000,%d0 | ...1/4
634 bges EM1CON1 | ...|X| >= 1 634 bges EM1CON1 | ...|X| >= 1/4
635 bra EM1SM 635 bra EM1SM
636 636
637 EM1CON1: 637 EM1CON1:
638 |--Step 1.3 638 |--Step 1.3
639 |--The case |X| >= 1/4 639 |--The case |X| >= 1/4
640 movew 4(%a0),%d0 | ...e 640 movew 4(%a0),%d0 | ...expo. and partial sig. of |X|
641 cmpil #0x4004C215,%d0 | ...7 641 cmpil #0x4004C215,%d0 | ...70log2 rounded up to 16 bits
642 bles EM1MAIN | ...1/4 <= | 642 bles EM1MAIN | ...1/4 <= |X| <= 70log2
643 bra EM1BIG 643 bra EM1BIG
644 644
645 EM1MAIN: 645 EM1MAIN:
646 |--Step 2. 646 |--Step 2.
647 |--This is the case: 1/4 <= |X| <= 70 log2. 647 |--This is the case: 1/4 <= |X| <= 70 log2.
648 fmovex (%a0),%fp0 | ...l 648 fmovex (%a0),%fp0 | ...load input from (a0)
649 649
650 fmovex %fp0,%fp1 650 fmovex %fp0,%fp1
651 fmuls #0x42B8AA3B,%fp0 651 fmuls #0x42B8AA3B,%fp0 | ...64/log2 * X
652 fmovemx %fp2-%fp2/%fp3,-(%a7) 652 fmovemx %fp2-%fp2/%fp3,-(%a7) | ...save fp2
653 | MOVE.W #$3F81,EM1A4 653 | MOVE.W #$3F81,EM1A4 ...prefetch in CB mode
654 fmovel %fp0,%d0 654 fmovel %fp0,%d0 | ...N = int( X * 64/log2 )
655 lea EXPTBL,%a1 655 lea EXPTBL,%a1
656 fmovel %d0,%fp0 656 fmovel %d0,%fp0 | ...convert to floating-format
657 657
658 movel %d0,L_SCR1(%a6) 658 movel %d0,L_SCR1(%a6) | ...save N temporarily
659 andil #0x3F,%d0 659 andil #0x3F,%d0 | ...D0 is J = N mod 64
660 lsll #4,%d0 660 lsll #4,%d0
661 addal %d0,%a1 661 addal %d0,%a1 | ...address of 2^(J/64)
662 movel L_SCR1(%a6),%d0 662 movel L_SCR1(%a6),%d0
663 asrl #6,%d0 663 asrl #6,%d0 | ...D0 is M
664 movel %d0,L_SCR1(%a6) 664 movel %d0,L_SCR1(%a6) | ...save a copy of M
665 | MOVE.W #$3FDC,L2 665 | MOVE.W #$3FDC,L2 ...prefetch L2 in CB mode
666 666
667 |--Step 3. 667 |--Step 3.
668 |--fp1,fp2 saved on the stack. fp0 is N, fp1 i 668 |--fp1,fp2 saved on the stack. fp0 is N, fp1 is X,
669 |--a0 points to 2^(J/64), D0 and a1 both conta 669 |--a0 points to 2^(J/64), D0 and a1 both contain M
670 fmovex %fp0,%fp2 670 fmovex %fp0,%fp2
671 fmuls #0xBC317218,%fp0 671 fmuls #0xBC317218,%fp0 | ...N * L1, L1 = lead(-log2/64)
672 fmulx L2,%fp2 | ...N 672 fmulx L2,%fp2 | ...N * L2, L1+L2 = -log2/64
673 faddx %fp1,%fp0 | ... 673 faddx %fp1,%fp0 | ...X + N*L1
674 faddx %fp2,%fp0 | ... 674 faddx %fp2,%fp0 | ...fp0 is R, reduced arg.
675 | MOVE.W #$3FC5,EM1A2 675 | MOVE.W #$3FC5,EM1A2 ...load EM1A2 in cache
676 addiw #0x3FFF,%d0 676 addiw #0x3FFF,%d0 | ...D0 is biased expo. of 2^M
677 677
678 |--Step 4. 678 |--Step 4.
679 |--WE NOW COMPUTE EXP(R)-1 BY A POLYNOMIAL 679 |--WE NOW COMPUTE EXP(R)-1 BY A POLYNOMIAL
680 |-- R + R*R*(A1 + R*(A2 + R*(A3 + R*(A4 + R*(A 680 |-- R + R*R*(A1 + R*(A2 + R*(A3 + R*(A4 + R*(A5 + R*A6)))))
681 |--TO FULLY UTILIZE THE PIPELINE, WE COMPUTE S 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*A 682 |--[R*S*(A2+S*(A4+S*A6))] + [R+S*(A1+S*(A3+S*A5))]
683 683
684 fmovex %fp0,%fp1 684 fmovex %fp0,%fp1
685 fmulx %fp1,%fp1 685 fmulx %fp1,%fp1 | ...fp1 IS S = R*R
686 686
687 fmoves #0x3950097B,%fp2 687 fmoves #0x3950097B,%fp2 | ...fp2 IS a6
688 | MOVE.W #0,2(%a1) ...loa 688 | MOVE.W #0,2(%a1) ...load 2^(J/64) in cache
689 689
690 fmulx %fp1,%fp2 690 fmulx %fp1,%fp2 | ...fp2 IS S*A6
691 fmovex %fp1,%fp3 691 fmovex %fp1,%fp3
692 fmuls #0x3AB60B6A,%fp3 692 fmuls #0x3AB60B6A,%fp3 | ...fp3 IS S*A5
693 693
694 faddd EM1A4,%fp2 | ...f 694 faddd EM1A4,%fp2 | ...fp2 IS A4+S*A6
695 faddd EM1A3,%fp3 | ...f 695 faddd EM1A3,%fp3 | ...fp3 IS A3+S*A5
696 movew %d0,SC(%a6) 696 movew %d0,SC(%a6) | ...SC is 2^(M) in extended
697 clrw SC+2(%a6) 697 clrw SC+2(%a6)
698 movel #0x80000000,SC+4(%a6) 698 movel #0x80000000,SC+4(%a6)
699 clrl SC+8(%a6) 699 clrl SC+8(%a6)
700 700
701 fmulx %fp1,%fp2 701 fmulx %fp1,%fp2 | ...fp2 IS S*(A4+S*A6)
702 movel L_SCR1(%a6),%d0 702 movel L_SCR1(%a6),%d0 | ...D0 is M
703 negw %d0 | ...D 703 negw %d0 | ...D0 is -M
704 fmulx %fp1,%fp3 704 fmulx %fp1,%fp3 | ...fp3 IS S*(A3+S*A5)
705 addiw #0x3FFF,%d0 | ...b 705 addiw #0x3FFF,%d0 | ...biased expo. of 2^(-M)
706 faddd EM1A2,%fp2 | ...f 706 faddd EM1A2,%fp2 | ...fp2 IS A2+S*(A4+S*A6)
707 fadds #0x3F000000,%fp3 707 fadds #0x3F000000,%fp3 | ...fp3 IS A1+S*(A3+S*A5)
708 708
709 fmulx %fp1,%fp2 709 fmulx %fp1,%fp2 | ...fp2 IS S*(A2+S*(A4+S*A6))
710 oriw #0x8000,%d0 | ...s 710 oriw #0x8000,%d0 | ...signed/expo. of -2^(-M)
711 movew %d0,ONEBYSC(%a6) 711 movew %d0,ONEBYSC(%a6) | ...OnebySc is -2^(-M)
712 clrw ONEBYSC+2(%a6) 712 clrw ONEBYSC+2(%a6)
713 movel #0x80000000,ONEBYSC+4( 713 movel #0x80000000,ONEBYSC+4(%a6)
714 clrl ONEBYSC+8(%a6) 714 clrl ONEBYSC+8(%a6)
715 fmulx %fp3,%fp1 715 fmulx %fp3,%fp1 | ...fp1 IS S*(A1+S*(A3+S*A5))
716 | ...fp3 716 | ...fp3 released
717 717
718 fmulx %fp0,%fp2 718 fmulx %fp0,%fp2 | ...fp2 IS R*S*(A2+S*(A4+S*A6))
719 faddx %fp1,%fp0 719 faddx %fp1,%fp0 | ...fp0 IS R+S*(A1+S*(A3+S*A5))
720 | ...fp1 720 | ...fp1 released
721 721
722 faddx %fp2,%fp0 722 faddx %fp2,%fp0 | ...fp0 IS EXP(R)-1
723 | ...fp2 723 | ...fp2 released
724 fmovemx (%a7)+,%fp2-%fp2/%fp3 | ...f 724 fmovemx (%a7)+,%fp2-%fp2/%fp3 | ...fp2 restored
725 725
726 |--Step 5 726 |--Step 5
727 |--Compute 2^(J/64)*p 727 |--Compute 2^(J/64)*p
728 728
729 fmulx (%a1),%fp0 | ...2 729 fmulx (%a1),%fp0 | ...2^(J/64)*(Exp(R)-1)
730 730
731 |--Step 6 731 |--Step 6
732 |--Step 6.1 732 |--Step 6.1
733 movel L_SCR1(%a6),%d0 733 movel L_SCR1(%a6),%d0 | ...retrieve M
734 cmpil #63,%d0 734 cmpil #63,%d0
735 bles MLE63 735 bles MLE63
736 |--Step 6.2 M >= 64 736 |--Step 6.2 M >= 64
737 fmoves 12(%a1),%fp1 | ...f 737 fmoves 12(%a1),%fp1 | ...fp1 is t
738 faddx ONEBYSC(%a6),%fp1 738 faddx ONEBYSC(%a6),%fp1 | ...fp1 is t+OnebySc
739 faddx %fp1,%fp0 739 faddx %fp1,%fp0 | ...p+(t+OnebySc), fp1 released
740 faddx (%a1),%fp0 | ...T 740 faddx (%a1),%fp0 | ...T+(p+(t+OnebySc))
741 bras EM1SCALE 741 bras EM1SCALE
742 MLE63: 742 MLE63:
743 |--Step 6.3 M <= 63 743 |--Step 6.3 M <= 63
744 cmpil #-3,%d0 744 cmpil #-3,%d0
745 bges MGEN3 745 bges MGEN3
746 MLTN3: 746 MLTN3:
747 |--Step 6.4 M <= -4 747 |--Step 6.4 M <= -4
748 fadds 12(%a1),%fp0 | ...p 748 fadds 12(%a1),%fp0 | ...p+t
749 faddx (%a1),%fp0 | ...T 749 faddx (%a1),%fp0 | ...T+(p+t)
750 faddx ONEBYSC(%a6),%fp0 750 faddx ONEBYSC(%a6),%fp0 | ...OnebySc + (T+(p+t))
751 bras EM1SCALE 751 bras EM1SCALE
752 MGEN3: 752 MGEN3:
753 |--Step 6.5 -3 <= M <= 63 753 |--Step 6.5 -3 <= M <= 63
754 fmovex (%a1)+,%fp1 | ...f 754 fmovex (%a1)+,%fp1 | ...fp1 is T
755 fadds (%a1),%fp0 | ...f 755 fadds (%a1),%fp0 | ...fp0 is p+t
756 faddx ONEBYSC(%a6),%fp1 756 faddx ONEBYSC(%a6),%fp1 | ...fp1 is T+OnebySc
757 faddx %fp1,%fp0 757 faddx %fp1,%fp0 | ...(T+OnebySc)+(p+t)
758 758
759 EM1SCALE: 759 EM1SCALE:
760 |--Step 6.6 760 |--Step 6.6
761 fmovel %d1,%FPCR 761 fmovel %d1,%FPCR
762 fmulx SC(%a6),%fp0 762 fmulx SC(%a6),%fp0
763 763
764 bra t_frcinx 764 bra t_frcinx
765 765
766 EM1SM: 766 EM1SM:
767 |--Step 7 |X| < 1/4. 767 |--Step 7 |X| < 1/4.
768 cmpil #0x3FBE0000,%d0 | ...2 768 cmpil #0x3FBE0000,%d0 | ...2^(-65)
769 bges EM1POLY 769 bges EM1POLY
770 770
771 EM1TINY: 771 EM1TINY:
772 |--Step 8 |X| < 2^(-65) 772 |--Step 8 |X| < 2^(-65)
773 cmpil #0x00330000,%d0 | ...2 773 cmpil #0x00330000,%d0 | ...2^(-16312)
774 blts EM12TINY 774 blts EM12TINY
775 |--Step 8.2 775 |--Step 8.2
776 movel #0x80010000,SC(%a6) 776 movel #0x80010000,SC(%a6) | ...SC is -2^(-16382)
777 movel #0x80000000,SC+4(%a6) 777 movel #0x80000000,SC+4(%a6)
778 clrl SC+8(%a6) 778 clrl SC+8(%a6)
779 fmovex (%a0),%fp0 779 fmovex (%a0),%fp0
780 fmovel %d1,%FPCR 780 fmovel %d1,%FPCR
781 faddx SC(%a6),%fp0 781 faddx SC(%a6),%fp0
782 782
783 bra t_frcinx 783 bra t_frcinx
784 784
785 EM12TINY: 785 EM12TINY:
786 |--Step 8.3 786 |--Step 8.3
787 fmovex (%a0),%fp0 787 fmovex (%a0),%fp0
788 fmuld TWO140,%fp0 788 fmuld TWO140,%fp0
789 movel #0x80010000,SC(%a6) 789 movel #0x80010000,SC(%a6)
790 movel #0x80000000,SC+4(%a6) 790 movel #0x80000000,SC+4(%a6)
791 clrl SC+8(%a6) 791 clrl SC+8(%a6)
792 faddx SC(%a6),%fp0 792 faddx SC(%a6),%fp0
793 fmovel %d1,%FPCR 793 fmovel %d1,%FPCR
794 fmuld TWON140,%fp0 794 fmuld TWON140,%fp0
795 795
796 bra t_frcinx 796 bra t_frcinx
797 797
798 EM1POLY: 798 EM1POLY:
799 |--Step 9 exp(X)-1 by a simple polynomia 799 |--Step 9 exp(X)-1 by a simple polynomial
800 fmovex (%a0),%fp0 | ...f 800 fmovex (%a0),%fp0 | ...fp0 is X
801 fmulx %fp0,%fp0 801 fmulx %fp0,%fp0 | ...fp0 is S := X*X
802 fmovemx %fp2-%fp2/%fp3,-(%a7) | ...s 802 fmovemx %fp2-%fp2/%fp3,-(%a7) | ...save fp2
803 fmoves #0x2F30CAA8,%fp1 803 fmoves #0x2F30CAA8,%fp1 | ...fp1 is B12
804 fmulx %fp0,%fp1 804 fmulx %fp0,%fp1 | ...fp1 is S*B12
805 fmoves #0x310F8290,%fp2 805 fmoves #0x310F8290,%fp2 | ...fp2 is B11
806 fadds #0x32D73220,%fp1 806 fadds #0x32D73220,%fp1 | ...fp1 is B10+S*B12
807 807
808 fmulx %fp0,%fp2 808 fmulx %fp0,%fp2 | ...fp2 is S*B11
809 fmulx %fp0,%fp1 809 fmulx %fp0,%fp1 | ...fp1 is S*(B10 + ...
810 810
811 fadds #0x3493F281,%fp2 811 fadds #0x3493F281,%fp2 | ...fp2 is B9+S*...
812 faddd EM1B8,%fp1 | ...f 812 faddd EM1B8,%fp1 | ...fp1 is B8+S*...
813 813
814 fmulx %fp0,%fp2 814 fmulx %fp0,%fp2 | ...fp2 is S*(B9+...
815 fmulx %fp0,%fp1 815 fmulx %fp0,%fp1 | ...fp1 is S*(B8+...
816 816
817 faddd EM1B7,%fp2 | ...f 817 faddd EM1B7,%fp2 | ...fp2 is B7+S*...
818 faddd EM1B6,%fp1 | ...f 818 faddd EM1B6,%fp1 | ...fp1 is B6+S*...
819 819
820 fmulx %fp0,%fp2 820 fmulx %fp0,%fp2 | ...fp2 is S*(B7+...
821 fmulx %fp0,%fp1 821 fmulx %fp0,%fp1 | ...fp1 is S*(B6+...
822 822
823 faddd EM1B5,%fp2 | ...f 823 faddd EM1B5,%fp2 | ...fp2 is B5+S*...
824 faddd EM1B4,%fp1 | ...f 824 faddd EM1B4,%fp1 | ...fp1 is B4+S*...
825 825
826 fmulx %fp0,%fp2 826 fmulx %fp0,%fp2 | ...fp2 is S*(B5+...
827 fmulx %fp0,%fp1 827 fmulx %fp0,%fp1 | ...fp1 is S*(B4+...
828 828
829 faddd EM1B3,%fp2 | ...f 829 faddd EM1B3,%fp2 | ...fp2 is B3+S*...
830 faddx EM1B2,%fp1 | ...f 830 faddx EM1B2,%fp1 | ...fp1 is B2+S*...
831 831
832 fmulx %fp0,%fp2 832 fmulx %fp0,%fp2 | ...fp2 is S*(B3+...
833 fmulx %fp0,%fp1 833 fmulx %fp0,%fp1 | ...fp1 is S*(B2+...
834 834
835 fmulx %fp0,%fp2 835 fmulx %fp0,%fp2 | ...fp2 is S*S*(B3+...)
836 fmulx (%a0),%fp1 | ...f 836 fmulx (%a0),%fp1 | ...fp1 is X*S*(B2...
837 837
838 fmuls #0x3F000000,%fp0 838 fmuls #0x3F000000,%fp0 | ...fp0 is S*B1
839 faddx %fp2,%fp1 839 faddx %fp2,%fp1 | ...fp1 is Q
840 | ...fp2 840 | ...fp2 released
841 841
842 fmovemx (%a7)+,%fp2-%fp2/%fp3 | ...f 842 fmovemx (%a7)+,%fp2-%fp2/%fp3 | ...fp2 restored
843 843
844 faddx %fp1,%fp0 844 faddx %fp1,%fp0 | ...fp0 is S*B1+Q
845 | ...fp1 845 | ...fp1 released
846 846
847 fmovel %d1,%FPCR 847 fmovel %d1,%FPCR
848 faddx (%a0),%fp0 848 faddx (%a0),%fp0
849 849
850 bra t_frcinx 850 bra t_frcinx
851 851
852 EM1BIG: 852 EM1BIG:
853 |--Step 10 |X| > 70 log2 853 |--Step 10 |X| > 70 log2
854 movel (%a0),%d0 854 movel (%a0),%d0
855 cmpil #0,%d0 855 cmpil #0,%d0
856 bgt EXPC1 856 bgt EXPC1
857 |--Step 10.2 857 |--Step 10.2
858 fmoves #0xBF800000,%fp0 858 fmoves #0xBF800000,%fp0 | ...fp0 is -1
859 fmovel %d1,%FPCR 859 fmovel %d1,%FPCR
860 fadds #0x00800000,%fp0 860 fadds #0x00800000,%fp0 | ...-1 + 2^(-126)
861 861
862 bra t_frcinx 862 bra t_frcinx
863 863
864 |end 864 |end
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