1 | 1 |
2 | stanh.sa 3.1 12/10/90 2 | stanh.sa 3.1 12/10/90
3 | 3 |
4 | The entry point sTanh computes the hyp 4 | The entry point sTanh computes the hyperbolic tangent of
5 | an input argument; sTanhd does the sam 5 | an input argument; sTanhd does the same except for denormalized
6 | input. 6 | input.
7 | 7 |
8 | Input: Double-extended number X in loc 8 | Input: Double-extended number X in location pointed to
9 | by address register a0. 9 | by address register a0.
10 | 10 |
11 | Output: The value tanh(X) returned in 11 | Output: The value tanh(X) returned in floating-point register Fp0.
12 | 12 |
13 | Accuracy and Monotonicity: The returne 13 | Accuracy and Monotonicity: The returned result is within 3 ulps in
14 | 64 significant bit, i.e. withi 14 | 64 significant bit, i.e. within 0.5001 ulp to 53 bits if the
15 | result is subsequently rounded 15 | result is subsequently rounded to double precision. The
16 | result is provably monotonic i 16 | result is provably monotonic in double precision.
17 | 17 |
18 | Speed: The program stanh takes approxi 18 | Speed: The program stanh takes approximately 270 cycles.
19 | 19 |
20 | Algorithm: 20 | Algorithm:
21 | 21 |
22 | TANH 22 | TANH
23 | 1. If |X| >= (5/2) log2 or |X| <= 2**( 23 | 1. If |X| >= (5/2) log2 or |X| <= 2**(-40), go to 3.
24 | 24 |
25 | 2. (2**(-40) < |X| < (5/2) log2) Calcu 25 | 2. (2**(-40) < |X| < (5/2) log2) Calculate tanh(X) by
26 | sgn := sign(X), y := 2|X|, z : 26 | sgn := sign(X), y := 2|X|, z := expm1(Y), and
27 | tanh(X) = sgn*( z/(2+z) ). 27 | tanh(X) = sgn*( z/(2+z) ).
28 | Exit. 28 | Exit.
29 | 29 |
30 | 3. (|X| <= 2**(-40) or |X| >= (5/2) lo 30 | 3. (|X| <= 2**(-40) or |X| >= (5/2) log2). If |X| < 1,
31 | go to 7. 31 | go to 7.
32 | 32 |
33 | 4. (|X| >= (5/2) log2) If |X| >= 50 lo 33 | 4. (|X| >= (5/2) log2) If |X| >= 50 log2, go to 6.
34 | 34 |
35 | 5. ((5/2) log2 <= |X| < 50 log2) Calcu 35 | 5. ((5/2) log2 <= |X| < 50 log2) Calculate tanh(X) by
36 | sgn := sign(X), y := 2|X|, z : 36 | sgn := sign(X), y := 2|X|, z := exp(Y),
37 | tanh(X) = sgn - [ sgn*2/(1+z) 37 | tanh(X) = sgn - [ sgn*2/(1+z) ].
38 | Exit. 38 | Exit.
39 | 39 |
40 | 6. (|X| >= 50 log2) Tanh(X) = +-1 (rou 40 | 6. (|X| >= 50 log2) Tanh(X) = +-1 (round to nearest). Thus, we
41 | calculate Tanh(X) by 41 | calculate Tanh(X) by
42 | sgn := sign(X), Tiny := 2**(-1 42 | sgn := sign(X), Tiny := 2**(-126),
43 | tanh(X) := sgn - sgn*Tiny. 43 | tanh(X) := sgn - sgn*Tiny.
44 | Exit. 44 | Exit.
45 | 45 |
46 | 7. (|X| < 2**(-40)). Tanh(X) = X. 46 | 7. (|X| < 2**(-40)). Tanh(X) = X. Exit.
47 | 47 |
48 48
49 | Copyright (C) Motorola, Inc. 1 49 | Copyright (C) Motorola, Inc. 1990
50 | All Rights Reserved 50 | All Rights Reserved
51 | 51 |
52 | For details on the license for this fi 52 | For details on the license for this file, please see the
53 | file, README, in this same directory. 53 | file, README, in this same directory.
54 54
55 |STANH idnt 2,1 | Motorola 040 Floating Po 55 |STANH idnt 2,1 | Motorola 040 Floating Point Software Package
56 56
57 |section 8 57 |section 8
58 58
59 #include "fpsp.h" 59 #include "fpsp.h"
60 60
61 .set X,FP_SCR5 61 .set X,FP_SCR5
62 .set XDCARE,X+2 62 .set XDCARE,X+2
63 .set XFRAC,X+4 63 .set XFRAC,X+4
64 64
65 .set SGN,L_SCR3 65 .set SGN,L_SCR3
66 66
67 .set V,FP_SCR6 67 .set V,FP_SCR6
68 68
69 BOUNDS1: .long 0x3FD78000,0x3FFFDDCE | 69 BOUNDS1: .long 0x3FD78000,0x3FFFDDCE | ... 2^(-40), (5/2)LOG2
70 70
71 |xref t_frcinx 71 |xref t_frcinx
72 |xref t_extdnrm 72 |xref t_extdnrm
73 |xref setox 73 |xref setox
74 |xref setoxm1 74 |xref setoxm1
75 75
76 .global stanhd 76 .global stanhd
77 stanhd: 77 stanhd:
78 |--TANH(X) = X FOR DENORMALIZED X 78 |--TANH(X) = X FOR DENORMALIZED X
79 79
80 bra t_extdnrm 80 bra t_extdnrm
81 81
82 .global stanh 82 .global stanh
83 stanh: 83 stanh:
84 fmovex (%a0),%fp0 | ...L 84 fmovex (%a0),%fp0 | ...LOAD INPUT
85 85
86 fmovex %fp0,X(%a6) 86 fmovex %fp0,X(%a6)
87 movel (%a0),%d0 87 movel (%a0),%d0
88 movew 4(%a0),%d0 88 movew 4(%a0),%d0
89 movel %d0,X(%a6) 89 movel %d0,X(%a6)
90 andl #0x7FFFFFFF,%d0 90 andl #0x7FFFFFFF,%d0
91 cmp2l BOUNDS1(%pc),%d0 91 cmp2l BOUNDS1(%pc),%d0 | ...2**(-40) < |X| < (5/2)LOG2 ?
92 bcss TANHBORS 92 bcss TANHBORS
93 93
94 |--THIS IS THE USUAL CASE 94 |--THIS IS THE USUAL CASE
95 |--Y = 2|X|, Z = EXPM1(Y), TANH(X) = SIGN(X) * 95 |--Y = 2|X|, Z = EXPM1(Y), TANH(X) = SIGN(X) * Z / (Z+2).
96 96
97 movel X(%a6),%d0 97 movel X(%a6),%d0
98 movel %d0,SGN(%a6) 98 movel %d0,SGN(%a6)
99 andl #0x7FFF0000,%d0 99 andl #0x7FFF0000,%d0
100 addl #0x00010000,%d0 | ...E 100 addl #0x00010000,%d0 | ...EXPONENT OF 2|X|
101 movel %d0,X(%a6) 101 movel %d0,X(%a6)
102 andl #0x80000000,SGN(%a6) 102 andl #0x80000000,SGN(%a6)
103 fmovex X(%a6),%fp0 103 fmovex X(%a6),%fp0 | ...FP0 IS Y = 2|X|
104 104
105 movel %d1,-(%a7) 105 movel %d1,-(%a7)
106 clrl %d1 106 clrl %d1
107 fmovemx %fp0-%fp0,(%a0) 107 fmovemx %fp0-%fp0,(%a0)
108 bsr setoxm1 | ...F 108 bsr setoxm1 | ...FP0 IS Z = EXPM1(Y)
109 movel (%a7)+,%d1 109 movel (%a7)+,%d1
110 110
111 fmovex %fp0,%fp1 111 fmovex %fp0,%fp1
112 fadds #0x40000000,%fp1 112 fadds #0x40000000,%fp1 | ...Z+2
113 movel SGN(%a6),%d0 113 movel SGN(%a6),%d0
114 fmovex %fp1,V(%a6) 114 fmovex %fp1,V(%a6)
115 eorl %d0,V(%a6) 115 eorl %d0,V(%a6)
116 116
117 fmovel %d1,%FPCR 117 fmovel %d1,%FPCR |restore users exceptions
118 fdivx V(%a6),%fp0 118 fdivx V(%a6),%fp0
119 bra t_frcinx 119 bra t_frcinx
120 120
121 TANHBORS: 121 TANHBORS:
122 cmpl #0x3FFF8000,%d0 122 cmpl #0x3FFF8000,%d0
123 blt TANHSM 123 blt TANHSM
124 124
125 cmpl #0x40048AA1,%d0 125 cmpl #0x40048AA1,%d0
126 bgt TANHHUGE 126 bgt TANHHUGE
127 127
128 |-- (5/2) LOG2 < |X| < 50 LOG2, 128 |-- (5/2) LOG2 < |X| < 50 LOG2,
129 |--TANH(X) = 1 - (2/[EXP(2X)+1]). LET Y = 2|X| 129 |--TANH(X) = 1 - (2/[EXP(2X)+1]). LET Y = 2|X|, SGN = SIGN(X),
130 |--TANH(X) = SGN - SGN*2/[EXP(Y)+1]. 130 |--TANH(X) = SGN - SGN*2/[EXP(Y)+1].
131 131
132 movel X(%a6),%d0 132 movel X(%a6),%d0
133 movel %d0,SGN(%a6) 133 movel %d0,SGN(%a6)
134 andl #0x7FFF0000,%d0 134 andl #0x7FFF0000,%d0
135 addl #0x00010000,%d0 | ...E 135 addl #0x00010000,%d0 | ...EXPO OF 2|X|
136 movel %d0,X(%a6) 136 movel %d0,X(%a6) | ...Y = 2|X|
137 andl #0x80000000,SGN(%a6) 137 andl #0x80000000,SGN(%a6)
138 movel SGN(%a6),%d0 138 movel SGN(%a6),%d0
139 fmovex X(%a6),%fp0 139 fmovex X(%a6),%fp0 | ...Y = 2|X|
140 140
141 movel %d1,-(%a7) 141 movel %d1,-(%a7)
142 clrl %d1 142 clrl %d1
143 fmovemx %fp0-%fp0,(%a0) 143 fmovemx %fp0-%fp0,(%a0)
144 bsr setox | ...F 144 bsr setox | ...FP0 IS EXP(Y)
145 movel (%a7)+,%d1 145 movel (%a7)+,%d1
146 movel SGN(%a6),%d0 146 movel SGN(%a6),%d0
147 fadds #0x3F800000,%fp0 147 fadds #0x3F800000,%fp0 | ...EXP(Y)+1
148 148
149 eorl #0xC0000000,%d0 | ...- 149 eorl #0xC0000000,%d0 | ...-SIGN(X)*2
150 fmoves %d0,%fp1 150 fmoves %d0,%fp1 | ...-SIGN(X)*2 IN SGL FMT
151 fdivx %fp0,%fp1 151 fdivx %fp0,%fp1 | ...-SIGN(X)2 / [EXP(Y)+1 ]
152 152
153 movel SGN(%a6),%d0 153 movel SGN(%a6),%d0
154 orl #0x3F800000,%d0 | ...S 154 orl #0x3F800000,%d0 | ...SGN
155 fmoves %d0,%fp0 155 fmoves %d0,%fp0 | ...SGN IN SGL FMT
156 156
157 fmovel %d1,%FPCR 157 fmovel %d1,%FPCR |restore users exceptions
158 faddx %fp1,%fp0 158 faddx %fp1,%fp0
159 159
160 bra t_frcinx 160 bra t_frcinx
161 161
162 TANHSM: 162 TANHSM:
163 movew #0x0000,XDCARE(%a6) 163 movew #0x0000,XDCARE(%a6)
164 164
165 fmovel %d1,%FPCR 165 fmovel %d1,%FPCR |restore users exceptions
166 fmovex X(%a6),%fp0 166 fmovex X(%a6),%fp0 |last inst - possible exception set
167 167
168 bra t_frcinx 168 bra t_frcinx
169 169
170 TANHHUGE: 170 TANHHUGE:
171 |---RETURN SGN(X) - SGN(X)EPS 171 |---RETURN SGN(X) - SGN(X)EPS
172 movel X(%a6),%d0 172 movel X(%a6),%d0
173 andl #0x80000000,%d0 173 andl #0x80000000,%d0
174 orl #0x3F800000,%d0 174 orl #0x3F800000,%d0
175 fmoves %d0,%fp0 175 fmoves %d0,%fp0
176 andl #0x80000000,%d0 176 andl #0x80000000,%d0
177 eorl #0x80800000,%d0 | ...- 177 eorl #0x80800000,%d0 | ...-SIGN(X)*EPS
178 178
179 fmovel %d1,%FPCR 179 fmovel %d1,%FPCR |restore users exceptions
180 fadds %d0,%fp0 180 fadds %d0,%fp0
181 181
182 bra t_frcinx 182 bra t_frcinx
183 183
184 |end 184 |end
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