1 | 1 |
2 | scosh.sa 3.1 12/10/90 2 | scosh.sa 3.1 12/10/90
3 | 3 |
4 | The entry point sCosh computes the hyp 4 | The entry point sCosh computes the hyperbolic cosine of
5 | an input argument; sCoshd does the sam 5 | an input argument; sCoshd 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 cosh(X) returned in 11 | Output: The value cosh(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 sCOSH takes approxi 18 | Speed: The program sCOSH takes approximately 250 cycles.
19 | 19 |
20 | Algorithm: 20 | Algorithm:
21 | 21 |
22 | COSH 22 | COSH
23 | 1. If |X| > 16380 log2, go to 3. 23 | 1. If |X| > 16380 log2, go to 3.
24 | 24 |
25 | 2. (|X| <= 16380 log2) Cosh(X) is obta 25 | 2. (|X| <= 16380 log2) Cosh(X) is obtained by the formulae
26 | y = |X|, z = exp(Y), and 26 | y = |X|, z = exp(Y), and
27 | cosh(X) = (1/2)*( z + 1/z ). 27 | cosh(X) = (1/2)*( z + 1/z ).
28 | Exit. 28 | Exit.
29 | 29 |
30 | 3. (|X| > 16380 log2). If |X| > 16480 30 | 3. (|X| > 16380 log2). If |X| > 16480 log2, go to 5.
31 | 31 |
32 | 4. (16380 log2 < |X| <= 16480 log2) 32 | 4. (16380 log2 < |X| <= 16480 log2)
33 | cosh(X) = sign(X) * exp(|X|)/2 33 | cosh(X) = sign(X) * exp(|X|)/2.
34 | However, invoking exp(|X|) may 34 | However, invoking exp(|X|) may cause premature overflow.
35 | Thus, we calculate sinh(X) as 35 | Thus, we calculate sinh(X) as follows:
36 | Y := |X| 36 | Y := |X|
37 | Fact := 2**(16380) 37 | Fact := 2**(16380)
38 | Y' := Y - 16381 log2 38 | Y' := Y - 16381 log2
39 | cosh(X) := Fact * exp(Y'). 39 | cosh(X) := Fact * exp(Y').
40 | Exit. 40 | Exit.
41 | 41 |
42 | 5. (|X| > 16480 log2) sinh(X) must ove 42 | 5. (|X| > 16480 log2) sinh(X) must overflow. Return
43 | Huge*Huge to generate overflow 43 | Huge*Huge to generate overflow and an infinity with
44 | the appropriate sign. Huge is 44 | the appropriate sign. Huge is the largest finite number in
45 | extended format. Exit. 45 | extended format. Exit.
46 | 46 |
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 |SCOSH idnt 2,1 | Motorola 040 Floating Po 55 |SCOSH idnt 2,1 | Motorola 040 Floating Point Software Package
56 56
57 |section 8 57 |section 8
58 58
59 |xref t_ovfl 59 |xref t_ovfl
60 |xref t_frcinx 60 |xref t_frcinx
61 |xref setox 61 |xref setox
62 62
63 T1: .long 0x40C62D38,0xD3D64634 | ... 1638 63 T1: .long 0x40C62D38,0xD3D64634 | ... 16381 LOG2 LEAD
64 T2: .long 0x3D6F90AE,0xB1E75CC7 | ... 1638 64 T2: .long 0x3D6F90AE,0xB1E75CC7 | ... 16381 LOG2 TRAIL
65 65
66 TWO16380: .long 0x7FFB0000,0x80000000,0x000000 66 TWO16380: .long 0x7FFB0000,0x80000000,0x00000000,0x00000000
67 67
68 .global scoshd 68 .global scoshd
69 scoshd: 69 scoshd:
70 |--COSH(X) = 1 FOR DENORMALIZED X 70 |--COSH(X) = 1 FOR DENORMALIZED X
71 71
72 fmoves #0x3F800000,%fp0 72 fmoves #0x3F800000,%fp0
73 73
74 fmovel %d1,%FPCR 74 fmovel %d1,%FPCR
75 fadds #0x00800000,%fp0 75 fadds #0x00800000,%fp0
76 bra t_frcinx 76 bra t_frcinx
77 77
78 .global scosh 78 .global scosh
79 scosh: 79 scosh:
80 fmovex (%a0),%fp0 | ...L 80 fmovex (%a0),%fp0 | ...LOAD INPUT
81 81
82 movel (%a0),%d0 82 movel (%a0),%d0
83 movew 4(%a0),%d0 83 movew 4(%a0),%d0
84 andil #0x7FFFFFFF,%d0 84 andil #0x7FFFFFFF,%d0
85 cmpil #0x400CB167,%d0 85 cmpil #0x400CB167,%d0
86 bgts COSHBIG 86 bgts COSHBIG
87 87
88 |--THIS IS THE USUAL CASE, |X| < 16380 LOG2 88 |--THIS IS THE USUAL CASE, |X| < 16380 LOG2
89 |--COSH(X) = (1/2) * ( EXP(X) + 1/EXP(X) ) 89 |--COSH(X) = (1/2) * ( EXP(X) + 1/EXP(X) )
90 90
91 fabsx %fp0 | ...| 91 fabsx %fp0 | ...|X|
92 92
93 movel %d1,-(%sp) 93 movel %d1,-(%sp)
94 clrl %d1 94 clrl %d1
95 fmovemx %fp0-%fp0,(%a0) |pass paramete 95 fmovemx %fp0-%fp0,(%a0) |pass parameter to setox
96 bsr setox | ...F 96 bsr setox | ...FP0 IS EXP(|X|)
97 fmuls #0x3F000000,%fp0 97 fmuls #0x3F000000,%fp0 | ...(1/2)EXP(|X|)
98 movel (%sp)+,%d1 98 movel (%sp)+,%d1
99 99
100 fmoves #0x3E800000,%fp1 100 fmoves #0x3E800000,%fp1 | ...(1/4)
101 fdivx %fp0,%fp1 101 fdivx %fp0,%fp1 | ...1/(2 EXP(|X|))
102 102
103 fmovel %d1,%FPCR 103 fmovel %d1,%FPCR
104 faddx %fp1,%fp0 104 faddx %fp1,%fp0
105 105
106 bra t_frcinx 106 bra t_frcinx
107 107
108 COSHBIG: 108 COSHBIG:
109 cmpil #0x400CB2B3,%d0 109 cmpil #0x400CB2B3,%d0
110 bgts COSHHUGE 110 bgts COSHHUGE
111 111
112 fabsx %fp0 112 fabsx %fp0
113 fsubd T1(%pc),%fp0 113 fsubd T1(%pc),%fp0 | ...(|X|-16381LOG2_LEAD)
114 fsubd T2(%pc),%fp0 114 fsubd T2(%pc),%fp0 | ...|X| - 16381 LOG2, ACCURATE
115 115
116 movel %d1,-(%sp) 116 movel %d1,-(%sp)
117 clrl %d1 117 clrl %d1
118 fmovemx %fp0-%fp0,(%a0) 118 fmovemx %fp0-%fp0,(%a0)
119 bsr setox 119 bsr setox
120 fmovel (%sp)+,%fpcr 120 fmovel (%sp)+,%fpcr
121 121
122 fmulx TWO16380(%pc),%fp0 122 fmulx TWO16380(%pc),%fp0
123 bra t_frcinx 123 bra t_frcinx
124 124
125 COSHHUGE: 125 COSHHUGE:
126 fmovel #0,%fpsr 126 fmovel #0,%fpsr |clr N bit if set by source
127 bclrb #7,(%a0) 127 bclrb #7,(%a0) |always return positive value
128 fmovemx (%a0),%fp0-%fp0 128 fmovemx (%a0),%fp0-%fp0
129 bra t_ovfl 129 bra t_ovfl
130 130
131 |end 131 |end
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