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1 /* origin: FreeBSD /usr/src/lib/msun/src/k_tan.c */ | 1 /* origin: FreeBSD /usr/src/lib/msun/src/k_tan.c */ |
2 /* | 2 /* |
3 * ==================================================== | 3 * ==================================================== |
4 * Copyright 2004 Sun Microsystems, Inc. All Rights Reserved. | 4 * Copyright 2004 Sun Microsystems, Inc. All Rights Reserved. |
5 * | 5 * |
6 * Permission to use, copy, modify, and distribute this | 6 * Permission to use, copy, modify, and distribute this |
7 * software is freely granted, provided that this notice | 7 * software is freely granted, provided that this notice |
8 * is preserved. | 8 * is preserved. |
9 * ==================================================== | 9 * ==================================================== |
10 */ | 10 */ |
11 /* __tan( x, y, k ) | 11 /* __tan( x, y, k ) |
12 * kernel tan function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854 | 12 * kernel tan function on ~[-pi/4, pi/4] (except on -0), pi/4 ~ 0.7854 |
13 * Input x is assumed to be bounded by ~pi/4 in magnitude. | 13 * Input x is assumed to be bounded by ~pi/4 in magnitude. |
14 * Input y is the tail of x. | 14 * Input y is the tail of x. |
15 * Input odd indicates whether tan (if odd = 0) or -1/tan (if odd = 1) is return
ed. | 15 * Input odd indicates whether tan (if odd = 0) or -1/tan (if odd = 1) is |
| 16 * returned. |
16 * | 17 * |
17 * Algorithm | 18 * Algorithm |
18 * 1. Since tan(-x) = -tan(x), we need only to consider positive x. | 19 * 1. Since tan(-x) = -tan(x), we need only to consider positive x. |
19 * 2. Callers must return tan(-0) = -0 without calling here since our | 20 * 2. Callers must return tan(-0) = -0 without calling here since our |
20 * odd polynomial is not evaluated in a way that preserves -0. | 21 * odd polynomial is not evaluated in a way that preserves -0. |
21 * Callers may do the optimization tan(x) ~ x for tiny x. | 22 * Callers may do the optimization tan(x) ~ x for tiny x. |
22 * 3. tan(x) is approximated by a odd polynomial of degree 27 on | 23 * 3. tan(x) is approximated by a odd polynomial of degree 27 on |
23 * [0,0.67434] | 24 * [0,0.67434] |
24 * 3 27 | 25 * 3 27 |
25 * tan(x) ~ x + T1*x + ... + T13*x | 26 * tan(x) ~ x + T1*x + ... + T13*x |
(...skipping 12 matching lines...) Expand all Loading... |
38 * 3 2 | 39 * 3 2 |
39 * tan(x+y) = x + (T1*x + (x *(r+y)+y)) | 40 * tan(x+y) = x + (T1*x + (x *(r+y)+y)) |
40 * | 41 * |
41 * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then | 42 * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then |
42 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) | 43 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) |
43 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) | 44 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) |
44 */ | 45 */ |
45 | 46 |
46 #include "libm.h" | 47 #include "libm.h" |
47 | 48 |
48 static const double T[] = { | 49 static const double T[] = |
49 3.33333333333334091986e-01, /* 3FD55555, 55555563 */ | 50 { |
50 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */ | 51 3.33333333333334091986e-01, /* 3FD55555, 55555563 */ |
51 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */ | 52 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */ |
52 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */ | 53 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */ |
53 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */ | 54 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */ |
54 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */ | 55 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */ |
55 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */ | 56 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */ |
56 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */ | 57 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */ |
57 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */ | 58 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */ |
58 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */ | 59 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */ |
59 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */ | 60 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */ |
60 -1.85586374855275456654e-05, /* BEF375CB, DB605373 */ | 61 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */ |
61 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */ | 62 -1.85586374855275456654e-05, /* BEF375CB, DB605373 */ |
| 63 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */ |
62 }, | 64 }, |
63 pio4 = 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ | 65 pio4 = 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ |
64 pio4lo = 3.06161699786838301793e-17; /* 3C81A626, 33145C07 */ | 66 pio4lo = 3.06161699786838301793e-17; /* 3C81A626, 33145C07 */ |
65 | 67 |
66 double __tan(double x, double y, int odd) | 68 double __tan(double x, double y, int odd) { |
67 { | 69 double_t z, r, v, w, s, a; |
68 » double_t z, r, v, w, s, a; | 70 double w0, a0; |
69 » double w0, a0; | 71 uint32_t hx; |
70 » uint32_t hx; | 72 int big, sign; |
71 » int big, sign; | |
72 | 73 |
73 » GET_HIGH_WORD(hx,x); | 74 GET_HIGH_WORD(hx, x); |
74 » big = (hx&0x7fffffff) >= 0x3FE59428; /* |x| >= 0.6744 */ | 75 big = (hx & 0x7fffffff) >= 0x3FE59428; /* |x| >= 0.6744 */ |
75 » if (big) { | 76 if (big) { |
76 » » sign = hx>>31; | 77 sign = hx >> 31; |
77 » » if (sign) { | 78 if (sign) { |
78 » » » x = -x; | 79 x = -x; |
79 » » » y = -y; | 80 y = -y; |
80 » » } | 81 } |
81 » » x = (pio4 - x) + (pio4lo - y); | 82 x = (pio4 - x) + (pio4lo - y); |
82 » » y = 0.0; | 83 y = 0.0; |
83 » } | 84 } |
84 » z = x * x; | 85 z = x * x; |
85 » w = z * z; | 86 w = z * z; |
86 » /* | 87 /* |
87 » * Break x^5*(T[1]+x^2*T[2]+...) into | 88 * Break x^5*(T[1]+x^2*T[2]+...) into |
88 » * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + | 89 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + |
89 » * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) | 90 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) |
90 » */ | 91 */ |
91 » r = T[1] + w*(T[3] + w*(T[5] + w*(T[7] + w*(T[9] + w*T[11])))); | 92 r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + w * T[11])))); |
92 » v = z*(T[2] + w*(T[4] + w*(T[6] + w*(T[8] + w*(T[10] + w*T[12]))))); | 93 v = z * |
93 » s = z * x; | 94 (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + w * T[12]))))); |
94 » r = y + z*(s*(r + v) + y) + s*T[0]; | 95 s = z * x; |
95 » w = x + r; | 96 r = y + z * (s * (r + v) + y) + s * T[0]; |
96 » if (big) { | 97 w = x + r; |
97 » » s = 1 - 2*odd; | 98 if (big) { |
98 » » v = s - 2.0 * (x + (r - w*w/(w + s))); | 99 s = 1 - 2 * odd; |
99 » » return sign ? -v : v; | 100 v = s - 2.0 * (x + (r - w * w / (w + s))); |
100 » } | 101 return sign ? -v : v; |
101 » if (!odd) | 102 } |
102 » » return w; | 103 if (!odd) |
103 » /* -1.0/(x+r) has up to 2ulp error, so compute it accurately */ | 104 return w; |
104 » w0 = w; | 105 /* -1.0/(x+r) has up to 2ulp error, so compute it accurately */ |
105 » SET_LOW_WORD(w0, 0); | 106 w0 = w; |
106 » v = r - (w0 - x); /* w0+v = r+x */ | 107 SET_LOW_WORD(w0, 0); |
107 » a0 = a = -1.0 / w; | 108 v = r - (w0 - x); /* w0+v = r+x */ |
108 » SET_LOW_WORD(a0, 0); | 109 a0 = a = -1.0 / w; |
109 » return a0 + a*(1.0 + a0*w0 + a0*v); | 110 SET_LOW_WORD(a0, 0); |
| 111 return a0 + a * (1.0 + a0 * w0 + a0 * v); |
110 } | 112 } |
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