author | convert-repo |
Tue, 31 Dec 2019 04:28:27 +0000 | |
changeset 3995 | d65488cfad3c |
parent 2380 | 9195eccdcbd9 |
permissions | -rw-r--r-- |
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#pragma ident "@(#)k_tan.c 1.5 04/04/22 SMI" |
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/* |
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* ==================================================== |
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* Copyright 2004 Sun Microsystems, Inc. All Rights Reserved. |
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* |
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* Permission to use, copy, modify, and distribute this |
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* software is freely granted, provided that this notice |
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* is preserved. |
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* ==================================================== |
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*/ |
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|
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/* INDENT OFF */ |
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/* __kernel_tan( x, y, k ) |
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* kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 |
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* Input x is assumed to be bounded by ~pi/4 in magnitude. |
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* Input y is the tail of x. |
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* Input k indicates whether tan (if k = 1) or -1/tan (if k = -1) is returned. |
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* |
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* Algorithm |
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* 1. Since tan(-x) = -tan(x), we need only to consider positive x. |
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* 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. |
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* 3. tan(x) is approximated by a odd polynomial of degree 27 on |
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* [0,0.67434] |
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* 3 27 |
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* tan(x) ~ x + T1*x + ... + T13*x |
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* where |
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* |
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* |tan(x) 2 4 26 | -59.2 |
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* |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 |
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* | x | |
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* |
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* Note: tan(x+y) = tan(x) + tan'(x)*y |
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* ~ tan(x) + (1+x*x)*y |
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* Therefore, for better accuracy in computing tan(x+y), let |
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* 3 2 2 2 2 |
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* r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) |
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* then |
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* 3 2 |
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* tan(x+y) = x + (T1*x + (x *(r+y)+y)) |
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* |
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* 4. For x in [0.67434,pi/4], let y = pi/4 - x, then |
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* tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) |
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* = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) |
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*/ |
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|
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#include "fdlibm.h" |
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|
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static const double xxx[] = { |
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3.33333333333334091986e-01, /* 3FD55555, 55555563 */ |
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1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */ |
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5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */ |
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2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */ |
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8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */ |
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3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */ |
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1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */ |
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5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */ |
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2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */ |
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7.81794442939557092300e-05, /* 3F147E88, A03792A6 */ |
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7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */ |
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-1.85586374855275456654e-05, /* BEF375CB, DB605373 */ |
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2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */ |
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/* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */ |
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/* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ |
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/* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ |
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}; |
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#define one xxx[13] |
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#define pio4 xxx[14] |
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#define pio4lo xxx[15] |
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#define T xxx |
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/* INDENT ON */ |
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|
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double |
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__kernel_tan(double x, double y, int iy) { |
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double z, r, v, w, s; |
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int ix, hx; |
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|
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hx = __HI(x); /* high word of x */ |
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ix = hx & 0x7fffffff; /* high word of |x| */ |
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if (ix < 0x3e300000) { /* x < 2**-28 */ |
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if ((int) x == 0) { /* generate inexact */ |
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if (((ix | __LO(x)) | (iy + 1)) == 0) |
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return one / fabs(x); |
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else { |
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if (iy == 1) |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
86 |
return x; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
87 |
else { /* compute -1 / (x+y) carefully */ |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
88 |
double a, t; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
89 |
|
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
90 |
z = w = x + y; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
91 |
__LO(z) = 0; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
92 |
v = y - (z - x); |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
93 |
t = a = -one / w; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
94 |
__LO(t) = 0; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
95 |
s = one + t * z; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
96 |
return t + a * (s + t * v); |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
97 |
} |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
98 |
} |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
99 |
} |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
100 |
} |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
101 |
if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */ |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
102 |
if (hx < 0) { |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
103 |
x = -x; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
104 |
y = -y; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
105 |
} |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
106 |
z = pio4 - x; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
107 |
w = pio4lo - y; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
108 |
x = z + w; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
109 |
y = 0.0; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
110 |
} |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
111 |
z = x * x; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
112 |
w = z * z; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
113 |
/* |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
114 |
* Break x^5*(T[1]+x^2*T[2]+...) into |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
115 |
* x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
116 |
* x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
117 |
*/ |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
118 |
r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
119 |
w * T[11])))); |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
120 |
v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
121 |
w * T[12]))))); |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
122 |
s = z * x; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
123 |
r = y + z * (s * (r + v) + y); |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
124 |
r += T[0] * s; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
125 |
w = x + r; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
126 |
if (ix >= 0x3FE59428) { |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
127 |
v = (double) iy; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
128 |
return (double) (1 - ((hx >> 30) & 2)) * |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
129 |
(v - 2.0 * (x - (w * w / (w + v) - r))); |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
130 |
} |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
131 |
if (iy == 1) |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
132 |
return w; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
133 |
else { |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
134 |
/* |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
135 |
* if allow error up to 2 ulp, simply return |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
136 |
* -1.0 / (x+r) here |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
137 |
*/ |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
138 |
/* compute -1.0 / (x+r) accurately */ |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
139 |
double a, t; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
140 |
z = w; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
141 |
__LO(z) = 0; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
142 |
v = r - (z - x); /* z+v = r+x */ |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
143 |
t = a = -1.0 / w; /* a = -1.0/w */ |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
144 |
__LO(t) = 0; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
145 |
s = 1.0 + t * z; |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
146 |
return t + a * (s + t * v); |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
147 |
} |
fa7400d022a0
Updated to https://swing.fit.cvut.cz/hg/stx.libjava/rev/31f8b995905e
Jan Vrany <jan.vrany@fit.cvut.cz>
parents:
diff
changeset
|
148 |
} |