SDL  2.0
k_tan.c
Go to the documentation of this file.
1 /*
2  * ====================================================
3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4  *
5  * Developed at SunPro, a Sun Microsystems, Inc. business.
6  * Permission to use, copy, modify, and distribute this
7  * software is freely granted, provided that this notice
8  * is preserved.
9  * ====================================================
10  */
11 
12 /* __kernel_tan( x, y, k )
13  * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
14  * Input x is assumed to be bounded by ~pi/4 in magnitude.
15  * Input y is the tail of x.
16  * Input k indicates whether tan (if k=1) or
17  * -1/tan (if k= -1) is returned.
18  *
19  * Algorithm
20  * 1. Since tan(-x) = -tan(x), we need only to consider positive x.
21  * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
22  * 3. tan(x) is approximated by a odd polynomial of degree 27 on
23  * [0,0.67434]
24  * 3 27
25  * tan(x) ~ x + T1*x + ... + T13*x
26  * where
27  *
28  * |tan(x) 2 4 26 | -59.2
29  * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
30  * | x |
31  *
32  * Note: tan(x+y) = tan(x) + tan'(x)*y
33  * ~ tan(x) + (1+x*x)*y
34  * Therefore, for better accuracy in computing tan(x+y), let
35  * 3 2 2 2 2
36  * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
37  * then
38  * 3 2
39  * tan(x+y) = x + (T1*x + (x *(r+y)+y))
40  *
41  * 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  * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
44  */
45 
46 #include "math_libm.h"
47 #include "math_private.h"
48 
49 static const double
50 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */
51 pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
52 pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
53 T[] = {
54  3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
55  1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
56  5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */
57  2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */
58  8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */
59  3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */
60  1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */
61  5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */
62  2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */
63  7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */
64  7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */
65  -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */
66  2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */
67 };
68 
69 double attribute_hidden __kernel_tan(double x, double y, int iy)
70 {
71  double z,r,v,w,s;
72  int32_t ix,hx;
73  GET_HIGH_WORD(hx,x);
74  ix = hx&0x7fffffff; /* high word of |x| */
75  if(ix<0x3e300000) /* x < 2**-28 */
76  {if((int)x==0) { /* generate inexact */
77  u_int32_t low;
78  GET_LOW_WORD(low,x);
79  if(((ix|low)|(iy+1))==0) return one/fabs(x);
80  else return (iy==1)? x: -one/x;
81  }
82  }
83  if(ix>=0x3FE59428) { /* |x|>=0.6744 */
84  if(hx<0) {x = -x; y = -y;}
85  z = pio4-x;
86  w = pio4lo-y;
87  x = z+w; y = 0.0;
88  }
89  z = x*x;
90  w = z*z;
91  /* Break x^5*(T[1]+x^2*T[2]+...) into
92  * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) +
93  * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12]))
94  */
95  r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11]))));
96  v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12])))));
97  s = z*x;
98  r = y + z*(s*(r+v)+y);
99  r += T[0]*s;
100  w = x+r;
101  if(ix>=0x3FE59428) {
102  v = (double)iy;
103  return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r)));
104  }
105  if(iy==1) return w;
106  else { /* if allow error up to 2 ulp,
107  simply return -1.0/(x+r) here */
108  /* compute -1.0/(x+r) accurately */
109  double a,t;
110  z = w;
111  SET_LOW_WORD(z,0);
112  v = r-(z - x); /* z+v = r+x */
113  t = a = -1.0/w; /* a = -1.0/w */
114  SET_LOW_WORD(t,0);
115  s = 1.0+t*z;
116  return t+a*(s+t*v);
117  }
118 }
#define GET_HIGH_WORD(i, d)
Definition: math_private.h:108
GLdouble GLdouble GLdouble r
Definition: SDL_opengl.h:2079
GLdouble GLdouble z
GLdouble s
Definition: SDL_opengl.h:2063
static const double pio4lo
Definition: k_tan.c:52
const GLdouble * v
Definition: SDL_opengl.h:2064
GLint GLint GLint GLint GLint x
Definition: SDL_opengl.h:1574
signed int int32_t
#define attribute_hidden
Definition: math_private.h:25
static const double T[]
Definition: k_tan.c:53
unsigned int u_int32_t
Definition: math_private.h:31
#define SET_LOW_WORD(d, v)
Definition: math_private.h:146
GLubyte GLubyte GLubyte GLubyte w
GLint GLint GLint GLint GLint GLint y
Definition: SDL_opengl.h:1574
#define GET_LOW_WORD(i, d)
Definition: math_private.h:117
static const double pio4
Definition: k_tan.c:51
double attribute_hidden __kernel_tan(double x, double y, int iy)
Definition: k_tan.c:69
static const double one
Definition: k_tan.c:50
GLboolean GLboolean GLboolean GLboolean a
double fabs(double x)
Definition: s_fabs.c:22
GLdouble GLdouble t
Definition: SDL_opengl.h:2071