diff options
| -rw-r--r-- | lib/libm/src/k_tan.c | 170 |
1 files changed, 101 insertions, 69 deletions
diff --git a/lib/libm/src/k_tan.c b/lib/libm/src/k_tan.c index 91146c2f767..33d63a1024d 100644 --- a/lib/libm/src/k_tan.c +++ b/lib/libm/src/k_tan.c @@ -5,7 +5,7 @@ * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this - * software is freely granted, provided that this notice + * software is freely granted, provided that this notice * is preserved. * ==================================================== */ @@ -18,25 +18,25 @@ static char rcsid[] = "$NetBSD: k_tan.c,v 1.8 1995/05/10 20:46:37 jtc Exp $"; * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 * Input x is assumed to be bounded by ~pi/4 in magnitude. * Input y is the tail of x. - * Input k indicates whether tan (if k=1) or + * Input k indicates whether tan (if k=1) or * -1/tan (if k= -1) is returned. * * Algorithm - * 1. Since tan(-x) = -tan(x), we need only to consider positive x. + * 1. Since tan(-x) = -tan(x), we need only to consider positive x. * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. * 3. tan(x) is approximated by a odd polynomial of degree 27 on * [0,0.67434] * 3 27 * tan(x) ~ x + T1*x + ... + T13*x * where - * + * * |tan(x) 2 4 26 | -59.2 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 - * | x | - * + * | x | + * * Note: tan(x+y) = tan(x) + tan'(x)*y * ~ tan(x) + (1+x*x)*y - * Therefore, for better accuracy in computing tan(x+y), let + * Therefore, for better accuracy in computing tan(x+y), let * 3 2 2 2 2 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) * then @@ -50,74 +50,106 @@ static char rcsid[] = "$NetBSD: k_tan.c,v 1.8 1995/05/10 20:46:37 jtc Exp $"; #include "math.h" #include "math_private.h" -static const double -one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ -pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ -pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */ -T[] = { - 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */ - 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */ - 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */ - 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */ - 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */ - 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */ - 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */ - 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */ - 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */ - 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */ - 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */ - -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */ - 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */ + +static const double xxx[] = { + 3.33333333333334091986e-01, /* 3FD55555, 55555563 */ + 1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */ + 5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */ + 2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */ + 8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */ + 3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */ + 1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */ + 5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */ + 2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */ + 7.81794442939557092300e-05, /* 3F147E88, A03792A6 */ + 7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */ + -1.85586374855275456654e-05, /* BEF375CB, DB605373 */ + 2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */ +/* one */ 1.00000000000000000000e+00, /* 3FF00000, 00000000 */ +/* pio4 */ 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */ +/* pio4lo */ 3.06161699786838301793e-17 /* 3C81A626, 33145C07 */ }; +#define one xxx[13] +#define pio4 xxx[14] +#define pio4lo xxx[15] +#define T xxx double __kernel_tan(double x, double y, int iy) { - double z,r,v,w,s; - int32_t ix,hx; - GET_HIGH_WORD(hx,x); - ix = hx&0x7fffffff; /* high word of |x| */ - if(ix<0x3e300000) /* x < 2**-28 */ - {if((int)x==0) { /* generate inexact */ - u_int32_t low; - GET_LOW_WORD(low,x); - if(((ix|low)|(iy+1))==0) return one/fabs(x); - else return (iy==1)? x: -one/x; - } - } - if(ix>=0x3FE59428) { /* |x|>=0.6744 */ - if(hx<0) {x = -x; y = -y;} - z = pio4-x; - w = pio4lo-y; - x = z+w; y = 0.0; + double z, r, v, w, s; + int32_t ix, hx; + + GET_HIGH_WORD(hx, x); /* high word of x */ + ix = hx & 0x7fffffff; /* high word of |x| */ + if (ix < 0x3e300000) { /* x < 2**-28 */ + if ((int) x == 0) { /* generate inexact */ + u_int32_t low; + GET_LOW_WORD(low, x); + if(((ix | low) | (iy + 1)) == 0) + return one / fabs(x); + else { + if (iy == 1) + return x; + else { /* compute -1 / (x+y) carefully */ + double a, t; + + z = w = x + y; + SET_LOW_WORD(z, 0); + v = y - (z - x); + t = a = -one / w; + SET_LOW_WORD(t, 0); + s = one + t * z; + return t + a * (s + t * v); + } + } + } + } + if (ix >= 0x3FE59428) { /* |x| >= 0.6744 */ + if (hx < 0) { + x = -x; + y = -y; + } + z = pio4 - x; + w = pio4lo - y; + x = z + w; + y = 0.0; } - z = x*x; - w = z*z; - /* Break x^5*(T[1]+x^2*T[2]+...) into - * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + - * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) - */ - r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11])))); - v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12]))))); - s = z*x; - r = y + z*(s*(r+v)+y); - r += T[0]*s; - w = x+r; - if(ix>=0x3FE59428) { - v = (double)iy; - return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r))); + z = x * x; + w = z * z; + /* + * Break x^5*(T[1]+x^2*T[2]+...) into + * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + + * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) + */ + r = T[1] + w * (T[3] + w * (T[5] + w * (T[7] + w * (T[9] + + w * T[11])))); + v = z * (T[2] + w * (T[4] + w * (T[6] + w * (T[8] + w * (T[10] + + w * T[12]))))); + s = z * x; + r = y + z * (s * (r + v) + y); + r += T[0] * s; + w = x + r; + if (ix >= 0x3FE59428) { + v = (double) iy; + return (double) (1 - ((hx >> 30) & 2)) * + (v - 2.0 * (x - (w * w / (w + v) - r))); } - if(iy==1) return w; - else { /* if allow error up to 2 ulp, - simply return -1.0/(x+r) here */ - /* compute -1.0/(x+r) accurately */ - double a,t; - z = w; - SET_LOW_WORD(z,0); - v = r-(z - x); /* z+v = r+x */ - t = a = -1.0/w; /* a = -1.0/w */ - SET_LOW_WORD(t,0); - s = 1.0+t*z; - return t+a*(s+t*v); + if (iy == 1) + return w; + else { + /* + * if allow error up to 2 ulp, simply return + * -1.0 / (x+r) here + */ + /* compute -1.0 / (x+r) accurately */ + double a, t; + z = w; + SET_LOW_WORD(z, 0); + v = r - (z - x); /* z+v = r+x */ + t = a = -1.0 / w; /* a = -1.0/w */ + SET_LOW_WORD(t, 0); + s = 1.0 + t * z; + return t + a * (s + t * v); } } |