double tan(double x) {
#include "utan.h"
#include "utan.tbl"
int ux,i,n;
double a,da,a2,b,db,c,dc,c1,cc1,c2,cc2,c3,cc3,fi,ffi,gi,pz,s,sy,
t,t1,t2,t3,t4,t5,t6,t7,t8,t9,t10,w,x2,xn,xx2,y,ya,yya,z0,z,zz,z2,zz2;
int p;
number num,v;
mp_no mpa,mpt1,mpt2;
#if 0
mp_no mpy;
#endif
int __branred(double, double *, double *);
int __mpranred(double, mp_no *, int);
/* x=+-INF, x=NaN */
num.d = x;
ux = num.i[HIGH_HALF];
if ((ux&0x7ff00000)==0x7ff00000) return x-x;
w=(x<ZERO) ? -x : x;
/* (I) The case abs(x) <= 1.259e-8 */
if (w<=g1.d) return x;
/* (II) The case 1.259e-8 < abs(x) <= 0.0608 */
if (w<=g2.d) {
/* First stage */
x2 = x*x;
t2 = x*x2*(d3.d+x2*(d5.d+x2*(d7.d+x2*(d9.d+x2*d11.d))));
if ((y=x+(t2-u1.d*t2)) == x+(t2+u1.d*t2)) return y;
/* Second stage */
c1 = x2*(a15.d+x2*(a17.d+x2*(a19.d+x2*(a21.d+x2*(a23.d+x2*(a25.d+
x2*a27.d))))));
EMULV(x,x,x2,xx2,t1,t2,t3,t4,t5)
ADD2(a13.d,aa13.d,c1,zero.d,c2,cc2,t1,t2)
MUL2(x2,xx2,c2,cc2,c1,cc1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(a11.d,aa11.d,c1,cc1,c2,cc2,t1,t2)
MUL2(x2,xx2,c2,cc2,c1,cc1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(a9.d ,aa9.d ,c1,cc1,c2,cc2,t1,t2)
MUL2(x2,xx2,c2,cc2,c1,cc1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(a7.d ,aa7.d ,c1,cc1,c2,cc2,t1,t2)
MUL2(x2,xx2,c2,cc2,c1,cc1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(a5.d ,aa5.d ,c1,cc1,c2,cc2,t1,t2)
MUL2(x2,xx2,c2,cc2,c1,cc1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(a3.d ,aa3.d ,c1,cc1,c2,cc2,t1,t2)
MUL2(x2,xx2,c2,cc2,c1,cc1,t1,t2,t3,t4,t5,t6,t7,t8)
MUL2(x ,zero.d,c1,cc1,c2,cc2,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(x ,zero.d,c2,cc2,c1,cc1,t1,t2)
if ((y=c1+(cc1-u2.d*c1)) == c1+(cc1+u2.d*c1)) return y;
return tanMp(x);
}
double
SECTION
__ieee754_atan2 (double y, double x)
{
int i, de, ux, dx, uy, dy;
static const int pr[MM] = { 6, 8, 10, 20, 32 };
double ax, ay, u, du, u9, ua, v, vv, dv, t1, t2, t3, t7, t8,
z, zz, cor, s1, ss1, s2, ss2;
#ifndef DLA_FMS
double t4, t5, t6;
#endif
number num;
static const int ep = 59768832, /* 57*16**5 */
em = -59768832; /* -57*16**5 */
/* x=NaN or y=NaN */
num.d = x;
ux = num.i[HIGH_HALF];
dx = num.i[LOW_HALF];
if ((ux & 0x7ff00000) == 0x7ff00000)
{
if (((ux & 0x000fffff) | dx) != 0x00000000)
return x + x;
}
num.d = y;
uy = num.i[HIGH_HALF];
dy = num.i[LOW_HALF];
if ((uy & 0x7ff00000) == 0x7ff00000)
{
if (((uy & 0x000fffff) | dy) != 0x00000000)
return y + y;
}
/* y=+-0 */
if (uy == 0x00000000)
{
if (dy == 0x00000000)
{
if ((ux & 0x80000000) == 0x00000000)
return 0;
else
return opi.d;
}
}
else if (uy == 0x80000000)
{
if (dy == 0x00000000)
{
if ((ux & 0x80000000) == 0x00000000)
return -0.0;
else
return mopi.d;
}
}
/* x=+-0 */
if (x == 0)
{
if ((uy & 0x80000000) == 0x00000000)
return hpi.d;
else
return mhpi.d;
}
/* x=+-INF */
if (ux == 0x7ff00000)
{
if (dx == 0x00000000)
{
if (uy == 0x7ff00000)
{
if (dy == 0x00000000)
return qpi.d;
}
else if (uy == 0xfff00000)
{
if (dy == 0x00000000)
return mqpi.d;
}
else
{
if ((uy & 0x80000000) == 0x00000000)
return 0;
else
return -0.0;
}
}
}
else if (ux == 0xfff00000)
{
if (dx == 0x00000000)
{
if (uy == 0x7ff00000)
{
if (dy == 0x00000000)
return tqpi.d;
}
else if (uy == 0xfff00000)
{
if (dy == 0x00000000)
return mtqpi.d;
}
else
{
if ((uy & 0x80000000) == 0x00000000)
return opi.d;
else
return mopi.d;
}
}
}
/* y=+-INF */
if (uy == 0x7ff00000)
{
if (dy == 0x00000000)
return hpi.d;
}
else if (uy == 0xfff00000)
{
if (dy == 0x00000000)
return mhpi.d;
}
/* either x/y or y/x is very close to zero */
ax = (x < 0) ? -x : x;
ay = (y < 0) ? -y : y;
de = (uy & 0x7ff00000) - (ux & 0x7ff00000);
if (de >= ep)
{
return ((y > 0) ? hpi.d : mhpi.d);
}
else if (de <= em)
{
if (x > 0)
{
if ((z = ay / ax) < TWOM1022)
return normalized (ax, ay, y, z);
else
return signArctan2 (y, z);
}
else
{
return ((y > 0) ? opi.d : mopi.d);
}
}
/* if either x or y is extremely close to zero, scale abs(x), abs(y). */
if (ax < twom500.d || ay < twom500.d)
{
ax *= two500.d;
ay *= two500.d;
}
/* Likewise for large x and y. */
if (ax > two500.d || ay > two500.d)
{
ax *= twom500.d;
ay *= twom500.d;
}
/* x,y which are neither special nor extreme */
if (ay < ax)
{
u = ay / ax;
EMULV (ax, u, v, vv, t1, t2, t3, t4, t5);
du = ((ay - v) - vv) / ax;
}
else
{
u = ax / ay;
EMULV (ay, u, v, vv, t1, t2, t3, t4, t5);
du = ((ax - v) - vv) / ay;
}
if (x > 0)
{
/* (i) x>0, abs(y)< abs(x): atan(ay/ax) */
if (ay < ax)
{
if (u < inv16.d)
{
v = u * u;
zz = du + u * v * (d3.d
+ v * (d5.d
+ v * (d7.d
+ v * (d9.d
+ v * (d11.d
+ v * d13.d)))));
if ((z = u + (zz - u1.d * u)) == u + (zz + u1.d * u))
return signArctan2 (y, z);
MUL2 (u, du, u, du, v, vv, t1, t2, t3, t4, t5, t6, t7, t8);
s1 = v * (f11.d + v * (f13.d
+ v * (f15.d + v * (f17.d + v * f19.d))));
ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
MUL2 (u, du, s1, ss1, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (u, du, s2, ss2, s1, ss1, t1, t2);
if ((z = s1 + (ss1 - u5.d * s1)) == s1 + (ss1 + u5.d * s1))
return signArctan2 (y, z);
return atan2Mp (x, y, pr);
}
i = (TWO52 + TWO8 * u) - TWO52;
i -= 16;
t3 = u - cij[i][0].d;
EADD (t3, du, v, dv);
t1 = cij[i][1].d;
t2 = cij[i][2].d;
zz = v * t2 + (dv * t2
+ v * v * (cij[i][3].d
+ v * (cij[i][4].d
+ v * (cij[i][5].d
+ v * cij[i][6].d))));
if (i < 112)
{
if (i < 48)
u9 = u91.d; /* u < 1/4 */
else
u9 = u92.d;
} /* 1/4 <= u < 1/2 */
else
{
if (i < 176)
u9 = u93.d; /* 1/2 <= u < 3/4 */
else
u9 = u94.d;
} /* 3/4 <= u <= 1 */
if ((z = t1 + (zz - u9 * t1)) == t1 + (zz + u9 * t1))
return signArctan2 (y, z);
t1 = u - hij[i][0].d;
EADD (t1, du, v, vv);
s1 = v * (hij[i][11].d
+ v * (hij[i][12].d
+ v * (hij[i][13].d
+ v * (hij[i][14].d
+ v * hij[i][15].d))));
ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
if ((z = s2 + (ss2 - ub.d * s2)) == s2 + (ss2 + ub.d * s2))
return signArctan2 (y, z);
return atan2Mp (x, y, pr);
}
/* (ii) x>0, abs(x)<=abs(y): pi/2-atan(ax/ay) */
if (u < inv16.d)
{
v = u * u;
zz = u * v * (d3.d
+ v * (d5.d
+ v * (d7.d
+ v * (d9.d
+ v * (d11.d
+ v * d13.d)))));
ESUB (hpi.d, u, t2, cor);
t3 = ((hpi1.d + cor) - du) - zz;
if ((z = t2 + (t3 - u2.d)) == t2 + (t3 + u2.d))
return signArctan2 (y, z);
MUL2 (u, du, u, du, v, vv, t1, t2, t3, t4, t5, t6, t7, t8);
s1 = v * (f11.d
+ v * (f13.d
+ v * (f15.d + v * (f17.d + v * f19.d))));
ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
MUL2 (u, du, s1, ss1, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (u, du, s2, ss2, s1, ss1, t1, t2);
SUB2 (hpi.d, hpi1.d, s1, ss1, s2, ss2, t1, t2);
if ((z = s2 + (ss2 - u6.d)) == s2 + (ss2 + u6.d))
return signArctan2 (y, z);
return atan2Mp (x, y, pr);
}
i = (TWO52 + TWO8 * u) - TWO52;
i -= 16;
v = (u - cij[i][0].d) + du;
zz = hpi1.d - v * (cij[i][2].d
+ v * (cij[i][3].d
+ v * (cij[i][4].d
+ v * (cij[i][5].d
+ v * cij[i][6].d))));
t1 = hpi.d - cij[i][1].d;
if (i < 112)
ua = ua1.d; /* w < 1/2 */
else
ua = ua2.d; /* w >= 1/2 */
if ((z = t1 + (zz - ua)) == t1 + (zz + ua))
return signArctan2 (y, z);
t1 = u - hij[i][0].d;
EADD (t1, du, v, vv);
s1 = v * (hij[i][11].d
+ v * (hij[i][12].d
+ v * (hij[i][13].d
+ v * (hij[i][14].d
+ v * hij[i][15].d))));
ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
SUB2 (hpi.d, hpi1.d, s2, ss2, s1, ss1, t1, t2);
if ((z = s1 + (ss1 - uc.d)) == s1 + (ss1 + uc.d))
return signArctan2 (y, z);
return atan2Mp (x, y, pr);
}
/* (iii) x<0, abs(x)< abs(y): pi/2+atan(ax/ay) */
if (ax < ay)
{
if (u < inv16.d)
{
v = u * u;
zz = u * v * (d3.d
+ v * (d5.d
+ v * (d7.d
+ v * (d9.d
+ v * (d11.d + v * d13.d)))));
EADD (hpi.d, u, t2, cor);
t3 = ((hpi1.d + cor) + du) + zz;
if ((z = t2 + (t3 - u3.d)) == t2 + (t3 + u3.d))
return signArctan2 (y, z);
MUL2 (u, du, u, du, v, vv, t1, t2, t3, t4, t5, t6, t7, t8);
s1 = v * (f11.d
+ v * (f13.d + v * (f15.d + v * (f17.d + v * f19.d))));
ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
MUL2 (u, du, s1, ss1, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (u, du, s2, ss2, s1, ss1, t1, t2);
ADD2 (hpi.d, hpi1.d, s1, ss1, s2, ss2, t1, t2);
if ((z = s2 + (ss2 - u7.d)) == s2 + (ss2 + u7.d))
return signArctan2 (y, z);
return atan2Mp (x, y, pr);
}
i = (TWO52 + TWO8 * u) - TWO52;
i -= 16;
v = (u - cij[i][0].d) + du;
zz = hpi1.d + v * (cij[i][2].d
+ v * (cij[i][3].d
+ v * (cij[i][4].d
+ v * (cij[i][5].d
+ v * cij[i][6].d))));
t1 = hpi.d + cij[i][1].d;
if (i < 112)
ua = ua1.d; /* w < 1/2 */
else
ua = ua2.d; /* w >= 1/2 */
if ((z = t1 + (zz - ua)) == t1 + (zz + ua))
return signArctan2 (y, z);
t1 = u - hij[i][0].d;
EADD (t1, du, v, vv);
s1 = v * (hij[i][11].d
+ v * (hij[i][12].d
+ v * (hij[i][13].d
+ v * (hij[i][14].d
+ v * hij[i][15].d))));
ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
ADD2 (hpi.d, hpi1.d, s2, ss2, s1, ss1, t1, t2);
if ((z = s1 + (ss1 - uc.d)) == s1 + (ss1 + uc.d))
return signArctan2 (y, z);
return atan2Mp (x, y, pr);
}
/* (iv) x<0, abs(y)<=abs(x): pi-atan(ax/ay) */
if (u < inv16.d)
{
v = u * u;
zz = u * v * (d3.d
+ v * (d5.d
+ v * (d7.d
+ v * (d9.d + v * (d11.d + v * d13.d)))));
ESUB (opi.d, u, t2, cor);
t3 = ((opi1.d + cor) - du) - zz;
if ((z = t2 + (t3 - u4.d)) == t2 + (t3 + u4.d))
return signArctan2 (y, z);
MUL2 (u, du, u, du, v, vv, t1, t2, t3, t4, t5, t6, t7, t8);
s1 = v * (f11.d + v * (f13.d + v * (f15.d + v * (f17.d + v * f19.d))));
ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
MUL2 (u, du, s1, ss1, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (u, du, s2, ss2, s1, ss1, t1, t2);
SUB2 (opi.d, opi1.d, s1, ss1, s2, ss2, t1, t2);
if ((z = s2 + (ss2 - u8.d)) == s2 + (ss2 + u8.d))
return signArctan2 (y, z);
return atan2Mp (x, y, pr);
}
i = (TWO52 + TWO8 * u) - TWO52;
i -= 16;
v = (u - cij[i][0].d) + du;
zz = opi1.d - v * (cij[i][2].d
+ v * (cij[i][3].d
+ v * (cij[i][4].d
+ v * (cij[i][5].d + v * cij[i][6].d))));
t1 = opi.d - cij[i][1].d;
if (i < 112)
ua = ua1.d; /* w < 1/2 */
else
ua = ua2.d; /* w >= 1/2 */
if ((z = t1 + (zz - ua)) == t1 + (zz + ua))
return signArctan2 (y, z);
t1 = u - hij[i][0].d;
EADD (t1, du, v, vv);
s1 = v * (hij[i][11].d
+ v * (hij[i][12].d
+ v * (hij[i][13].d
+ v * (hij[i][14].d + v * hij[i][15].d))));
ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
SUB2 (opi.d, opi1.d, s2, ss2, s1, ss1, t1, t2);
if ((z = s1 + (ss1 - uc.d)) == s1 + (ss1 + uc.d))
return signArctan2 (y, z);
return atan2Mp (x, y, pr);
}
/* routine computes the correctly rounded (to nearest) value of atan(x). */
double atan(double x) {
double cor,s1,ss1,s2,ss2,t1,t2,t3,t4,t5,t6,t7,t8,t9,t10,u,u2,u3,
v,vv,w,ww,y,yy,z,zz;
#if 0
double y1,y2;
#endif
int i,ux,dx;
#if 0
int p;
#endif
static const int pr[M]={6,8,10,32};
number num;
#if 0
mp_no mpt1,mpx,mpy,mpy1,mpy2,mperr;
#endif
num.d = x; ux = num.i[HIGH_HALF]; dx = num.i[LOW_HALF];
/* x=NaN */
if (((ux&0x7ff00000)==0x7ff00000) && (((ux&0x000fffff)|dx)!=0x00000000))
return x+x;
/* Regular values of x, including denormals +-0 and +-INF */
u = (x<ZERO) ? -x : x;
if (u<C) {
if (u<B) {
if (u<A) { /* u < A */
return x; }
else { /* A <= u < B */
v=x*x; yy=x*v*(d3.d+v*(d5.d+v*(d7.d+v*(d9.d+v*(d11.d+v*d13.d)))));
if ((y=x+(yy-U1*x)) == x+(yy+U1*x)) return y;
EMULV(x,x,v,vv,t1,t2,t3,t4,t5) /* v+vv=x^2 */
s1=v*(f11.d+v*(f13.d+v*(f15.d+v*(f17.d+v*f19.d))));
ADD2(f9.d,ff9.d,s1,ZERO,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(f7.d,ff7.d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(f5.d,ff5.d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(f3.d,ff3.d,s1,ss1,s2,ss2,t1,t2)
MUL2(v,vv,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
MUL2(x,ZERO,s1,ss1,s2,ss2,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(x,ZERO,s2,ss2,s1,ss1,t1,t2)
if ((y=s1+(ss1-U5*s1)) == s1+(ss1+U5*s1)) return y;
return atanMp(x,pr);
} }
else { /* B <= u < C */
i=(TWO52+TWO8*u)-TWO52; i-=16;
z=u-cij[i][0].d;
yy=z*(cij[i][2].d+z*(cij[i][3].d+z*(cij[i][4].d+
z*(cij[i][5].d+z* cij[i][6].d))));
t1=cij[i][1].d;
if (i<112) {
if (i<48) u2=U21; /* u < 1/4 */
else u2=U22; } /* 1/4 <= u < 1/2 */
else {
if (i<176) u2=U23; /* 1/2 <= u < 3/4 */
else u2=U24; } /* 3/4 <= u <= 1 */
if ((y=t1+(yy-u2*t1)) == t1+(yy+u2*t1)) return __signArctan(x,y);
z=u-hij[i][0].d;
s1=z*(hij[i][11].d+z*(hij[i][12].d+z*(hij[i][13].d+
z*(hij[i][14].d+z* hij[i][15].d))));
ADD2(hij[i][9].d,hij[i][10].d,s1,ZERO,s2,ss2,t1,t2)
MUL2(z,ZERO,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(hij[i][7].d,hij[i][8].d,s1,ss1,s2,ss2,t1,t2)
MUL2(z,ZERO,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(hij[i][5].d,hij[i][6].d,s1,ss1,s2,ss2,t1,t2)
MUL2(z,ZERO,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(hij[i][3].d,hij[i][4].d,s1,ss1,s2,ss2,t1,t2)
MUL2(z,ZERO,s2,ss2,s1,ss1,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(hij[i][1].d,hij[i][2].d,s1,ss1,s2,ss2,t1,t2)
if ((y=s2+(ss2-U6*s2)) == s2+(ss2+U6*s2)) return __signArctan(x,y);
return atanMp(x,pr);
}
}
double
SECTION
__ieee754_log(double x) {
#define M 4
static const int pr[M]={8,10,18,32};
int i,j,n,ux,dx,p;
#if 0
int k;
#endif
double dbl_n,u,p0,q,r0,w,nln2a,luai,lubi,lvaj,lvbj,
sij,ssij,ttij,A,B,B0,y,y1,y2,polI,polII,sa,sb,
t1,t2,t7,t8,t,ra,rb,ww,
a0,aa0,s1,s2,ss2,s3,ss3,a1,aa1,a,aa,b,bb,c;
#ifndef DLA_FMS
double t3,t4,t5,t6;
#endif
number num;
mp_no mpx,mpy,mpy1,mpy2,mperr;
#include "ulog.tbl"
#include "ulog.h"
/* Treating special values of x ( x<=0, x=INF, x=NaN etc.). */
num.d = x; ux = num.i[HIGH_HALF]; dx = num.i[LOW_HALF];
n=0;
if (__builtin_expect(ux < 0x00100000, 0)) {
if (__builtin_expect(((ux & 0x7fffffff) | dx) == 0, 0))
return MHALF/ZERO; /* return -INF */
if (__builtin_expect(ux < 0, 0))
return (x-x)/ZERO; /* return NaN */
n -= 54; x *= two54.d; /* scale x */
num.d = x;
}
if (__builtin_expect(ux >= 0x7ff00000, 0))
return x+x; /* INF or NaN */
/* Regular values of x */
w = x-ONE;
if (__builtin_expect(ABS(w) > U03, 1)) { goto case_03; }
/*--- Stage I, the case abs(x-1) < 0.03 */
t8 = MHALF*w;
EMULV(t8,w,a,aa,t1,t2,t3,t4,t5)
EADD(w,a,b,bb)
/* Evaluate polynomial II */
polII = (b0.d+w*(b1.d+w*(b2.d+w*(b3.d+w*(b4.d+
w*(b5.d+w*(b6.d+w*(b7.d+w*b8.d))))))))*w*w*w;
c = (aa+bb)+polII;
/* End stage I, case abs(x-1) < 0.03 */
if ((y=b+(c+b*E2)) == b+(c-b*E2)) return y;
/*--- Stage II, the case abs(x-1) < 0.03 */
a = d11.d+w*(d12.d+w*(d13.d+w*(d14.d+w*(d15.d+w*(d16.d+
w*(d17.d+w*(d18.d+w*(d19.d+w*d20.d))))))));
EMULV(w,a,s2,ss2,t1,t2,t3,t4,t5)
ADD2(d10.d,dd10.d,s2,ss2,s3,ss3,t1,t2)
MUL2(w,ZERO,s3,ss3,s2,ss2,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(d9.d,dd9.d,s2,ss2,s3,ss3,t1,t2)
MUL2(w,ZERO,s3,ss3,s2,ss2,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(d8.d,dd8.d,s2,ss2,s3,ss3,t1,t2)
MUL2(w,ZERO,s3,ss3,s2,ss2,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(d7.d,dd7.d,s2,ss2,s3,ss3,t1,t2)
MUL2(w,ZERO,s3,ss3,s2,ss2,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(d6.d,dd6.d,s2,ss2,s3,ss3,t1,t2)
MUL2(w,ZERO,s3,ss3,s2,ss2,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(d5.d,dd5.d,s2,ss2,s3,ss3,t1,t2)
MUL2(w,ZERO,s3,ss3,s2,ss2,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(d4.d,dd4.d,s2,ss2,s3,ss3,t1,t2)
MUL2(w,ZERO,s3,ss3,s2,ss2,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(d3.d,dd3.d,s2,ss2,s3,ss3,t1,t2)
MUL2(w,ZERO,s3,ss3,s2,ss2,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(d2.d,dd2.d,s2,ss2,s3,ss3,t1,t2)
MUL2(w,ZERO,s3,ss3,s2,ss2,t1,t2,t3,t4,t5,t6,t7,t8)
MUL2(w,ZERO,s2,ss2,s3,ss3,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(w,ZERO, s3,ss3, b, bb,t1,t2)
/* End stage II, case abs(x-1) < 0.03 */
if ((y=b+(bb+b*E4)) == b+(bb-b*E4)) return y;
goto stage_n;
/*--- Stage I, the case abs(x-1) > 0.03 */
case_03:
/* Find n,u such that x = u*2**n, 1/sqrt(2) < u < sqrt(2) */
n += (num.i[HIGH_HALF] >> 20) - 1023;
num.i[HIGH_HALF] = (num.i[HIGH_HALF] & 0x000fffff) | 0x3ff00000;
if (num.d > SQRT_2) { num.d *= HALF; n++; }
u = num.d; dbl_n = (double) n;
/* Find i such that ui=1+(i-75)/2**8 is closest to u (i= 0,1,2,...,181) */
num.d += h1.d;
i = (num.i[HIGH_HALF] & 0x000fffff) >> 12;
/* Find j such that vj=1+(j-180)/2**16 is closest to v=u/ui (j= 0,...,361) */
num.d = u*Iu[i].d + h2.d;
j = (num.i[HIGH_HALF] & 0x000fffff) >> 4;
/* Compute w=(u-ui*vj)/(ui*vj) */
p0=(ONE+(i-75)*DEL_U)*(ONE+(j-180)*DEL_V);
q=u-p0; r0=Iu[i].d*Iv[j].d; w=q*r0;
/* Evaluate polynomial I */
polI = w+(a2.d+a3.d*w)*w*w;
/* Add up everything */
nln2a = dbl_n*LN2A;
luai = Lu[i][0].d; lubi = Lu[i][1].d;
lvaj = Lv[j][0].d; lvbj = Lv[j][1].d;
EADD(luai,lvaj,sij,ssij)
EADD(nln2a,sij,A ,ttij)
B0 = (((lubi+lvbj)+ssij)+ttij)+dbl_n*LN2B;
B = polI+B0;
/* End stage I, case abs(x-1) >= 0.03 */
if ((y=A+(B+E1)) == A+(B-E1)) return y;
/*--- Stage II, the case abs(x-1) > 0.03 */
/* Improve the accuracy of r0 */
EMULV(p0,r0,sa,sb,t1,t2,t3,t4,t5)
t=r0*((ONE-sa)-sb);
EADD(r0,t,ra,rb)
/* Compute w */
MUL2(q,ZERO,ra,rb,w,ww,t1,t2,t3,t4,t5,t6,t7,t8)
EADD(A,B0,a0,aa0)
/* Evaluate polynomial III */
s1 = (c3.d+(c4.d+c5.d*w)*w)*w;
EADD(c2.d,s1,s2,ss2)
MUL2(s2,ss2,w,ww,s3,ss3,t1,t2,t3,t4,t5,t6,t7,t8)
MUL2(s3,ss3,w,ww,s2,ss2,t1,t2,t3,t4,t5,t6,t7,t8)
ADD2(s2,ss2,w,ww,s3,ss3,t1,t2)
ADD2(s3,ss3,a0,aa0,a1,aa1,t1,t2)
/* End stage II, case abs(x-1) >= 0.03 */
if ((y=a1+(aa1+E3)) == a1+(aa1-E3)) return y;
/* Final stages. Use multi-precision arithmetic. */
stage_n:
for (i=0; i<M; i++) {
p = pr[i];
__dbl_mp(x,&mpx,p); __dbl_mp(y,&mpy,p);
__mplog(&mpx,&mpy,p);
__dbl_mp(e[i].d,&mperr,p);
__add(&mpy,&mperr,&mpy1,p); __sub(&mpy,&mperr,&mpy2,p);
__mp_dbl(&mpy1,&y1,p); __mp_dbl(&mpy2,&y2,p);
if (y1==y2) return y1;
}
return y1;
}
double
__ieee754_sqrt (double x)
{
#include "uroot.h"
static const double
rt0 = 9.99999999859990725855365213134618E-01,
rt1 = 4.99999999495955425917856814202739E-01,
rt2 = 3.75017500867345182581453026130850E-01,
rt3 = 3.12523626554518656309172508769531E-01;
static const double big = 134217728.0;
double y, t, del, res, res1, hy, z, zz, p, hx, tx, ty, s;
mynumber a, c = { { 0, 0 } };
int4 k;
a.x = x;
k = a.i[HIGH_HALF];
a.i[HIGH_HALF] = (k & 0x001fffff) | 0x3fe00000;
t = inroot[(k & 0x001fffff) >> 14];
s = a.x;
/*----------------- 2^-1022 <= | x |< 2^1024 -----------------*/
if (k > 0x000fffff && k < 0x7ff00000)
{
int rm = __fegetround ();
fenv_t env;
libc_feholdexcept_setround (&env, FE_TONEAREST);
double ret;
y = 1.0 - t * (t * s);
t = t * (rt0 + y * (rt1 + y * (rt2 + y * rt3)));
c.i[HIGH_HALF] = 0x20000000 + ((k & 0x7fe00000) >> 1);
y = t * s;
hy = (y + big) - big;
del = 0.5 * t * ((s - hy * hy) - (y - hy) * (y + hy));
res = y + del;
if (res == (res + 1.002 * ((y - res) + del)))
ret = res * c.x;
else
{
res1 = res + 1.5 * ((y - res) + del);
EMULV (res, res1, z, zz, p, hx, tx, hy, ty); /* (z+zz)=res*res1 */
res = ((((z - s) + zz) < 0) ? max (res, res1) :
min (res, res1));
ret = res * c.x;
}
math_force_eval (ret);
libc_fesetenv (&env);
double dret = x / ret;
if (dret != ret)
{
double force_inexact = 1.0 / 3.0;
math_force_eval (force_inexact);
/* The square root is inexact, ret is the round-to-nearest
value which may need adjusting for other rounding
modes. */
switch (rm)
{
#ifdef FE_UPWARD
case FE_UPWARD:
if (dret > ret)
ret = (res + 0x1p-1022) * c.x;
break;
#endif
#ifdef FE_DOWNWARD
case FE_DOWNWARD:
#endif
#ifdef FE_TOWARDZERO
case FE_TOWARDZERO:
#endif
#if defined FE_DOWNWARD || defined FE_TOWARDZERO
if (dret < ret)
ret = (res - 0x1p-1022) * c.x;
break;
#endif
default:
break;
}
}
/* Otherwise (x / ret == ret), either the square root was exact or
the division was inexact. */
return ret;
}
/* routine computes the correctly rounded (to nearest) value of atan(x). */
double
atan (double x)
{
double cor, s1, ss1, s2, ss2, t1, t2, t3, t7, t8, t9, t10, u, u2, u3,
v, vv, w, ww, y, yy, z, zz;
#ifndef DLA_FMS
double t4, t5, t6;
#endif
int i, ux, dx;
static const int pr[M] = { 6, 8, 10, 32 };
number num;
num.d = x;
ux = num.i[HIGH_HALF];
dx = num.i[LOW_HALF];
/* x=NaN */
if (((ux & 0x7ff00000) == 0x7ff00000)
&& (((ux & 0x000fffff) | dx) != 0x00000000))
return x + x;
/* Regular values of x, including denormals +-0 and +-INF */
SET_RESTORE_ROUND (FE_TONEAREST);
u = (x < 0) ? -x : x;
if (u < C)
{
if (u < B)
{
if (u < A)
{
math_check_force_underflow_nonneg (u);
return x;
}
else
{ /* A <= u < B */
v = x * x;
yy = d11.d + v * d13.d;
yy = d9.d + v * yy;
yy = d7.d + v * yy;
yy = d5.d + v * yy;
yy = d3.d + v * yy;
yy *= x * v;
if ((y = x + (yy - U1 * x)) == x + (yy + U1 * x))
return y;
EMULV (x, x, v, vv, t1, t2, t3, t4, t5); /* v+vv=x^2 */
s1 = f17.d + v * f19.d;
s1 = f15.d + v * s1;
s1 = f13.d + v * s1;
s1 = f11.d + v * s1;
s1 *= v;
ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
MUL2 (x, 0, s1, ss1, s2, ss2, t1, t2, t3, t4, t5, t6, t7,
t8);
ADD2 (x, 0, s2, ss2, s1, ss1, t1, t2);
if ((y = s1 + (ss1 - U5 * s1)) == s1 + (ss1 + U5 * s1))
return y;
return atanMp (x, pr);
}
}
else
{ /* B <= u < C */
i = (TWO52 + TWO8 * u) - TWO52;
i -= 16;
z = u - cij[i][0].d;
yy = cij[i][5].d + z * cij[i][6].d;
yy = cij[i][4].d + z * yy;
yy = cij[i][3].d + z * yy;
yy = cij[i][2].d + z * yy;
yy *= z;
t1 = cij[i][1].d;
if (i < 112)
{
if (i < 48)
u2 = U21; /* u < 1/4 */
else
u2 = U22;
} /* 1/4 <= u < 1/2 */
else
{
if (i < 176)
u2 = U23; /* 1/2 <= u < 3/4 */
else
u2 = U24;
} /* 3/4 <= u <= 1 */
if ((y = t1 + (yy - u2 * t1)) == t1 + (yy + u2 * t1))
return __signArctan (x, y);
z = u - hij[i][0].d;
s1 = hij[i][14].d + z * hij[i][15].d;
s1 = hij[i][13].d + z * s1;
s1 = hij[i][12].d + z * s1;
s1 = hij[i][11].d + z * s1;
s1 *= z;
ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (z, 0, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
if ((y = s2 + (ss2 - U6 * s2)) == s2 + (ss2 + U6 * s2))
return __signArctan (x, y);
return atanMp (x, pr);
}
}
else
{
if (u < D)
{ /* C <= u < D */
w = 1 / u;
EMULV (w, u, t1, t2, t3, t4, t5, t6, t7);
ww = w * ((1 - t1) - t2);
i = (TWO52 + TWO8 * w) - TWO52;
i -= 16;
z = (w - cij[i][0].d) + ww;
yy = cij[i][5].d + z * cij[i][6].d;
yy = cij[i][4].d + z * yy;
yy = cij[i][3].d + z * yy;
yy = cij[i][2].d + z * yy;
yy = HPI1 - z * yy;
t1 = HPI - cij[i][1].d;
if (i < 112)
u3 = U31; /* w < 1/2 */
else
u3 = U32; /* w >= 1/2 */
if ((y = t1 + (yy - u3)) == t1 + (yy + u3))
return __signArctan (x, y);
DIV2 (1, 0, u, 0, w, ww, t1, t2, t3, t4, t5, t6, t7, t8, t9,
t10);
t1 = w - hij[i][0].d;
EADD (t1, ww, z, zz);
s1 = hij[i][14].d + z * hij[i][15].d;
s1 = hij[i][13].d + z * s1;
s1 = hij[i][12].d + z * s1;
s1 = hij[i][11].d + z * s1;
s1 *= z;
ADD2 (hij[i][9].d, hij[i][10].d, s1, 0, s2, ss2, t1, t2);
MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][7].d, hij[i][8].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][5].d, hij[i][6].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][3].d, hij[i][4].d, s1, ss1, s2, ss2, t1, t2);
MUL2 (z, zz, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (hij[i][1].d, hij[i][2].d, s1, ss1, s2, ss2, t1, t2);
SUB2 (HPI, HPI1, s2, ss2, s1, ss1, t1, t2);
if ((y = s1 + (ss1 - U7)) == s1 + (ss1 + U7))
return __signArctan (x, y);
return atanMp (x, pr);
}
else
{
if (u < E)
{ /* D <= u < E */
w = 1 / u;
v = w * w;
EMULV (w, u, t1, t2, t3, t4, t5, t6, t7);
yy = d11.d + v * d13.d;
yy = d9.d + v * yy;
yy = d7.d + v * yy;
yy = d5.d + v * yy;
yy = d3.d + v * yy;
yy *= w * v;
ww = w * ((1 - t1) - t2);
ESUB (HPI, w, t3, cor);
yy = ((HPI1 + cor) - ww) - yy;
if ((y = t3 + (yy - U4)) == t3 + (yy + U4))
return __signArctan (x, y);
DIV2 (1, 0, u, 0, w, ww, t1, t2, t3, t4, t5, t6, t7, t8,
t9, t10);
MUL2 (w, ww, w, ww, v, vv, t1, t2, t3, t4, t5, t6, t7, t8);
s1 = f17.d + v * f19.d;
s1 = f15.d + v * s1;
s1 = f13.d + v * s1;
s1 = f11.d + v * s1;
s1 *= v;
ADD2 (f9.d, ff9.d, s1, 0, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f7.d, ff7.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f5.d, ff5.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (f3.d, ff3.d, s1, ss1, s2, ss2, t1, t2);
MUL2 (v, vv, s2, ss2, s1, ss1, t1, t2, t3, t4, t5, t6, t7, t8);
MUL2 (w, ww, s1, ss1, s2, ss2, t1, t2, t3, t4, t5, t6, t7, t8);
ADD2 (w, ww, s2, ss2, s1, ss1, t1, t2);
SUB2 (HPI, HPI1, s1, ss1, s2, ss2, t1, t2);
if ((y = s2 + (ss2 - U8)) == s2 + (ss2 + U8))
return __signArctan (x, y);
return atanMp (x, pr);
}
else
{
/* u >= E */
if (x > 0)
return HPI;
else
return MHPI;
}
}
}
}
