* Optimized by Bruce D. Evans.
*/
#include "libm.h"
-#if LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
-/* ld80 version of __rem_pio2(x,y)
+#if (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
+/* ld80 and ld128 version of __rem_pio2(x,y)
*
* return the remainder of x rem pi/2 in y[0]+y[1]
* use __rem_pio2_large() for large x
*/
-#define BIAS (LDBL_MAX_EXP - 1)
+static const long double toint = 1.5/LDBL_EPSILON;
+#if LDBL_MANT_DIG == 64
+/* u ~< 0x1p25*pi/2 */
+#define SMALL(u) (((u.i.se & 0x7fffU)<<16 | u.i.m>>48) < ((0x3fff + 25)<<16 | 0x921f>>1 | 0x8000))
+#define QUOBITS(x) ((uint32_t)(int32_t)x & 0x7fffffff)
+#define ROUND1 22
+#define ROUND2 61
+#define NX 3
+#define NY 2
/*
* invpio2: 64 bits of 2/pi
* pio2_1: first 39 bits of pi/2
* pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
*/
static const double
-zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
-two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
pio2_1 = 1.57079632679597125389e+00, /* 0x3FF921FB, 0x54444000 */
pio2_2 = -1.07463465549783099519e-12, /* -0x12e7b967674000.0p-92 */
pio2_3 = 6.36831716351370313614e-25; /* 0x18a2e037074000.0p-133 */
-
static const long double
+pio4 = 0x1.921fb54442d1846ap-1L,
invpio2 = 6.36619772367581343076e-01L, /* 0xa2f9836e4e44152a.0p-64 */
pio2_1t = -1.07463465549719416346e-12L, /* -0x973dcb3b399d747f.0p-103 */
pio2_2t = 6.36831716351095013979e-25L, /* 0xc51701b839a25205.0p-144 */
pio2_3t = -2.75299651904407171810e-37L; /* -0xbb5bf6c7ddd660ce.0p-185 */
+#elif LDBL_MANT_DIG == 113
+/* u ~< 0x1p45*pi/2 */
+#define SMALL(u) (((u.i.se & 0x7fffU)<<16 | u.i.top) < ((0x3fff + 45)<<16 | 0x921f))
+#define QUOBITS(x) ((uint32_t)(int64_t)x & 0x7fffffff)
+#define ROUND1 51
+#define ROUND2 119
+#define NX 5
+#define NY 3
+static const long double
+pio4 = 0x1.921fb54442d18469898cc51701b8p-1L,
+invpio2 = 6.3661977236758134307553505349005747e-01L, /* 0x145f306dc9c882a53f84eafa3ea6a.0p-113 */
+pio2_1 = 1.5707963267948966192292994253909555e+00L, /* 0x1921fb54442d18469800000000000.0p-112 */
+pio2_1t = 2.0222662487959507323996846200947577e-21L, /* 0x13198a2e03707344a4093822299f3.0p-181 */
+pio2_2 = 2.0222662487959507323994779168837751e-21L, /* 0x13198a2e03707344a400000000000.0p-181 */
+pio2_2t = 2.0670321098263988236496903051604844e-43L, /* 0x127044533e63a0105df531d89cd91.0p-254 */
+pio2_3 = 2.0670321098263988236499468110329591e-43L, /* 0x127044533e63a0105e00000000000.0p-254 */
+pio2_3t = -2.5650587247459238361625433492959285e-65L; /* -0x159c4ec64ddaeb5f78671cbfb2210.0p-327 */
+#endif
int __rem_pio2l(long double x, long double *y)
{
- union IEEEl2bits u,u1;
+ union ldshape u,uz;
long double z,w,t,r,fn;
- double tx[3],ty[2];
- int e0,ex,i,j,nx,n;
- int16_t expsign;
+ double tx[NX],ty[NY];
+ int ex,ey,n,i;
- u.e = x;
- expsign = u.xbits.expsign;
- ex = expsign & 0x7fff;
- if (ex < BIAS + 25 || (ex == BIAS + 25 && u.bits.manh < 0xc90fdaa2)) {
- union IEEEl2bits u2;
- int ex1;
-
- /* |x| ~< 2^25*(pi/2), medium size */
- /* Use a specialized rint() to get fn. Assume round-to-nearest. */
- fn = x*invpio2 + 0x1.8p63;
- fn = fn - 0x1.8p63;
-// FIXME
-//#ifdef HAVE_EFFICIENT_IRINT
-// n = irint(fn);
-//#else
- n = fn;
-//#endif
+ u.f = x;
+ ex = u.i.se & 0x7fff;
+ if (SMALL(u)) {
+ /* rint(x/(pi/2)) */
+ fn = x*invpio2 + toint - toint;
+ n = QUOBITS(fn);
r = x-fn*pio2_1;
- w = fn*pio2_1t; /* 1st round good to 102 bit */
- j = ex;
+ w = fn*pio2_1t; /* 1st round good to 102/180 bits (ld80/ld128) */
+ /* Matters with directed rounding. */
+ if (predict_false(r - w < -pio4)) {
+ n--;
+ fn--;
+ r = x - fn*pio2_1;
+ w = fn*pio2_1t;
+ } else if (predict_false(r - w > pio4)) {
+ n++;
+ fn++;
+ r = x - fn*pio2_1;
+ w = fn*pio2_1t;
+ }
y[0] = r-w;
- u2.e = y[0];
- ex1 = u2.xbits.expsign & 0x7fff;
- i = j-ex1;
- if (i > 22) { /* 2nd iteration needed, good to 141 */
+ u.f = y[0];
+ ey = u.i.se & 0x7fff;
+ if (ex - ey > ROUND1) { /* 2nd iteration needed, good to 141/248 (ld80/ld128) */
t = r;
w = fn*pio2_2;
r = t-w;
w = fn*pio2_2t-((t-r)-w);
y[0] = r-w;
- u2.e = y[0];
- ex1 = u2.xbits.expsign & 0x7fff;
- i = j-ex1;
- if (i > 61) { /* 3rd iteration need, 180 bits acc */
- t = r; /* will cover all possible cases */
+ u.f = y[0];
+ ey = u.i.se & 0x7fff;
+ if (ex - ey > ROUND2) { /* 3rd iteration, good to 180/316 bits */
+ t = r; /* will cover all possible cases (not verified for ld128) */
w = fn*pio2_3;
r = t-w;
w = fn*pio2_3t-((t-r)-w);
y[0] = y[1] = x - x;
return 0;
}
- /* set z = scalbn(|x|,ilogb(x)-23) */
- u1.e = x;
- e0 = ex - BIAS - 23; /* e0 = ilogb(|x|)-23; */
- u1.xbits.expsign = ex - e0;
- z = u1.e;
- for (i=0; i<2; i++) {
+ /* set z = scalbn(|x|,-ilogb(x)+23) */
+ uz.f = x;
+ uz.i.se = 0x3fff + 23;
+ z = uz.f;
+ for (i=0; i < NX - 1; i++) {
tx[i] = (double)(int32_t)z;
- z = (z-tx[i])*two24;
+ z = (z-tx[i])*0x1p24;
}
- tx[2] = z;
- nx = 3;
- while (tx[nx-1] == zero)
- nx--; /* skip zero term */
- n = __rem_pio2_large(tx,ty,e0,nx,2);
- r = (long double)ty[0] + ty[1];
- w = ty[1] - (r - ty[0]);
- if (expsign < 0) {
+ tx[i] = z;
+ while (tx[i] == 0)
+ i--;
+ n = __rem_pio2_large(tx, ty, ex-0x3fff-23, i+1, NY);
+ w = ty[1];
+ if (NY == 3)
+ w += ty[2];
+ r = ty[0] + w;
+ /* TODO: for ld128 this does not follow the recommendation of the
+ comments of __rem_pio2_large which seem wrong if |ty[0]| > |ty[1]+ty[2]| */
+ w -= r - ty[0];
+ if (u.i.se >> 15) {
y[0] = -r;
y[1] = -w;
return -n;
projects / oweals / musl.git / blobdiff