/* origin: FreeBSD /usr/src/lib/msun/ld80/e_rem_pio2.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
* Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*
* Optimized by Bruce D. Evans.
*/
#include "libm.h"
#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
*/
#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 TOINT 0x1.8p63
#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_1t: pi/2 - pio2_1
* pio2_2: second 39 bits of pi/2
* pio2_2t: pi/2 - (pio2_1+pio2_2)
* pio2_3: third 39 bits of pi/2
* pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
*/
static const double
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
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 TOINT 0x1.8p112
#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
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 ldshape u,uz;
long double z,w,t,r,fn;
double tx[NX],ty[NY];
int ex,ey,n,i;
u.f = x;
ex = u.i.se & 0x7fff;
if (SMALL(u)) {
/* rint(x/(pi/2)), Assume round-to-nearest. */
fn = x*invpio2 + TOINT - TOINT;
n = QUOBITS(fn);
r = x-fn*pio2_1;
w = fn*pio2_1t; /* 1st round good to 102/180 bits (ld80/ld128) */
y[0] = r-w;
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;
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] = r-w;
}
}
y[1] = (r - y[0]) - w;
return n;
}
/*
* all other (large) arguments
*/
if (ex == 0x7fff) { /* x is inf or NaN */
y[0] = y[1] = x - x;
return 0;
}
/* 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])*0x1p24;
}
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;
}
y[0] = r;
y[1] = w;
return n;
}
#endif
