Index: head/lib/msun/ld128/e_powl.c =================================================================== --- head/lib/msun/ld128/e_powl.c (revision 342650) +++ head/lib/msun/ld128/e_powl.c (revision 342651) @@ -1,443 +1,443 @@ /*- * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunPro, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== */ /* * Copyright (c) 2008 Stephen L. Moshier * * Permission to use, copy, modify, and distribute this software for any * purpose with or without fee is hereby granted, provided that the above * copyright notice and this permission notice appear in all copies. * * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */ /* powl(x,y) return x**y * * n * Method: Let x = 2 * (1+f) * 1. Compute and return log2(x) in two pieces: * log2(x) = w1 + w2, * where w1 has 113-53 = 60 bit trailing zeros. - * 2. Perform y*log2(x) = n+y' by simulating muti-precision + * 2. Perform y*log2(x) = n+y' by simulating multi-precision * arithmetic, where |y'|<=0.5. * 3. Return x**y = 2**n*exp(y'*log2) * * Special cases: * 1. (anything) ** 0 is 1 * 2. (anything) ** 1 is itself * 3. (anything) ** NAN is NAN * 4. NAN ** (anything except 0) is NAN * 5. +-(|x| > 1) ** +INF is +INF * 6. +-(|x| > 1) ** -INF is +0 * 7. +-(|x| < 1) ** +INF is +0 * 8. +-(|x| < 1) ** -INF is +INF * 9. +-1 ** +-INF is NAN * 10. +0 ** (+anything except 0, NAN) is +0 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 * 12. +0 ** (-anything except 0, NAN) is +INF * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) * 15. +INF ** (+anything except 0,NAN) is +INF * 16. +INF ** (-anything except 0,NAN) is +0 * 17. -INF ** (anything) = -0 ** (-anything) * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) * 19. (-anything except 0 and inf) ** (non-integer) is NAN * */ #include __FBSDID("$FreeBSD$"); #include #include #include "math_private.h" static const long double bp[] = { 1.0L, 1.5L, }; /* log_2(1.5) */ static const long double dp_h[] = { 0.0, 5.8496250072115607565592654282227158546448E-1L }; /* Low part of log_2(1.5) */ static const long double dp_l[] = { 0.0, 1.0579781240112554492329533686862998106046E-16L }; static const long double zero = 0.0L, one = 1.0L, two = 2.0L, two113 = 1.0384593717069655257060992658440192E34L, huge = 1.0e3000L, tiny = 1.0e-3000L; /* 3/2 log x = 3 z + z^3 + z^3 (z^2 R(z^2)) z = (x-1)/(x+1) 1 <= x <= 1.25 Peak relative error 2.3e-37 */ static const long double LN[] = { -3.0779177200290054398792536829702930623200E1L, 6.5135778082209159921251824580292116201640E1L, -4.6312921812152436921591152809994014413540E1L, 1.2510208195629420304615674658258363295208E1L, -9.9266909031921425609179910128531667336670E-1L }; static const long double LD[] = { -5.129862866715009066465422805058933131960E1L, 1.452015077564081884387441590064272782044E2L, -1.524043275549860505277434040464085593165E2L, 7.236063513651544224319663428634139768808E1L, -1.494198912340228235853027849917095580053E1L /* 1.0E0 */ }; /* exp(x) = 1 + x - x / (1 - 2 / (x - x^2 R(x^2))) 0 <= x <= 0.5 Peak relative error 5.7e-38 */ static const long double PN[] = { 5.081801691915377692446852383385968225675E8L, 9.360895299872484512023336636427675327355E6L, 4.213701282274196030811629773097579432957E4L, 5.201006511142748908655720086041570288182E1L, 9.088368420359444263703202925095675982530E-3L, }; static const long double PD[] = { 3.049081015149226615468111430031590411682E9L, 1.069833887183886839966085436512368982758E8L, 8.259257717868875207333991924545445705394E5L, 1.872583833284143212651746812884298360922E3L, /* 1.0E0 */ }; static const long double /* ln 2 */ lg2 = 6.9314718055994530941723212145817656807550E-1L, lg2_h = 6.9314718055994528622676398299518041312695E-1L, lg2_l = 2.3190468138462996154948554638754786504121E-17L, ovt = 8.0085662595372944372e-0017L, /* 2/(3*log(2)) */ cp = 9.6179669392597560490661645400126142495110E-1L, cp_h = 9.6179669392597555432899980587535537779331E-1L, cp_l = 5.0577616648125906047157785230014751039424E-17L; long double powl(long double x, long double y) { long double z, ax, z_h, z_l, p_h, p_l; long double yy1, t1, t2, r, s, t, u, v, w; long double s2, s_h, s_l, t_h, t_l; int32_t i, j, k, yisint, n; u_int32_t ix, iy; int32_t hx, hy; ieee_quad_shape_type o, p, q; p.value = x; hx = p.parts32.mswhi; ix = hx & 0x7fffffff; q.value = y; hy = q.parts32.mswhi; iy = hy & 0x7fffffff; /* y==zero: x**0 = 1 */ if ((iy | q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0) return one; /* 1.0**y = 1; -1.0**+-Inf = 1 */ if (x == one) return one; if (x == -1.0L && iy == 0x7fff0000 && (q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0) return one; /* +-NaN return x+y */ if ((ix > 0x7fff0000) || ((ix == 0x7fff0000) && ((p.parts32.mswlo | p.parts32.lswhi | p.parts32.lswlo) != 0)) || (iy > 0x7fff0000) || ((iy == 0x7fff0000) && ((q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) != 0))) return nan_mix(x, y); /* determine if y is an odd int when x < 0 * yisint = 0 ... y is not an integer * yisint = 1 ... y is an odd int * yisint = 2 ... y is an even int */ yisint = 0; if (hx < 0) { if (iy >= 0x40700000) /* 2^113 */ yisint = 2; /* even integer y */ else if (iy >= 0x3fff0000) /* 1.0 */ { if (floorl (y) == y) { z = 0.5 * y; if (floorl (z) == z) yisint = 2; else yisint = 1; } } } /* special value of y */ if ((q.parts32.mswlo | q.parts32.lswhi | q.parts32.lswlo) == 0) { if (iy == 0x7fff0000) /* y is +-inf */ { if (((ix - 0x3fff0000) | p.parts32.mswlo | p.parts32.lswhi | p.parts32.lswlo) == 0) return y - y; /* +-1**inf is NaN */ else if (ix >= 0x3fff0000) /* (|x|>1)**+-inf = inf,0 */ return (hy >= 0) ? y : zero; else /* (|x|<1)**-,+inf = inf,0 */ return (hy < 0) ? -y : zero; } if (iy == 0x3fff0000) { /* y is +-1 */ if (hy < 0) return one / x; else return x; } if (hy == 0x40000000) return x * x; /* y is 2 */ if (hy == 0x3ffe0000) { /* y is 0.5 */ if (hx >= 0) /* x >= +0 */ return sqrtl (x); } } ax = fabsl (x); /* special value of x */ if ((p.parts32.mswlo | p.parts32.lswhi | p.parts32.lswlo) == 0) { if (ix == 0x7fff0000 || ix == 0 || ix == 0x3fff0000) { z = ax; /*x is +-0,+-inf,+-1 */ if (hy < 0) z = one / z; /* z = (1/|x|) */ if (hx < 0) { if (((ix - 0x3fff0000) | yisint) == 0) { z = (z - z) / (z - z); /* (-1)**non-int is NaN */ } else if (yisint == 1) z = -z; /* (x<0)**odd = -(|x|**odd) */ } return z; } } /* (x<0)**(non-int) is NaN */ if (((((u_int32_t) hx >> 31) - 1) | yisint) == 0) return (x - x) / (x - x); /* |y| is huge. 2^-16495 = 1/2 of smallest representable value. If (1 - 1/131072)^y underflows, y > 1.4986e9 */ if (iy > 0x401d654b) { /* if (1 - 2^-113)^y underflows, y > 1.1873e38 */ if (iy > 0x407d654b) { if (ix <= 0x3ffeffff) return (hy < 0) ? huge * huge : tiny * tiny; if (ix >= 0x3fff0000) return (hy > 0) ? huge * huge : tiny * tiny; } /* over/underflow if x is not close to one */ if (ix < 0x3ffeffff) return (hy < 0) ? huge * huge : tiny * tiny; if (ix > 0x3fff0000) return (hy > 0) ? huge * huge : tiny * tiny; } n = 0; /* take care subnormal number */ if (ix < 0x00010000) { ax *= two113; n -= 113; o.value = ax; ix = o.parts32.mswhi; } n += ((ix) >> 16) - 0x3fff; j = ix & 0x0000ffff; /* determine interval */ ix = j | 0x3fff0000; /* normalize ix */ if (j <= 0x3988) k = 0; /* |x|> 31) - 1) | (yisint - 1)) == 0) s = -one; /* (-ve)**(odd int) */ /* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */ yy1 = y; o.value = yy1; o.parts32.lswlo = 0; o.parts32.lswhi &= 0xf8000000; yy1 = o.value; p_l = (y - yy1) * t1 + y * t2; p_h = yy1 * t1; z = p_l + p_h; o.value = z; j = o.parts32.mswhi; if (j >= 0x400d0000) /* z >= 16384 */ { /* if z > 16384 */ if (((j - 0x400d0000) | o.parts32.mswlo | o.parts32.lswhi | o.parts32.lswlo) != 0) return s * huge * huge; /* overflow */ else { if (p_l + ovt > z - p_h) return s * huge * huge; /* overflow */ } } else if ((j & 0x7fffffff) >= 0x400d01b9) /* z <= -16495 */ { /* z < -16495 */ if (((j - 0xc00d01bc) | o.parts32.mswlo | o.parts32.lswhi | o.parts32.lswlo) != 0) return s * tiny * tiny; /* underflow */ else { if (p_l <= z - p_h) return s * tiny * tiny; /* underflow */ } } /* compute 2**(p_h+p_l) */ i = j & 0x7fffffff; k = (i >> 16) - 0x3fff; n = 0; if (i > 0x3ffe0000) { /* if |z| > 0.5, set n = [z+0.5] */ n = floorl (z + 0.5L); t = n; p_h -= t; } t = p_l + p_h; o.value = t; o.parts32.lswlo = 0; o.parts32.lswhi &= 0xf8000000; t = o.value; u = t * lg2_h; v = (p_l - (t - p_h)) * lg2 + t * lg2_l; z = u + v; w = v - (z - u); /* exp(z) */ t = z * z; u = PN[0] + t * (PN[1] + t * (PN[2] + t * (PN[3] + t * PN[4]))); v = PD[0] + t * (PD[1] + t * (PD[2] + t * (PD[3] + t))); t1 = z - t * u / v; r = (z * t1) / (t1 - two) - (w + z * w); z = one - (r - z); o.value = z; j = o.parts32.mswhi; j += (n << 16); if ((j >> 16) <= 0) z = scalbnl (z, n); /* subnormal output */ else { o.parts32.mswhi = j; z = o.value; } return s * z; } Index: head/lib/msun/src/k_rem_pio2.c =================================================================== --- head/lib/msun/src/k_rem_pio2.c (revision 342650) +++ head/lib/msun/src/k_rem_pio2.c (revision 342651) @@ -1,443 +1,443 @@ /* @(#)k_rem_pio2.c 1.3 95/01/18 */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * 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. * ==================================================== */ #include __FBSDID("$FreeBSD$"); /* * __kernel_rem_pio2(x,y,e0,nx,prec) * double x[],y[]; int e0,nx,prec; * * __kernel_rem_pio2 return the last three digits of N with * y = x - N*pi/2 * so that |y| < pi/2. * * The method is to compute the integer (mod 8) and fraction parts of * (2/pi)*x without doing the full multiplication. In general we * skip the part of the product that are known to be a huge integer ( * more accurately, = 0 mod 8 ). Thus the number of operations are * independent of the exponent of the input. * * (2/pi) is represented by an array of 24-bit integers in ipio2[]. * * Input parameters: * x[] The input value (must be positive) is broken into nx * pieces of 24-bit integers in double precision format. * x[i] will be the i-th 24 bit of x. The scaled exponent * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 * match x's up to 24 bits. * * Example of breaking a double positive z into x[0]+x[1]+x[2]: * e0 = ilogb(z)-23 * z = scalbn(z,-e0) * for i = 0,1,2 * x[i] = floor(z) * z = (z-x[i])*2**24 * * * y[] output result in an array of double precision numbers. * The dimension of y[] is: * 24-bit precision 1 * 53-bit precision 2 * 64-bit precision 2 * 113-bit precision 3 * The actual value is the sum of them. Thus for 113-bit - * precison, one may have to do something like: + * precision, one may have to do something like: * * long double t,w,r_head, r_tail; * t = (long double)y[2] + (long double)y[1]; * w = (long double)y[0]; * r_head = t+w; * r_tail = w - (r_head - t); * * e0 The exponent of x[0]. Must be <= 16360 or you need to * expand the ipio2 table. * * nx dimension of x[] * * prec an integer indicating the precision: * 0 24 bits (single) * 1 53 bits (double) * 2 64 bits (extended) * 3 113 bits (quad) * * External function: * double scalbn(), floor(); * * * Here is the description of some local variables: * * jk jk+1 is the initial number of terms of ipio2[] needed * in the computation. The minimum and recommended value * for jk is 3,4,4,6 for single, double, extended, and quad. * jk+1 must be 2 larger than you might expect so that our * recomputation test works. (Up to 24 bits in the integer * part (the 24 bits of it that we compute) and 23 bits in * the fraction part may be lost to cancellation before we * recompute.) * * jz local integer variable indicating the number of * terms of ipio2[] used. * * jx nx - 1 * * jv index for pointing to the suitable ipio2[] for the * computation. In general, we want * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 * is an integer. Thus * e0-3-24*jv >= 0 or (e0-3)/24 >= jv * Hence jv = max(0,(e0-3)/24). * * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. * * q[] double array with integral value, representing the * 24-bits chunk of the product of x and 2/pi. * * q0 the corresponding exponent of q[0]. Note that the * exponent for q[i] would be q0-24*i. * * PIo2[] double precision array, obtained by cutting pi/2 * into 24 bits chunks. * * f[] ipio2[] in floating point * * iq[] integer array by breaking up q[] in 24-bits chunk. * * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] * * ih integer. If >0 it indicates q[] is >= 0.5, hence * it also indicates the *sign* of the result. * */ /* * Constants: * The hexadecimal values are the intended ones for the following * constants. The decimal values may be used, provided that the * compiler will convert from decimal to binary accurately enough * to produce the hexadecimal values shown. */ #include #include "math.h" #include "math_private.h" static const int init_jk[] = {3,4,4,6}; /* initial value for jk */ /* * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi * * integer array, contains the (24*i)-th to (24*i+23)-th * bit of 2/pi after binary point. The corresponding * floating value is * * ipio2[i] * 2^(-24(i+1)). * * NB: This table must have at least (e0-3)/24 + jk terms. * For quad precision (e0 <= 16360, jk = 6), this is 686. */ static const int32_t ipio2[] = { 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, 0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, 0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, 0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41, 0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8, 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF, 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5, 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, 0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, 0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, 0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B, #if LDBL_MAX_EXP > 1024 #if LDBL_MAX_EXP > 16384 #error "ipio2 table needs to be expanded" #endif 0x47C419, 0xC367CD, 0xDCE809, 0x2A8359, 0xC4768B, 0x961CA6, 0xDDAF44, 0xD15719, 0x053EA5, 0xFF0705, 0x3F7E33, 0xE832C2, 0xDE4F98, 0x327DBB, 0xC33D26, 0xEF6B1E, 0x5EF89F, 0x3A1F35, 0xCAF27F, 0x1D87F1, 0x21907C, 0x7C246A, 0xFA6ED5, 0x772D30, 0x433B15, 0xC614B5, 0x9D19C3, 0xC2C4AD, 0x414D2C, 0x5D000C, 0x467D86, 0x2D71E3, 0x9AC69B, 0x006233, 0x7CD2B4, 0x97A7B4, 0xD55537, 0xF63ED7, 0x1810A3, 0xFC764D, 0x2A9D64, 0xABD770, 0xF87C63, 0x57B07A, 0xE71517, 0x5649C0, 0xD9D63B, 0x3884A7, 0xCB2324, 0x778AD6, 0x23545A, 0xB91F00, 0x1B0AF1, 0xDFCE19, 0xFF319F, 0x6A1E66, 0x615799, 0x47FBAC, 0xD87F7E, 0xB76522, 0x89E832, 0x60BFE6, 0xCDC4EF, 0x09366C, 0xD43F5D, 0xD7DE16, 0xDE3B58, 0x929BDE, 0x2822D2, 0xE88628, 0x4D58E2, 0x32CAC6, 0x16E308, 0xCB7DE0, 0x50C017, 0xA71DF3, 0x5BE018, 0x34132E, 0x621283, 0x014883, 0x5B8EF5, 0x7FB0AD, 0xF2E91E, 0x434A48, 0xD36710, 0xD8DDAA, 0x425FAE, 0xCE616A, 0xA4280A, 0xB499D3, 0xF2A606, 0x7F775C, 0x83C2A3, 0x883C61, 0x78738A, 0x5A8CAF, 0xBDD76F, 0x63A62D, 0xCBBFF4, 0xEF818D, 0x67C126, 0x45CA55, 0x36D9CA, 0xD2A828, 0x8D61C2, 0x77C912, 0x142604, 0x9B4612, 0xC459C4, 0x44C5C8, 0x91B24D, 0xF31700, 0xAD43D4, 0xE54929, 0x10D5FD, 0xFCBE00, 0xCC941E, 0xEECE70, 0xF53E13, 0x80F1EC, 0xC3E7B3, 0x28F8C7, 0x940593, 0x3E71C1, 0xB3092E, 0xF3450B, 0x9C1288, 0x7B20AB, 0x9FB52E, 0xC29247, 0x2F327B, 0x6D550C, 0x90A772, 0x1FE76B, 0x96CB31, 0x4A1679, 0xE27941, 0x89DFF4, 0x9794E8, 0x84E6E2, 0x973199, 0x6BED88, 0x365F5F, 0x0EFDBB, 0xB49A48, 0x6CA467, 0x427271, 0x325D8D, 0xB8159F, 0x09E5BC, 0x25318D, 0x3974F7, 0x1C0530, 0x010C0D, 0x68084B, 0x58EE2C, 0x90AA47, 0x02E774, 0x24D6BD, 0xA67DF7, 0x72486E, 0xEF169F, 0xA6948E, 0xF691B4, 0x5153D1, 0xF20ACF, 0x339820, 0x7E4BF5, 0x6863B2, 0x5F3EDD, 0x035D40, 0x7F8985, 0x295255, 0xC06437, 0x10D86D, 0x324832, 0x754C5B, 0xD4714E, 0x6E5445, 0xC1090B, 0x69F52A, 0xD56614, 0x9D0727, 0x50045D, 0xDB3BB4, 0xC576EA, 0x17F987, 0x7D6B49, 0xBA271D, 0x296996, 0xACCCC6, 0x5414AD, 0x6AE290, 0x89D988, 0x50722C, 0xBEA404, 0x940777, 0x7030F3, 0x27FC00, 0xA871EA, 0x49C266, 0x3DE064, 0x83DD97, 0x973FA3, 0xFD9443, 0x8C860D, 0xDE4131, 0x9D3992, 0x8C70DD, 0xE7B717, 0x3BDF08, 0x2B3715, 0xA0805C, 0x93805A, 0x921110, 0xD8E80F, 0xAF806C, 0x4BFFDB, 0x0F9038, 0x761859, 0x15A562, 0xBBCB61, 0xB989C7, 0xBD4010, 0x04F2D2, 0x277549, 0xF6B6EB, 0xBB22DB, 0xAA140A, 0x2F2689, 0x768364, 0x333B09, 0x1A940E, 0xAA3A51, 0xC2A31D, 0xAEEDAF, 0x12265C, 0x4DC26D, 0x9C7A2D, 0x9756C0, 0x833F03, 0xF6F009, 0x8C402B, 0x99316D, 0x07B439, 0x15200C, 0x5BC3D8, 0xC492F5, 0x4BADC6, 0xA5CA4E, 0xCD37A7, 0x36A9E6, 0x9492AB, 0x6842DD, 0xDE6319, 0xEF8C76, 0x528B68, 0x37DBFC, 0xABA1AE, 0x3115DF, 0xA1AE00, 0xDAFB0C, 0x664D64, 0xB705ED, 0x306529, 0xBF5657, 0x3AFF47, 0xB9F96A, 0xF3BE75, 0xDF9328, 0x3080AB, 0xF68C66, 0x15CB04, 0x0622FA, 0x1DE4D9, 0xA4B33D, 0x8F1B57, 0x09CD36, 0xE9424E, 0xA4BE13, 0xB52333, 0x1AAAF0, 0xA8654F, 0xA5C1D2, 0x0F3F0B, 0xCD785B, 0x76F923, 0x048B7B, 0x721789, 0x53A6C6, 0xE26E6F, 0x00EBEF, 0x584A9B, 0xB7DAC4, 0xBA66AA, 0xCFCF76, 0x1D02D1, 0x2DF1B1, 0xC1998C, 0x77ADC3, 0xDA4886, 0xA05DF7, 0xF480C6, 0x2FF0AC, 0x9AECDD, 0xBC5C3F, 0x6DDED0, 0x1FC790, 0xB6DB2A, 0x3A25A3, 0x9AAF00, 0x9353AD, 0x0457B6, 0xB42D29, 0x7E804B, 0xA707DA, 0x0EAA76, 0xA1597B, 0x2A1216, 0x2DB7DC, 0xFDE5FA, 0xFEDB89, 0xFDBE89, 0x6C76E4, 0xFCA906, 0x70803E, 0x156E85, 0xFF87FD, 0x073E28, 0x336761, 0x86182A, 0xEABD4D, 0xAFE7B3, 0x6E6D8F, 0x396795, 0x5BBF31, 0x48D784, 0x16DF30, 0x432DC7, 0x356125, 0xCE70C9, 0xB8CB30, 0xFD6CBF, 0xA200A4, 0xE46C05, 0xA0DD5A, 0x476F21, 0xD21262, 0x845CB9, 0x496170, 0xE0566B, 0x015299, 0x375550, 0xB7D51E, 0xC4F133, 0x5F6E13, 0xE4305D, 0xA92E85, 0xC3B21D, 0x3632A1, 0xA4B708, 0xD4B1EA, 0x21F716, 0xE4698F, 0x77FF27, 0x80030C, 0x2D408D, 0xA0CD4F, 0x99A520, 0xD3A2B3, 0x0A5D2F, 0x42F9B4, 0xCBDA11, 0xD0BE7D, 0xC1DB9B, 0xBD17AB, 0x81A2CA, 0x5C6A08, 0x17552E, 0x550027, 0xF0147F, 0x8607E1, 0x640B14, 0x8D4196, 0xDEBE87, 0x2AFDDA, 0xB6256B, 0x34897B, 0xFEF305, 0x9EBFB9, 0x4F6A68, 0xA82A4A, 0x5AC44F, 0xBCF82D, 0x985AD7, 0x95C7F4, 0x8D4D0D, 0xA63A20, 0x5F57A4, 0xB13F14, 0x953880, 0x0120CC, 0x86DD71, 0xB6DEC9, 0xF560BF, 0x11654D, 0x6B0701, 0xACB08C, 0xD0C0B2, 0x485551, 0x0EFB1E, 0xC37295, 0x3B06A3, 0x3540C0, 0x7BDC06, 0xCC45E0, 0xFA294E, 0xC8CAD6, 0x41F3E8, 0xDE647C, 0xD8649B, 0x31BED9, 0xC397A4, 0xD45877, 0xC5E369, 0x13DAF0, 0x3C3ABA, 0x461846, 0x5F7555, 0xF5BDD2, 0xC6926E, 0x5D2EAC, 0xED440E, 0x423E1C, 0x87C461, 0xE9FD29, 0xF3D6E7, 0xCA7C22, 0x35916F, 0xC5E008, 0x8DD7FF, 0xE26A6E, 0xC6FDB0, 0xC10893, 0x745D7C, 0xB2AD6B, 0x9D6ECD, 0x7B723E, 0x6A11C6, 0xA9CFF7, 0xDF7329, 0xBAC9B5, 0x5100B7, 0x0DB2E2, 0x24BA74, 0x607DE5, 0x8AD874, 0x2C150D, 0x0C1881, 0x94667E, 0x162901, 0x767A9F, 0xBEFDFD, 0xEF4556, 0x367ED9, 0x13D9EC, 0xB9BA8B, 0xFC97C4, 0x27A831, 0xC36EF1, 0x36C594, 0x56A8D8, 0xB5A8B4, 0x0ECCCF, 0x2D8912, 0x34576F, 0x89562C, 0xE3CE99, 0xB920D6, 0xAA5E6B, 0x9C2A3E, 0xCC5F11, 0x4A0BFD, 0xFBF4E1, 0x6D3B8E, 0x2C86E2, 0x84D4E9, 0xA9B4FC, 0xD1EEEF, 0xC9352E, 0x61392F, 0x442138, 0xC8D91B, 0x0AFC81, 0x6A4AFB, 0xD81C2F, 0x84B453, 0x8C994E, 0xCC2254, 0xDC552A, 0xD6C6C0, 0x96190B, 0xB8701A, 0x649569, 0x605A26, 0xEE523F, 0x0F117F, 0x11B5F4, 0xF5CBFC, 0x2DBC34, 0xEEBC34, 0xCC5DE8, 0x605EDD, 0x9B8E67, 0xEF3392, 0xB817C9, 0x9B5861, 0xBC57E1, 0xC68351, 0x103ED8, 0x4871DD, 0xDD1C2D, 0xA118AF, 0x462C21, 0xD7F359, 0x987AD9, 0xC0549E, 0xFA864F, 0xFC0656, 0xAE79E5, 0x362289, 0x22AD38, 0xDC9367, 0xAAE855, 0x382682, 0x9BE7CA, 0xA40D51, 0xB13399, 0x0ED7A9, 0x480569, 0xF0B265, 0xA7887F, 0x974C88, 0x36D1F9, 0xB39221, 0x4A827B, 0x21CF98, 0xDC9F40, 0x5547DC, 0x3A74E1, 0x42EB67, 0xDF9DFE, 0x5FD45E, 0xA4677B, 0x7AACBA, 0xA2F655, 0x23882B, 0x55BA41, 0x086E59, 0x862A21, 0x834739, 0xE6E389, 0xD49EE5, 0x40FB49, 0xE956FF, 0xCA0F1C, 0x8A59C5, 0x2BFA94, 0xC5C1D3, 0xCFC50F, 0xAE5ADB, 0x86C547, 0x624385, 0x3B8621, 0x94792C, 0x876110, 0x7B4C2A, 0x1A2C80, 0x12BF43, 0x902688, 0x893C78, 0xE4C4A8, 0x7BDBE5, 0xC23AC4, 0xEAF426, 0x8A67F7, 0xBF920D, 0x2BA365, 0xB1933D, 0x0B7CBD, 0xDC51A4, 0x63DD27, 0xDDE169, 0x19949A, 0x9529A8, 0x28CE68, 0xB4ED09, 0x209F44, 0xCA984E, 0x638270, 0x237C7E, 0x32B90F, 0x8EF5A7, 0xE75614, 0x08F121, 0x2A9DB5, 0x4D7E6F, 0x5119A5, 0xABF9B5, 0xD6DF82, 0x61DD96, 0x023616, 0x9F3AC4, 0xA1A283, 0x6DED72, 0x7A8D39, 0xA9B882, 0x5C326B, 0x5B2746, 0xED3400, 0x7700D2, 0x55F4FC, 0x4D5901, 0x8071E0, #endif }; static const double PIo2[] = { 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ }; static const double zero = 0.0, one = 1.0, two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec) { int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; double z,fw,f[20],fq[20],q[20]; /* initialize jk*/ jk = init_jk[prec]; jp = jk; /* determine jx,jv,q0, note that 3>q0 */ jx = nx-1; jv = (e0-3)/24; if(jv<0) jv=0; q0 = e0-24*(jv+1); /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ j = jv-jx; m = jx+jk; for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j]; /* compute q[0],q[1],...q[jk] */ for (i=0;i<=jk;i++) { for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw; } jz = jk; recompute: /* distill q[] into iq[] reversingly */ for(i=0,j=jz,z=q[jz];j>0;i++,j--) { fw = (double)((int32_t)(twon24* z)); iq[i] = (int32_t)(z-two24*fw); z = q[j-1]+fw; } /* compute n */ z = scalbn(z,q0); /* actual value of z */ z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */ n = (int32_t) z; z -= (double)n; ih = 0; if(q0>0) { /* need iq[jz-1] to determine n */ i = (iq[jz-1]>>(24-q0)); n += i; iq[jz-1] -= i<<(24-q0); ih = iq[jz-1]>>(23-q0); } else if(q0==0) ih = iq[jz-1]>>23; else if(z>=0.5) ih=2; if(ih>0) { /* q > 0.5 */ n += 1; carry = 0; for(i=0;i0) { /* rare case: chance is 1 in 12 */ switch(q0) { case 1: iq[jz-1] &= 0x7fffff; break; case 2: iq[jz-1] &= 0x3fffff; break; } } if(ih==2) { z = one - z; if(carry!=0) z -= scalbn(one,q0); } } /* check if recomputation is needed */ if(z==zero) { j = 0; for (i=jz-1;i>=jk;i--) j |= iq[i]; if(j==0) { /* need recomputation */ for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */ for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */ f[jx+i] = (double) ipio2[jv+i]; for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw; } jz += k; goto recompute; } } /* chop off zero terms */ if(z==0.0) { jz -= 1; q0 -= 24; while(iq[jz]==0) { jz--; q0-=24;} } else { /* break z into 24-bit if necessary */ z = scalbn(z,-q0); if(z>=two24) { fw = (double)((int32_t)(twon24*z)); iq[jz] = (int32_t)(z-two24*fw); jz += 1; q0 += 24; iq[jz] = (int32_t) fw; } else iq[jz] = (int32_t) z ; } /* convert integer "bit" chunk to floating-point value */ fw = scalbn(one,q0); for(i=jz;i>=0;i--) { q[i] = fw*(double)iq[i]; fw*=twon24; } /* compute PIo2[0,...,jp]*q[jz,...,0] */ for(i=jz;i>=0;i--) { for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; fq[jz-i] = fw; } /* compress fq[] into y[] */ switch(prec) { case 0: fw = 0.0; for (i=jz;i>=0;i--) fw += fq[i]; y[0] = (ih==0)? fw: -fw; break; case 1: case 2: fw = 0.0; for (i=jz;i>=0;i--) fw += fq[i]; STRICT_ASSIGN(double,fw,fw); y[0] = (ih==0)? fw: -fw; fw = fq[0]-fw; for (i=1;i<=jz;i++) fw += fq[i]; y[1] = (ih==0)? fw: -fw; break; case 3: /* painful */ for (i=jz;i>0;i--) { fw = fq[i-1]+fq[i]; fq[i] += fq[i-1]-fw; fq[i-1] = fw; } for (i=jz;i>1;i--) { fw = fq[i-1]+fq[i]; fq[i] += fq[i-1]-fw; fq[i-1] = fw; } for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; if(ih==0) { y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; } else { y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; } } return n&7; }