443 lines
16 KiB
C
443 lines
16 KiB
C
|
/* origin: FreeBSD /usr/src/lib/msun/src/k_rem_pio2.c */
|
||
|
|
/*
|
||
|
|
* ====================================================
|
||
|
|
* 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.
|
||
|
|
* ====================================================
|
||
|
|
*/
|
||
|
|
/*
|
||
|
|
* __rem_pio2_large(x,y,e0,nx,prec)
|
||
|
|
* double x[],y[]; int e0,nx,prec;
|
||
|
|
*
|
||
|
|
* __rem_pio2_large 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[] ouput 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:
|
||
|
|
*
|
||
|
|
* 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 cancelation 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 "libm.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
|
||
|
|
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 */
|
||
|
|
};
|
||
|
|
|
||
|
|
int __rem_pio2_large(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 ? 0.0 : (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)(0x1p-24*z);
|
||
|
|
iq[i] = (int32_t)(z - 0x1p24*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; i<jz; i++) { /* compute 1-q */
|
||
|
|
j = iq[i];
|
||
|
|
if (carry == 0) {
|
||
|
|
if (j != 0) {
|
||
|
|
carry = 1;
|
||
|
|
iq[i] = 0x1000000 - j;
|
||
|
|
}
|
||
|
|
} else
|
||
|
|
iq[i] = 0xffffff - j;
|
||
|
|
}
|
||
|
|
if (q0 > 0) { /* 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 = 1.0 - z;
|
||
|
|
if (carry != 0)
|
||
|
|
z -= scalbn(1.0,q0);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
/* check if recomputation is needed */
|
||
|
|
if (z == 0.0) {
|
||
|
|
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 >= 0x1p24) {
|
||
|
|
fw = (double)(int32_t)(0x1p-24*z);
|
||
|
|
iq[jz] = (int32_t)(z - 0x1p24*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(1.0,q0);
|
||
|
|
for (i=jz; i>=0; i--) {
|
||
|
|
q[i] = fw*(double)iq[i];
|
||
|
|
fw *= 0x1p-24;
|
||
|
|
}
|
||
|
|
|
||
|
|
/* 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];
|
||
|
|
// TODO: drop excess precision here once double_t is used
|
||
|
|
fw = (double)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;
|
||
|
|
}
|