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Diffstat (limited to 'libquadmath/math/rem_pio2q.c')
| -rw-r--r-- | libquadmath/math/rem_pio2q.c | 587 |
1 files changed, 587 insertions, 0 deletions
diff --git a/libquadmath/math/rem_pio2q.c b/libquadmath/math/rem_pio2q.c new file mode 100644 index 000000000..47ee8ef20 --- /dev/null +++ b/libquadmath/math/rem_pio2q.c @@ -0,0 +1,587 @@ +#include "quadmath-imp.h" +#include <math.h> + + +/* @(#)k_rem_pio2.c 5.1 93/09/24 */ +/* + * ==================================================== + * 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. + * ==================================================== + */ + +/* + * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) + * double x[],y[]; int e0,nx,prec; int ipio2[]; + * + * __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[] 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 + * 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] + * + * 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) + * + * ipio2[] + * 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)). + * + * 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 recommended value is 2,3,4, + * 6 for single, double, extended,and quad. + * + * 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. + */ + + +static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ + +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 */ + + +static int +__quadmath_kernel_rem_pio2 (double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2) +{ + 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;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 = 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]; + 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--) { +#if __FLT_EVAL_METHOD__ != 0 + volatile +#endif + double fv = (double)(fq[i-1]+fq[i]); + fq[i] += fq[i-1]-fv; + fq[i-1] = fv; + } + for (i=jz;i>1;i--) { +#if __FLT_EVAL_METHOD__ != 0 + volatile +#endif + double fv = (double)(fq[i-1]+fq[i]); + fq[i] += fq[i-1]-fv; + fq[i-1] = fv; + } + 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; +} + + + + + +/* Quad-precision floating point argument reduction. + Copyright (C) 1999 Free Software Foundation, Inc. + This file is part of the GNU C Library. + Contributed by Jakub Jelinek <jj@ultra.linux.cz> + + The GNU C Library is free software; you can redistribute it and/or + modify it under the terms of the GNU Lesser General Public + License as published by the Free Software Foundation; either + version 2.1 of the License, or (at your option) any later version. + + The GNU C Library is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + Lesser General Public License for more details. + + You should have received a copy of the GNU Lesser General Public + License along with the GNU C Library; if not, write to the Free + Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA + 02111-1307 USA. */ + +/* + * Table of constants for 2/pi, 5628 hexadecimal digits of 2/pi + */ +static const int32_t two_over_pi[] = { +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, +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, +0xe13f89, 0xb295f3, 0x64a8f1, 0xaea74b, 0x38fc4c, 0xeab2bb, +0x47270b, 0xabc3a7, 0x34ba60, 0x52dd34, 0xf8563a, 0xeb7e8a, +0x31bb36, 0x5895b7, 0x47f7a9, 0x94c3aa, 0xd39225, 0x1e7f3e, +0xd8974e, 0xbba94f, 0xd8ae01, 0xe661b4, 0x393d8e, 0xa523aa, +0x33068e, 0x1633b5, 0x3bb188, 0x1d3a9d, 0x4013d0, 0xcc1be5, +0xf862e7, 0x3bf28f, 0x39b5bf, 0x0bc235, 0x22747e, 0xa247c0, +0xd52d1f, 0x19add3, 0x9094df, 0x9311d0, 0xb42b25, 0x496db2, +0xe264b2, 0x5ef135, 0x3bc6a4, 0x1a4ad0, 0xaac92e, 0x64e886, +0x573091, 0x982cfb, 0x311b1a, 0x08728b, 0xbdcee1, 0x60e142, +0xeb641d, 0xd0bba3, 0xe559d4, 0x597b8c, 0x2a4483, 0xf332ba, +0xf84867, 0x2c8d1b, 0x2fa9b0, 0x50f3dd, 0xf9f573, 0xdb61b4, +0xfe233e, 0x6c41a6, 0xeea318, 0x775a26, 0xbc5e5c, 0xcea708, +0x94dc57, 0xe20196, 0xf1e839, 0xbe4851, 0x5d2d2f, 0x4e9555, +0xd96ec2, 0xe7d755, 0x6304e0, 0xc02e0e, 0xfc40a0, 0xbbf9b3, +0x7125a7, 0x222dfb, 0xf619d8, 0x838c1c, 0x6619e6, 0xb20d55, +0xbb5137, 0x79e809, 0xaf9149, 0x0d73de, 0x0b0da5, 0xce7f58, +0xac1934, 0x724667, 0x7a1a13, 0x9e26bc, 0x4555e7, 0x585cb5, +0x711d14, 0x486991, 0x480d60, 0x56adab, 0xd62f64, 0x96ee0c, +0x212ff3, 0x5d6d88, 0xa67684, 0x95651e, 0xab9e0a, 0x4ddefe, +0x571010, 0x836a39, 0xf8ea31, 0x9e381d, 0xeac8b1, 0xcac96b, +0x37f21e, 0xd505e9, 0x984743, 0x9fc56c, 0x0331b7, 0x3b8bf8, +0x86e56a, 0x8dc343, 0x6230e7, 0x93cfd5, 0x6a8f2d, 0x733005, +0x1af021, 0xa09fcb, 0x7415a1, 0xd56b23, 0x6ff725, 0x2f4bc7, +0xb8a591, 0x7fac59, 0x5c55de, 0x212c38, 0xb13296, 0x5cff50, +0x366262, 0xfa7b16, 0xf4d9a6, 0x2acfe7, 0xf07403, 0xd4d604, +0x6fd916, 0x31b1bf, 0xcbb450, 0x5bd7c8, 0x0ce194, 0x6bd643, +0x4fd91c, 0xdf4543, 0x5f3453, 0xe2b5aa, 0xc9aec8, 0x131485, +0xf9d2bf, 0xbadb9e, 0x76f5b9, 0xaf15cf, 0xca3182, 0x14b56d, +0xe9fe4d, 0x50fc35, 0xf5aed5, 0xa2d0c1, 0xc96057, 0x192eb6, +0xe91d92, 0x07d144, 0xaea3c6, 0x343566, 0x26d5b4, 0x3161e2, +0x37f1a2, 0x209eff, 0x958e23, 0x493798, 0x35f4a6, 0x4bdc02, +0xc2be13, 0xbe80a0, 0x0b72a3, 0x115c5f, 0x1e1bd1, 0x0db4d3, +0x869e85, 0x96976b, 0x2ac91f, 0x8a26c2, 0x3070f0, 0x041412, +0xfc9fa5, 0xf72a38, 0x9c6878, 0xe2aa76, 0x50cfe1, 0x559274, +0x934e38, 0x0a92f7, 0x5533f0, 0xa63db4, 0x399971, 0xe2b755, +0xa98a7c, 0x008f19, 0xac54d2, 0x2ea0b4, 0xf5f3e0, 0x60c849, +0xffd269, 0xae52ce, 0x7a5fdd, 0xe9ce06, 0xfb0ae8, 0xa50cce, +0xea9d3e, 0x3766dd, 0xb834f5, 0x0da090, 0x846f88, 0x4ae3d5, +0x099a03, 0x2eae2d, 0xfcb40a, 0xfb9b33, 0xe281dd, 0x1b16ba, +0xd8c0af, 0xd96b97, 0xb52dc9, 0x9c277f, 0x5951d5, 0x21ccd6, +0xb6496b, 0x584562, 0xb3baf2, 0xa1a5c4, 0x7ca2cf, 0xa9b93d, +0x7b7b89, 0x483d38, +}; + +static const __float128 c[] = { +/* 93 bits of pi/2 */ +#define PI_2_1 c[0] + 1.57079632679489661923132169155131424e+00Q, /* 3fff921fb54442d18469898cc5100000 */ + +/* pi/2 - PI_2_1 */ +#define PI_2_1t c[1] + 8.84372056613570112025531863263659260e-29Q, /* 3fa1c06e0e68948127044533e63a0106 */ +}; + + +int32_t +__quadmath_rem_pio2q (__float128 x, __float128 *y) +{ + __float128 z, w, t; + double tx[8]; + int64_t exp, n, ix, hx; + uint64_t lx; + + GET_FLT128_WORDS64 (hx, lx, x); + ix = hx & 0x7fffffffffffffffLL; + if (ix <= 0x3ffe921fb54442d1LL) /* x in <-pi/4, pi/4> */ + { + y[0] = x; + y[1] = 0; + return 0; + } + + if (ix < 0x40002d97c7f3321dLL) /* |x| in <pi/4, 3pi/4) */ + { + if (hx > 0) + { + /* 113 + 93 bit PI is ok */ + z = x - PI_2_1; + y[0] = z - PI_2_1t; + y[1] = (z - y[0]) - PI_2_1t; + return 1; + } + else + { + /* 113 + 93 bit PI is ok */ + z = x + PI_2_1; + y[0] = z + PI_2_1t; + y[1] = (z - y[0]) + PI_2_1t; + return -1; + } + } + + if (ix >= 0x7fff000000000000LL) /* x is +=oo or NaN */ + { + y[0] = x - x; + y[1] = y[0]; + return 0; + } + + /* Handle large arguments. + We split the 113 bits of the mantissa into 5 24bit integers + stored in a double array. */ + exp = (ix >> 48) - 16383 - 23; + + /* This is faster than doing this in floating point, because we + have to convert it to integers anyway and like this we can keep + both integer and floating point units busy. */ + tx [0] = (double)(((ix >> 25) & 0x7fffff) | 0x800000); + tx [1] = (double)((ix >> 1) & 0xffffff); + tx [2] = (double)(((ix << 23) | (lx >> 41)) & 0xffffff); + tx [3] = (double)((lx >> 17) & 0xffffff); + tx [4] = (double)((lx << 7) & 0xffffff); + + n = __quadmath_kernel_rem_pio2 (tx, tx + 5, exp, + ((lx << 7) & 0xffffff) ? 5 : 4, + 3, two_over_pi); + + /* The result is now stored in 3 double values, we need to convert it into + two __float128 values. */ + t = (__float128) tx [6] + (__float128) tx [7]; + w = (__float128) tx [5]; + + if (hx >= 0) + { + y[0] = w + t; + y[1] = t - (y[0] - w); + return n; + } + else + { + y[0] = -(w + t); + y[1] = -t - (y[0] + w); + return -n; + } +} |