/* @(#)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. * ==================================================== */ #if defined(LIBM_SCCS) && !defined(lint) static char rcsid[] = "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $"; #endif /* * __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. */ #include #include 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 */ int __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:; #if __FLT_EVAL_METHOD__ != 0 volatile #endif double fv = 0.0; for (i = jz; i >= 0; i--) fv += fq[i]; y[0] = (ih == 0) ? fv : -fv; fv = fq[0] - fv; for (i = 1; i <= jz; i++) fv += fq[i]; y[1] = (ih == 0) ? fv : -fv; 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; }