k_rem_pio2.c 8.5 KB

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  1. /* @(#)k_rem_pio2.c 5.1 93/09/24 */
  2. /*
  3. * ====================================================
  4. * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  5. *
  6. * Developed at SunPro, a Sun Microsystems, Inc. business.
  7. * Permission to use, copy, modify, and distribute this
  8. * software is freely granted, provided that this notice
  9. * is preserved.
  10. * ====================================================
  11. */
  12. #if defined(LIBM_SCCS) && !defined(lint)
  13. static char rcsid[] = "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $";
  14. #endif
  15. /*
  16. * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
  17. * double x[],y[]; int e0,nx,prec; int ipio2[];
  18. *
  19. * __kernel_rem_pio2 return the last three digits of N with
  20. * y = x - N*pi/2
  21. * so that |y| < pi/2.
  22. *
  23. * The method is to compute the integer (mod 8) and fraction parts of
  24. * (2/pi)*x without doing the full multiplication. In general we
  25. * skip the part of the product that are known to be a huge integer (
  26. * more accurately, = 0 mod 8 ). Thus the number of operations are
  27. * independent of the exponent of the input.
  28. *
  29. * (2/pi) is represented by an array of 24-bit integers in ipio2[].
  30. *
  31. * Input parameters:
  32. * x[] The input value (must be positive) is broken into nx
  33. * pieces of 24-bit integers in double precision format.
  34. * x[i] will be the i-th 24 bit of x. The scaled exponent
  35. * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
  36. * match x's up to 24 bits.
  37. *
  38. * Example of breaking a double positive z into x[0]+x[1]+x[2]:
  39. * e0 = ilogb(z)-23
  40. * z = scalbn(z,-e0)
  41. * for i = 0,1,2
  42. * x[i] = floor(z)
  43. * z = (z-x[i])*2**24
  44. *
  45. *
  46. * y[] ouput result in an array of double precision numbers.
  47. * The dimension of y[] is:
  48. * 24-bit precision 1
  49. * 53-bit precision 2
  50. * 64-bit precision 2
  51. * 113-bit precision 3
  52. * The actual value is the sum of them. Thus for 113-bit
  53. * precison, one may have to do something like:
  54. *
  55. * long double t,w,r_head, r_tail;
  56. * t = (long double)y[2] + (long double)y[1];
  57. * w = (long double)y[0];
  58. * r_head = t+w;
  59. * r_tail = w - (r_head - t);
  60. *
  61. * e0 The exponent of x[0]
  62. *
  63. * nx dimension of x[]
  64. *
  65. * prec an integer indicating the precision:
  66. * 0 24 bits (single)
  67. * 1 53 bits (double)
  68. * 2 64 bits (extended)
  69. * 3 113 bits (quad)
  70. *
  71. * ipio2[]
  72. * integer array, contains the (24*i)-th to (24*i+23)-th
  73. * bit of 2/pi after binary point. The corresponding
  74. * floating value is
  75. *
  76. * ipio2[i] * 2^(-24(i+1)).
  77. *
  78. * External function:
  79. * double scalbn(), floor();
  80. *
  81. *
  82. * Here is the description of some local variables:
  83. *
  84. * jk jk+1 is the initial number of terms of ipio2[] needed
  85. * in the computation. The recommended value is 2,3,4,
  86. * 6 for single, double, extended,and quad.
  87. *
  88. * jz local integer variable indicating the number of
  89. * terms of ipio2[] used.
  90. *
  91. * jx nx - 1
  92. *
  93. * jv index for pointing to the suitable ipio2[] for the
  94. * computation. In general, we want
  95. * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
  96. * is an integer. Thus
  97. * e0-3-24*jv >= 0 or (e0-3)/24 >= jv
  98. * Hence jv = max(0,(e0-3)/24).
  99. *
  100. * jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
  101. *
  102. * q[] double array with integral value, representing the
  103. * 24-bits chunk of the product of x and 2/pi.
  104. *
  105. * q0 the corresponding exponent of q[0]. Note that the
  106. * exponent for q[i] would be q0-24*i.
  107. *
  108. * PIo2[] double precision array, obtained by cutting pi/2
  109. * into 24 bits chunks.
  110. *
  111. * f[] ipio2[] in floating point
  112. *
  113. * iq[] integer array by breaking up q[] in 24-bits chunk.
  114. *
  115. * fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
  116. *
  117. * ih integer. If >0 it indicates q[] is >= 0.5, hence
  118. * it also indicates the *sign* of the result.
  119. *
  120. */
  121. /*
  122. * Constants:
  123. * The hexadecimal values are the intended ones for the following
  124. * constants. The decimal values may be used, provided that the
  125. * compiler will convert from decimal to binary accurately enough
  126. * to produce the hexadecimal values shown.
  127. */
  128. #include "math.h"
  129. #include "math_private.h"
  130. #ifdef __STDC__
  131. static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
  132. #else
  133. static int init_jk[] = {2,3,4,6};
  134. #endif
  135. #ifdef __STDC__
  136. static const double PIo2[] = {
  137. #else
  138. static double PIo2[] = {
  139. #endif
  140. 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
  141. 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
  142. 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
  143. 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
  144. 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
  145. 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
  146. 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
  147. 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
  148. };
  149. #ifdef __STDC__
  150. static const double
  151. #else
  152. static double
  153. #endif
  154. zero = 0.0,
  155. one = 1.0,
  156. two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
  157. twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
  158. #ifdef __STDC__
  159. int attribute_hidden __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)
  160. #else
  161. int attribute_hidden __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
  162. double x[], y[]; int e0,nx,prec; int32_t ipio2[];
  163. #endif
  164. {
  165. int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
  166. double z,fw,f[20],fq[20],q[20];
  167. /* initialize jk*/
  168. jk = init_jk[prec];
  169. jp = jk;
  170. /* determine jx,jv,q0, note that 3>q0 */
  171. jx = nx-1;
  172. jv = (e0-3)/24; if(jv<0) jv=0;
  173. q0 = e0-24*(jv+1);
  174. /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
  175. j = jv-jx; m = jx+jk;
  176. for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
  177. /* compute q[0],q[1],...q[jk] */
  178. for (i=0;i<=jk;i++) {
  179. for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
  180. }
  181. jz = jk;
  182. recompute:
  183. /* distill q[] into iq[] reversingly */
  184. for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
  185. fw = (double)((int32_t)(twon24* z));
  186. iq[i] = (int32_t)(z-two24*fw);
  187. z = q[j-1]+fw;
  188. }
  189. /* compute n */
  190. z = scalbn(z,q0); /* actual value of z */
  191. z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
  192. n = (int32_t) z;
  193. z -= (double)n;
  194. ih = 0;
  195. if(q0>0) { /* need iq[jz-1] to determine n */
  196. i = (iq[jz-1]>>(24-q0)); n += i;
  197. iq[jz-1] -= i<<(24-q0);
  198. ih = iq[jz-1]>>(23-q0);
  199. }
  200. else if(q0==0) ih = iq[jz-1]>>23;
  201. else if(z>=0.5) ih=2;
  202. if(ih>0) { /* q > 0.5 */
  203. n += 1; carry = 0;
  204. for(i=0;i<jz ;i++) { /* compute 1-q */
  205. j = iq[i];
  206. if(carry==0) {
  207. if(j!=0) {
  208. carry = 1; iq[i] = 0x1000000- j;
  209. }
  210. } else iq[i] = 0xffffff - j;
  211. }
  212. if(q0>0) { /* rare case: chance is 1 in 12 */
  213. switch(q0) {
  214. case 1:
  215. iq[jz-1] &= 0x7fffff; break;
  216. case 2:
  217. iq[jz-1] &= 0x3fffff; break;
  218. }
  219. }
  220. if(ih==2) {
  221. z = one - z;
  222. if(carry!=0) z -= scalbn(one,q0);
  223. }
  224. }
  225. /* check if recomputation is needed */
  226. if(z==zero) {
  227. j = 0;
  228. for (i=jz-1;i>=jk;i--) j |= iq[i];
  229. if(j==0) { /* need recomputation */
  230. for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
  231. for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
  232. f[jx+i] = (double) ipio2[jv+i];
  233. for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
  234. q[i] = fw;
  235. }
  236. jz += k;
  237. goto recompute;
  238. }
  239. }
  240. /* chop off zero terms */
  241. if(z==0.0) {
  242. jz -= 1; q0 -= 24;
  243. while(iq[jz]==0) { jz--; q0-=24;}
  244. } else { /* break z into 24-bit if necessary */
  245. z = scalbn(z,-q0);
  246. if(z>=two24) {
  247. fw = (double)((int32_t)(twon24*z));
  248. iq[jz] = (int32_t)(z-two24*fw);
  249. jz += 1; q0 += 24;
  250. iq[jz] = (int32_t) fw;
  251. } else iq[jz] = (int32_t) z ;
  252. }
  253. /* convert integer "bit" chunk to floating-point value */
  254. fw = scalbn(one,q0);
  255. for(i=jz;i>=0;i--) {
  256. q[i] = fw*(double)iq[i]; fw*=twon24;
  257. }
  258. /* compute PIo2[0,...,jp]*q[jz,...,0] */
  259. for(i=jz;i>=0;i--) {
  260. for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
  261. fq[jz-i] = fw;
  262. }
  263. /* compress fq[] into y[] */
  264. switch(prec) {
  265. case 0:
  266. fw = 0.0;
  267. for (i=jz;i>=0;i--) fw += fq[i];
  268. y[0] = (ih==0)? fw: -fw;
  269. break;
  270. case 1:
  271. case 2:
  272. fw = 0.0;
  273. for (i=jz;i>=0;i--) fw += fq[i];
  274. y[0] = (ih==0)? fw: -fw;
  275. fw = fq[0]-fw;
  276. for (i=1;i<=jz;i++) fw += fq[i];
  277. y[1] = (ih==0)? fw: -fw;
  278. break;
  279. case 3: /* painful */
  280. for (i=jz;i>0;i--) {
  281. fw = fq[i-1]+fq[i];
  282. fq[i] += fq[i-1]-fw;
  283. fq[i-1] = fw;
  284. }
  285. for (i=jz;i>1;i--) {
  286. fw = fq[i-1]+fq[i];
  287. fq[i] += fq[i-1]-fw;
  288. fq[i-1] = fw;
  289. }
  290. for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
  291. if(ih==0) {
  292. y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
  293. } else {
  294. y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
  295. }
  296. }
  297. return n&7;
  298. }