k_rem_pio2.c 8.0 KB

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