sysdeps/ieee754/dbl-64/s_sin.c

   1 /*
   2  * IBM Accurate Mathematical Library
   3  * written by International Business Machines Corp.
   4  * Copyright (C) 2001-2019 Free Software Foundation, Inc.
   5  *
   6  * This program is free software; you can redistribute it and/or modify
   7  * it under the terms of the GNU Lesser General Public License as published by
   8  * the Free Software Foundation; either version 2.1 of the License, or
   9  * (at your option) any later version.
  10  *
  11  * This program is distributed in the hope that it will be useful,
  12  * but WITHOUT ANY WARRANTY; without even the implied warranty of
  13  * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  14  * GNU Lesser General Public License for more details.
  15  *
  16  * You should have received a copy of the GNU  Lesser General Public License
  17  * along with this program; if not, see <http://www.gnu.org/licenses/>.
  18  */
  19 /****************************************************************************/
  20 /*                                                                          */
  21 /* MODULE_NAME:usncs.c                                                      */
  22 /*                                                                          */
  23 /* FUNCTIONS: usin                                                          */
  24 /*            ucos                                                          */
  25 /* FILES NEEDED: dla.h endian.h mpa.h mydefs.h  usncs.h                     */
  26 /*               branred.c sincos.tbl                                       */
  27 /*                                                                          */
  28 /* An ultimate sin and cos routine. Given an IEEE double machine number x   */
  29 /* it computes sin(x) or cos(x) with ~0.55 ULP.                             */
  30 /* Assumption: Machine arithmetic operations are performed in               */
  31 /* round to nearest mode of IEEE 754 standard.                              */
  32 /*                                                                          */
  33 /****************************************************************************/
  34
  35
  36 #include <errno.h>
  37 #include <float.h>
  38 #include "endian.h"
  39 #include "mydefs.h"
  40 #include "usncs.h"
  41 #include "MathLib.h"
  42 #include <math.h>
  43 #include <math_private.h>
  44 #include <fenv_private.h>
  45 #include <math-underflow.h>
  46 #include <libm-alias-double.h>
  47 #include <fenv.h>
  48
  49 /* Helper macros to compute sin of the input values.  */
  50 #define POLYNOMIAL2(xx) ((((s5 * (xx) + s4) * (xx) + s3) * (xx) + s2) * (xx))
  51
  52 #define POLYNOMIAL(xx) (POLYNOMIAL2 (xx) + s1)
  53
  54 /* The computed polynomial is a variation of the Taylor series expansion for
  55    sin(a):
  56
  57    a - a^3/3! + a^5/5! - a^7/7! + a^9/9! + (1 - a^2) * da / 2
  58
  59    The constants s1, s2, s3, etc. are pre-computed values of 1/3!, 1/5! and so
  60    on.  The result is returned to LHS.  */
  61 #define TAYLOR_SIN(xx, a, da) \
  62 ({                                                                            \
  63   double t = ((POLYNOMIAL (xx)  * (a) - 0.5 * (da))  * (xx) + (da));          \
  64   double res = (a) + t;                                                       \
  65   res;                                                                        \
  66 })
  67
  68 #define SINCOS_TABLE_LOOKUP(u, sn, ssn, cs, ccs) \
  69 ({                                                                            \
  70   int4 k = u.i[LOW_HALF] << 2;                                                \
  71   sn = __sincostab.x[k];                                                      \
  72   ssn = __sincostab.x[k + 1];                                                 \
  73   cs = __sincostab.x[k + 2];                                                  \
  74   ccs = __sincostab.x[k + 3];                                                 \
  75 })
  76
  77 #ifndef SECTION
  78 # define SECTION
  79 #endif
  80
  81 extern const union
  82 {
  83   int4 i[880];
  84   double x[440];
  85 } __sincostab attribute_hidden;
  86
  87 static const double
  88   sn3 = -1.66666666666664880952546298448555E-01,
  89   sn5 = 8.33333214285722277379541354343671E-03,
  90   cs2 = 4.99999999999999999999950396842453E-01,
  91   cs4 = -4.16666666666664434524222570944589E-02,
  92   cs6 = 1.38888874007937613028114285595617E-03;
  93
  94 int __branred (double x, double *a, double *aa);
  95
  96 /* Given a number partitioned into X and DX, this function computes the cosine
  97    of the number by combining the sin and cos of X (as computed by a variation
  98    of the Taylor series) with the values looked up from the sin/cos table to
  99    get the result.  */
 100 static inline double
 101 __always_inline
 102 do_cos (double x, double dx)
 103 {
 104   mynumber u;
 105
 106   if (x < 0)
 107     dx = -dx;
 108
 109   u.x = big + fabs (x);
 110   x = fabs (x) - (u.x - big) + dx;
 111
 112   double xx, s, sn, ssn, c, cs, ccs, cor;
 113   xx = x * x;
 114   s = x + x * xx * (sn3 + xx * sn5);
 115   c = xx * (cs2 + xx * (cs4 + xx * cs6));
 116   SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
 117   cor = (ccs - s * ssn - cs * c) - sn * s;
 118   return cs + cor;
 119 }
 120
 121 /* Given a number partitioned into X and DX, this function computes the sine of
 122    the number by combining the sin and cos of X (as computed by a variation of
 123    the Taylor series) with the values looked up from the sin/cos table to get
 124    the result.  */
 125 static inline double
 126 __always_inline
 127 do_sin (double x, double dx)
 128 {
 129   double xold = x;
 130   /* Max ULP is 0.501 if |x| < 0.126, otherwise ULP is 0.518.  */
 131   if (fabs (x) < 0.126)
 132     return TAYLOR_SIN (x * x, x, dx);
 133
 134   mynumber u;
 135
 136   if (x <= 0)
 137     dx = -dx;
 138   u.x = big + fabs (x);
 139   x = fabs (x) - (u.x - big);
 140
 141   double xx, s, sn, ssn, c, cs, ccs, cor;
 142   xx = x * x;
 143   s = x + (dx + x * xx * (sn3 + xx * sn5));
 144   c = x * dx + xx * (cs2 + xx * (cs4 + xx * cs6));
 145   SINCOS_TABLE_LOOKUP (u, sn, ssn, cs, ccs);
 146   cor = (ssn + s * ccs - sn * c) + cs * s;
 147   return copysign (sn + cor, xold);
 148 }
 149
 150 /* Reduce range of x to within PI/2 with abs (x) < 105414350.  The high part
 151    is written to *a, the low part to *da.  Range reduction is accurate to 136
 152    bits so that when x is large and *a very close to zero, all 53 bits of *a
 153    are correct.  */
 154 static inline int4
 155 __always_inline
 156 reduce_sincos (double x, double *a, double *da)
 157 {
 158   mynumber v;
 159
 160   double t = (x * hpinv + toint);
 161   double xn = t - toint;
 162   v.x = t;
 163   double y = (x - xn * mp1) - xn * mp2;
 164   int4 n = v.i[LOW_HALF] & 3;
 165
 166   double b, db, t1, t2;
 167   t1 = xn * pp3;
 168   t2 = y - t1;
 169   db = (y - t2) - t1;
 170
 171   t1 = xn * pp4;
 172   b = t2 - t1;
 173   db += (t2 - b) - t1;
 174
 175   *a = b;
 176   *da = db;
 177   return n;
 178 }
 179
 180 /* Compute sin or cos (A + DA) for the given quadrant N.  */
 181 static double
 182 __always_inline
 183 do_sincos (double a, double da, int4 n)
 184 {
 185   double retval;
 186
 187   if (n & 1)
 188     /* Max ULP is 0.513.  */
 189     retval = do_cos (a, da);
 190   else
 191     /* Max ULP is 0.501 if xx < 0.01588, otherwise ULP is 0.518.  */
 192     retval = do_sin (a, da);
 193
 194   return (n & 2) ? -retval : retval;
 195 }
 196
 197
 198 /*******************************************************************/
 199 /* An ultimate sin routine. Given an IEEE double machine number x   */
 200 /* it computes the correctly rounded (to nearest) value of sin(x)  */
 201 /*******************************************************************/
 202 #ifndef IN_SINCOS
 203 double
 204 SECTION
 205 __sin (double x)
 206 {
 207   double t, a, da;
 208   mynumber u;
 209   int4 k, m, n;
 210   double retval = 0;
 211
 212   SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
 213
 214   u.x = x;
 215   m = u.i[HIGH_HALF];
 216   k = 0x7fffffff & m;           /* no sign           */
 217   if (k < 0x3e500000)           /* if x->0 =>sin(x)=x */
 218     {
 219       math_check_force_underflow (x);
 220       retval = x;
 221     }
 222 /*--------------------------- 2^-26<|x|< 0.855469---------------------- */
 223   else if (k < 0x3feb6000)
 224     {
 225       /* Max ULP is 0.548.  */
 226       retval = do_sin (x, 0);
 227     }                           /*   else  if (k < 0x3feb6000)    */
 228
 229 /*----------------------- 0.855469  <|x|<2.426265  ----------------------*/
 230   else if (k < 0x400368fd)
 231     {
 232       t = hp0 - fabs (x);
 233       /* Max ULP is 0.51.  */
 234       retval = copysign (do_cos (t, hp1), x);
 235     }                           /*   else  if (k < 0x400368fd)    */
 236
 237 /*-------------------------- 2.426265<|x|< 105414350 ----------------------*/
 238   else if (k < 0x419921FB)
 239     {
 240       n = reduce_sincos (x, &a, &da);
 241       retval = do_sincos (a, da, n);
 242     }                           /*   else  if (k <  0x419921FB )    */
 243
 244 /* --------------------105414350 <|x| <2^1024------------------------------*/
 245   else if (k < 0x7ff00000)
 246     {
 247       n = __branred (x, &a, &da);
 248       retval = do_sincos (a, da, n);
 249     }
 250 /*--------------------- |x| > 2^1024 ----------------------------------*/
 251   else
 252     {
 253       if (k == 0x7ff00000 && u.i[LOW_HALF] == 0)
 254         __set_errno (EDOM);
 255       retval = x / x;
 256     }
 257
 258   return retval;
 259 }
 260
 261
 262 /*******************************************************************/
 263 /* An ultimate cos routine. Given an IEEE double machine number x   */
 264 /* it computes the correctly rounded (to nearest) value of cos(x)  */
 265 /*******************************************************************/
 266
 267 double
 268 SECTION
 269 __cos (double x)
 270 {
 271   double y, a, da;
 272   mynumber u;
 273   int4 k, m, n;
 274
 275   double retval = 0;
 276
 277   SET_RESTORE_ROUND_53BIT (FE_TONEAREST);
 278
 279   u.x = x;
 280   m = u.i[HIGH_HALF];
 281   k = 0x7fffffff & m;
 282
 283   /* |x|<2^-27 => cos(x)=1 */
 284   if (k < 0x3e400000)
 285     retval = 1.0;
 286
 287   else if (k < 0x3feb6000)
 288     {                           /* 2^-27 < |x| < 0.855469 */
 289       /* Max ULP is 0.51.  */
 290       retval = do_cos (x, 0);
 291     }                           /*   else  if (k < 0x3feb6000)    */
 292
 293   else if (k < 0x400368fd)
 294     { /* 0.855469  <|x|<2.426265  */ ;
 295       y = hp0 - fabs (x);
 296       a = y + hp1;
 297       da = (y - a) + hp1;
 298       /* Max ULP is 0.501 if xx < 0.01588 or 0.518 otherwise.
 299          Range reduction uses 106 bits here which is sufficient.  */
 300       retval = do_sin (a, da);
 301     }                           /*   else  if (k < 0x400368fd)    */
 302
 303   else if (k < 0x419921FB)
 304     {                           /* 2.426265<|x|< 105414350 */
 305       n = reduce_sincos (x, &a, &da);
 306       retval = do_sincos (a, da, n + 1);
 307     }                           /*   else  if (k <  0x419921FB )    */
 308
 309   /* 105414350 <|x| <2^1024 */
 310   else if (k < 0x7ff00000)
 311     {
 312       n = __branred (x, &a, &da);
 313       retval = do_sincos (a, da, n + 1);
 314     }
 315
 316   else
 317     {
 318       if (k == 0x7ff00000 && u.i[LOW_HALF] == 0)
 319         __set_errno (EDOM);
 320       retval = x / x;           /* |x| > 2^1024 */
 321     }
 322
 323   return retval;
 324 }
 325
 326 #ifndef __cos
 327 libm_alias_double (__cos, cos)
 328 #endif
 329 #ifndef __sin
 330 libm_alias_double (__sin, sin)
 331 #endif
 332
 333 #endif