usr/src/lib/libm/common/C/__sincos.c

   1 /*
   2  * CDDL HEADER START
   3  *
   4  * The contents of this file are subject to the terms of the
   5  * Common Development and Distribution License (the "License").
   6  * You may not use this file except in compliance with the License.
   7  *
   8  * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
   9  * or http://www.opensolaris.org/os/licensing.
  10  * See the License for the specific language governing permissions
  11  * and limitations under the License.
  12  *
  13  * When distributing Covered Code, include this CDDL HEADER in each
  14  * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
  15  * If applicable, add the following below this CDDL HEADER, with the
  16  * fields enclosed by brackets "[]" replaced with your own identifying
  17  * information: Portions Copyright [yyyy] [name of copyright owner]
  18  *
  19  * CDDL HEADER END
  20  */
  21 /*
  22  * Copyright 2011 Nexenta Systems, Inc.  All rights reserved.
  23  */
  24 /*
  25  * Copyright 2005 Sun Microsystems, Inc.  All rights reserved.
  26  * Use is subject to license terms.
  27  */
  28
  29 /* INDENT OFF */
  30 /*
  31  * double __k_sincos(double x, double y, double *c);
  32  * kernel sincos function on [-pi/4, pi/4], pi/4 ~ 0.785398164
  33  * Input x is assumed to be bounded by ~pi/4 in magnitude.
  34  * Input y is the tail of x.
  35  * return sin(x) with *c = cos(x)
  36  *
  37  * Accurate Table look-up algorithm by K.C. Ng, May, 1995.
  38  *
  39  * 1. Reduce x to x>0 by sin(-x)=-sin(x),cos(-x)=cos(x).
  40  * 2. For 0<= x < pi/4, let i = (64*x chopped)-10. Let d = x - a[i], where
  41  *    a[i] is a double that is close to (i+10.5)/64 and such that
  42  *    sin(a[i]) and cos(a[i]) is close to a double (with error less
  43  *    than 2**-8 ulp). Then
  44  *      cos(x) = cos(a[i]+d) = cos(a[i])cos(d) - sin(a[i])*sin(d)
  45  *             = TBL_cos_a[i]*(1+QQ1*d^2+QQ2*d^4) -
  46  *                      TBL_sin_a[i]*(d+PP1*d^3+PP2*d^5)
  47  *             = TBL_cos_a[i] + (TBL_cos_a[i]*d^2*(QQ1+QQ2*d^2) -
  48  *                      TBL_sin_a[i]*(d+PP1*d^3+PP2*d^5))
  49  *      sin(x) = sin(a[i]+d) = sin(a[i])cos(d) + cos(a[i])*sin(d)
  50  *             = TBL_sin_a[i]*(1+QQ1*d^2+QQ2*d^4) +
  51  *                      TBL_cos_a[i]*(d+PP1*d^3+PP2*d^5)
  52  *             = TBL_sin_a[i] + (TBL_sin_a[i]*d^2*(QQ1+QQ2*d^2) +
  53  *                      TBL_cos_a[i]*(d+PP1*d^3+PP2*d^5))
  54  *
  55  *    For |y| less than 10.5/64 = 0.1640625, use
  56  *      sin(y) = y + y^3*(p1+y^2*(p2+y^2*(p3+y^2*p4)))
  57  *      cos(y) = 1 + y^2*(q1+y^2*(q2+y^2*(q3+y^2*q4)))
  58  *
  59  *    For |y| less than 0.008, use
  60  *      sin(y) = y + y^3*(pp1+y^2*pp2)
  61  *      cos(y) = 1 + y^2*(qq1+y^2*qq2)
  62  *
  63  * Accuracy:
  64  *      TRIG(x) returns trig(x) nearly rounded (less than 1 ulp)
  65  */
  66
  67 #include "libm.h"
  68
  69 static const double sc[] = {
  70 /* ONE  = */  1.0,
  71 /* NONE = */ -1.0,
  72 /*
  73  * |sin(x) - (x+pp1*x^3+pp2*x^5)| <= 2^-58.79 for |x| < 0.008
  74  */
  75 /* PP1  = */ -0.166666666666316558867252052378889521480627858683055567,
  76 /* PP2  = */   .008333315652997472323564894248466758248475374977974017927,
  77 /*
  78  * |(sin(x) - (x+p1*x^3+...+p4*x^9)|
  79  * |------------------------------ | <= 2^-57.63 for |x| < 0.1953125
  80  * |                 x             |
  81  */
  82 /* P1   = */ -1.666666666666629669805215138920301589656e-0001,
  83 /* P2   = */  8.333333332390951295683993455280336376663e-0003,
  84 /* P3   = */ -1.984126237997976692791551778230098403960e-0004,
  85 /* P4   = */  2.753403624854277237649987622848330351110e-0006,
  86 /*
  87  * |cos(x) - (1+qq1*x^2+qq2*x^4)| <= 2^-55.99 for |x| <= 0.008 (0x3f80624d)
  88  */
  89 /* QQ1  = */ -0.4999999999975492381842911981948418542742729,
  90 /* QQ2  = */  0.041666542904352059294545209158357640398771740,
  91 /*
  92  * |cos(x) - (1+q1*x^2+...+q4*x^8)| <= 2^-55.86 for |x| <= 0.1640625 (10.5/64)
  93  */
  94 /* Q1   = */ -0.5,
  95 /* Q2   = */  4.166666666500350703680945520860748617445e-0002,
  96 /* Q3   = */ -1.388888596436972210694266290577848696006e-0003,
  97 /* Q4   = */  2.478563078858589473679519517892953492192e-0005,
  98 };
  99 /* INDENT ON */
 100
 101 #define ONE     sc[0]
 102 #define NONE    sc[1]
 103 #define PP1     sc[2]
 104 #define PP2     sc[3]
 105 #define P1      sc[4]
 106 #define P2      sc[5]
 107 #define P3      sc[6]
 108 #define P4      sc[7]
 109 #define QQ1     sc[8]
 110 #define QQ2     sc[9]
 111 #define Q1      sc[10]
 112 #define Q2      sc[11]
 113 #define Q3      sc[12]
 114 #define Q4      sc[13]
 115
 116 extern const double _TBL_sincos[], _TBL_sincosx[];
 117
 118 double
 119 __k_sincos(double x, double y, double *c) {
 120         double  z, w, s, v, p, q;
 121         int     i, j, n, hx, ix;
 122
 123         hx = ((int *)&x)[HIWORD];
 124         ix = hx & ~0x80000000;
 125
 126         if (ix <= 0x3fc50000) { /* |x| < 10.5/64 = 0.164062500 */
 127                 if (ix < 0x3e400000) {  /* |x| < 2**-27 */
 128                         if ((int)x == 0)
 129                                 *c = ONE;
 130                         return (x + y);
 131                 } else {
 132                         z = x * x;
 133                         if (ix < 0x3f800000) {  /* |x| < 0.008 */
 134                                 q = z * (QQ1 + z * QQ2);
 135                                 p = (x * z) * (PP1 + z * PP2) + y;
 136                         } else {
 137                                 q = z * ((Q1 + z * Q2) + (z * z) * (Q3 +
 138                                     z * Q4));
 139                                 p = (x * z) * ((P1 + z * P2) + (z * z) * (P3 +
 140                                     z * P4)) + y;
 141                         }
 142                         *c = ONE + q;
 143                         return (x + p);
 144                 }
 145         } else {                /* 0.164062500 < |x| < ~pi/4 */
 146                 n = ix >> 20;
 147                 i = (((ix >> 12) & 0xff) | 0x100) >> (0x401 - n);
 148                 j = i - 10;
 149                 if (hx < 0)
 150                         v = -y - (_TBL_sincosx[j] + x);
 151                 else
 152                         v = y - (_TBL_sincosx[j] - x);
 153                 s = v * v;
 154                 j <<= 1;
 155                 w = _TBL_sincos[j];
 156                 z = _TBL_sincos[j+1];
 157                 p = s * (PP1 + s * PP2);
 158                 q = s * (QQ1 + s * QQ2);
 159                 p = v + v * p;
 160                 *c = z - (w * p - z * q);
 161                 s = w * q + z * p;
 162                 return ((hx >= 0)? w + s : -(w + s));
 163         }
 164 }