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 /*
  23  * Copyright 2011 Nexenta Systems, Inc.  All rights reserved.
  24  */
  25 /*
  26  * Copyright 2006 Sun Microsystems, Inc.  All rights reserved.
  27  * Use is subject to license terms.
  28  */
  29 
  30 /* INDENT OFF */
  31 /*
  32  * __k_tanl( long double x;  long double y; int k )
  33  * kernel tan/cotan function on [-pi/4, pi/4], pi/4 ~ 0.785398164
  34  * Input x is assumed to be bounded by ~pi/4 in magnitude.
  35  * Input y is the tail of x.
  36  * Input k indicate -- tan if k=0; else -1/tan
  37  *
  38  * Table look up algorithm
  39  *      1. by tan(-x) = -tan(x), need only to consider positive x
  40  *      2. if x < 5/32 = [0x3ffc4000, 0] = 0.15625 , then
  41  *           if x < 2^-57 (hx < 0x3fc40000 0), set w=x with inexact if x !=  0
  42  *           else
  43  *              z = x*x;
  44  *              w = x + (y+(x*z)*(t1+z*(t2+z*(t3+z*(t4+z*(t5+z*t6))))))
  45  *         return (k == 0 ? w : 1/w);
  46  *      3. else
  47  *              ht = (hx + 0x400)&0x7ffff800        (round x to a break point t)
  48  *              lt = 0
  49  *              i  = (hy-0x3ffc4000)>>11; (i<=64)
  50  *              x' = (x - t)+y                  (|x'| ~<= 2^-7)
  51  *         By
  52  *              tan(t+x')
  53  *                = (tan(t)+tan(x'))/(1-tan(x')tan(t))
  54  *         We have
  55  *                           sin(x')+tan(t)*(tan(t)*sin(x'))
  56  *                = tan(t) + -------------------------------    for k=0
  57  *                              cos(x') - tan(t)*sin(x')
  58  *
  59  *                           cos(x') - tan(t)*sin(x')
  60  *                = - --------------------------------------    for k=1
  61  *                     tan(t) + tan(t)*(cos(x')-1) + sin(x')
  62  *
  63  *
  64  *         where        tan(t) is from the table,
  65  *                      sin(x') = x + pp1*x^3 + ...+ pp5*x^11
  66  *                      cos(x') = 1 + qq1*x^2 + ...+ qq5*x^10
  67  */
  68 
  69 #include "libm.h"
  70 
  71 #include <sys/isa_defs.h>
  72 
  73 extern const long double _TBL_tanl_hi[], _TBL_tanl_lo[];
  74 static const long double
  75 one     = 1.0,
  76 /*
  77  * |sin(x) - (x+pp1*x^3+...+ pp5*x^11)| <= 2^-122.32 for |x|<1/64
  78  */
  79 pp1     = -1.666666666666666666666666666586782940810e-0001L,
  80 pp2     =  8.333333333333333333333003723660929317540e-0003L,
  81 pp3     = -1.984126984126984076045903483778337804470e-0004L,
  82 pp4     =  2.755731922361906641319723106210900949413e-0006L,
  83 pp5     = -2.505198398570947019093998469135012057673e-0008L,
  84 /*
  85  *                   2           10        -123.84
  86  * |cos(x) - (1+qq1*x +...+ qq5*x  )| <= 2        for |x|<=1/128
  87  */
  88 qq1     = -4.999999999999999999999999999999378373641e-0001L,
  89 qq2     =  4.166666666666666666666665478399327703130e-0002L,
  90 qq3     = -1.388888888888888888058211230618051613494e-0003L,
  91 qq4     =  2.480158730156105377771585658905303111866e-0005L,
  92 qq5     = -2.755728099762526325736488376695157008736e-0007L,
  93 /*
  94  * |tan(x) - (x+t1*x^3+...+t6*x^13)|
  95  * |------------------------------ | <= 2^-59.73 for |x|<0.15625
  96  * |                x              |
  97  */
  98 t1      =  3.333333333333333333333333333333423342490e-0001L,
  99 t2      =  1.333333333333333333333333333093838744537e-0001L,
 100 t3      =  5.396825396825396825396827906318682662250e-0002L,
 101 t4      =  2.186948853615520282185576976994418486911e-0002L,
 102 t5      =  8.863235529902196573354554519991152936246e-0003L,
 103 t6      =  3.592128036572480064652191427543994878790e-0003L,
 104 t7      =  1.455834387051455257856833807581901305474e-0003L,
 105 t8      =  5.900274409318599857829983256201725587477e-0004L,
 106 t9      =  2.391291152117265181501116961901122362937e-0004L,
 107 t10     =  9.691533169382729742394024173194981882375e-0005L,
 108 t11     =  3.927994733186415603228178184225780859951e-0005L,
 109 t12     =  1.588300018848323824227640064883334101288e-0005L,
 110 t13     =  6.916271223396808311166202285131722231723e-0006L;
 111 /* INDENT ON */
 112 long double
 113 __k_tanl(long double x, long double y, int k) {
 114         long double a, t, z, w = 0.0, s, c;
 115         int *pt = (int *) &t, *px = (int *) &x;
 116         int i, j, hx, ix;
 117 
 118         t = 1.0;
 119 #if defined(__i386) || defined(__amd64)
 120         XTOI(px, hx);
 121 #else
 122         hx = px[0];
 123 #endif
 124         ix = hx & 0x7fffffff;
 125         if (ix < 0x3ffc4000) {
 126                 if (ix < 0x3fc60000) {
 127                         if ((i = (int) x) == 0) /* generate inexact */
 128                                 w = x;
 129                 } else {
 130                         z = x * x;
 131                         if (ix < 0x3ff30000) /* 2**-12 */
 132                                 t = z * (t1 + z * (t2 + z * (t3 + z * t4)));
 133                         else
 134                                 t = z * (t1 + z * (t2 + z * (t3 + z * (t4 +
 135                                         z * (t5 + z * (t6 + z * (t7 + z *
 136                                         (t8 + z * (t9 + z * (t10 + z * (t11 +
 137                                         z * (t12 + z * t13))))))))))));
 138                         t = y + x * t;
 139                         w = x + t;
 140                 }
 141                 return (k == 0 ? w : -one / w);
 142         }
 143         j = (ix + 0x400) & 0x7ffff800;
 144         i = (j - 0x3ffc4000) >> 11;
 145 #if defined(__i386) || defined(__amd64)
 146         ITOX(j, pt);
 147 #else
 148         pt[0] = j;
 149 #endif
 150         if (hx > 0)
 151                 x = y - (t - x);
 152         else
 153                 x = (-y) - (t + x);
 154         a = _TBL_tanl_hi[i];
 155         z = x * x;
 156         /* cos(x)-1 */
 157         t = z * (qq1 + z * (qq2 + z * (qq3 + z * (qq4 + z * qq5))));
 158         /* sin(x) */
 159         s = x * (one + z * (pp1 + z * (pp2 + z * (pp3 + z * (pp4 + z *
 160                 pp5)))));
 161         if (k == 0) {
 162                 w = a * s;
 163                 t = _TBL_tanl_lo[i] + (s + a * w) / (one - (w - t));
 164                 return (hx < 0 ? -a - t : a + t);
 165         } else {
 166                 w = s + a * t;
 167                 c = w + _TBL_tanl_lo[i];
 168                 z = (one - (a * s - t));
 169                 return (hx >= 0 ? z / (-a - c) : z / (a + c));
 170         }
 171 }