sysdeps/generic/trig.h

   1 /*
   2  * Copyright (c) 1987, 1993
   3  *      The Regents of the University of California.  All rights reserved.
   4  *
   5  * Redistribution and use in source and binary forms, with or without
   6  * modification, are permitted provided that the following conditions
   7  * are met:
   8  * 1. Redistributions of source code must retain the above copyright
   9  *    notice, this list of conditions and the following disclaimer.
  10  * 2. Redistributions in binary form must reproduce the above copyright
  11  *    notice, this list of conditions and the following disclaimer in the
  12  *    documentation and/or other materials provided with the distribution.
  13  * 3. All advertising materials mentioning features or use of this software
  14  *    must display the following acknowledgement:
  15  *      This product includes software developed by the University of
  16  *      California, Berkeley and its contributors.
  17  * 4. Neither the name of the University nor the names of its contributors
  18  *    may be used to endorse or promote products derived from this software
  19  *    without specific prior written permission.
  20  *
  21  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
  22  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  23  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  24  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
  25  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  26  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  27  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  28  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  29  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  30  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  31  * SUCH DAMAGE.
  32  *
  33  *      @(#)trig.h      8.1 (Berkeley) 6/4/93
  34  */
  35
  36 #include "mathimpl.h"
  37
  38 vc(thresh, 2.6117239648121182150E-1 ,b863,3f85,6ea0,6b02, -1, .85B8636B026EA0)
  39 vc(PIo4,   7.8539816339744830676E-1 ,0fda,4049,68c2,a221,  0, .C90FDAA22168C2)
  40 vc(PIo2,   1.5707963267948966135E0  ,0fda,40c9,68c2,a221,  1, .C90FDAA22168C2)
  41 vc(PI3o4,  2.3561944901923449203E0  ,cbe3,4116,0e92,f999,  2, .96CBE3F9990E92)
  42 vc(PI,     3.1415926535897932270E0  ,0fda,4149,68c2,a221,  2, .C90FDAA22168C2)
  43 vc(PI2,    6.2831853071795864540E0  ,0fda,41c9,68c2,a221,  3, .C90FDAA22168C2)
  44
  45 ic(thresh, 2.6117239648121182150E-1 , -2, 1.0B70C6D604DD4)
  46 ic(PIo4,   7.8539816339744827900E-1 , -1, 1.921FB54442D18)
  47 ic(PIo2,   1.5707963267948965580E0  ,  0, 1.921FB54442D18)
  48 ic(PI3o4,  2.3561944901923448370E0  ,  1, 1.2D97C7F3321D2)
  49 ic(PI,     3.1415926535897931160E0  ,  1, 1.921FB54442D18)
  50 ic(PI2,    6.2831853071795862320E0  ,  2, 1.921FB54442D18)
  51
  52 #ifdef vccast
  53 #define thresh  vccast(thresh)
  54 #define PIo4    vccast(PIo4)
  55 #define PIo2    vccast(PIo2)
  56 #define PI3o4   vccast(PI3o4)
  57 #define PI      vccast(PI)
  58 #define PI2     vccast(PI2)
  59 #endif
  60
  61 #ifdef national
  62 static long fmaxx[]     = { 0xffffffff, 0x7fefffff};
  63 #define   fmax    (*(double*)fmaxx)
  64 #endif  /* national */
  65
  66 static const double
  67         zero = 0,
  68         one = 1,
  69         negone = -1,
  70         half = 1.0/2.0,
  71         small = 1E-10,  /* 1+small**2 == 1; better values for small:
  72                          *              small   = 1.5E-9 for VAX D
  73                          *                      = 1.2E-8 for IEEE Double
  74                          *                      = 2.8E-10 for IEEE Extended
  75                          */
  76         big = 1E20;     /* big := 1/(small**2) */
  77
  78 /* sin__S(x*x) ... re-implemented as a macro
  79  * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
  80  * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X)
  81  * CODED IN C BY K.C. NG, 1/21/85;
  82  * REVISED BY K.C. NG on 8/13/85.
  83  *
  84  *          sin(x*k) - x
  85  * RETURN  --------------- on [-PI/4,PI/4] , where k=pi/PI, PI is the rounded
  86  *                  x
  87  * value of pi in machine precision:
  88  *
  89  *      Decimal:
  90  *              pi = 3.141592653589793 23846264338327 .....
  91  *    53 bits   PI = 3.141592653589793 115997963 ..... ,
  92  *    56 bits   PI = 3.141592653589793 227020265 ..... ,
  93  *
  94  *      Hexadecimal:
  95  *              pi = 3.243F6A8885A308D313198A2E....
  96  *    53 bits   PI = 3.243F6A8885A30  =  2 * 1.921FB54442D18
  97  *    56 bits   PI = 3.243F6A8885A308 =  4 * .C90FDAA22168C2
  98  *
  99  * Method:
 100  *      1. Let z=x*x. Create a polynomial approximation to
 101  *          (sin(k*x)-x)/x  =  z*(S0 + S1*z^1 + ... + S5*z^5).
 102  *      Then
 103  *      sin__S(x*x) = z*(S0 + S1*z^1 + ... + S5*z^5)
 104  *
 105  *      The coefficient S's are obtained by a special Remez algorithm.
 106  *
 107  * Accuracy:
 108  *      In the absence of rounding error, the approximation has absolute error
 109  *      less than 2**(-61.11) for VAX D FORMAT, 2**(-57.45) for IEEE DOUBLE.
 110  *
 111  * Constants:
 112  * The hexadecimal values are the intended ones for the following constants.
 113  * The decimal values may be used, provided that the compiler will convert
 114  * from decimal to binary accurately enough to produce the hexadecimal values
 115  * shown.
 116  *
 117  */
 118
 119 vc(S0, -1.6666666666666646660E-1  ,aaaa,bf2a,aa71,aaaa,  -2, -.AAAAAAAAAAAA71)
 120 vc(S1,  8.3333333333297230413E-3  ,8888,3d08,477f,8888,  -6,  .8888888888477F)
 121 vc(S2, -1.9841269838362403710E-4  ,0d00,ba50,1057,cf8a, -12, -.D00D00CF8A1057)
 122 vc(S3,  2.7557318019967078930E-6  ,ef1c,3738,bedc,a326, -18,  .B8EF1CA326BEDC)
 123 vc(S4, -2.5051841873876551398E-8  ,3195,b3d7,e1d3,374c, -25, -.D73195374CE1D3)
 124 vc(S5,  1.6028995389845827653E-10 ,3d9c,3030,cccc,6d26, -32,  .B03D9C6D26CCCC)
 125 vc(S6, -6.2723499671769283121E-13 ,8d0b,ac30,ea82,7561, -40, -.B08D0B7561EA82)
 126
 127 ic(S0, -1.6666666666666463126E-1  ,  -3, -1.555555555550C)
 128 ic(S1,  8.3333333332992771264E-3  ,  -7,  1.111111110C461)
 129 ic(S2, -1.9841269816180999116E-4  , -13, -1.A01A019746345)
 130 ic(S3,  2.7557309793219876880E-6  , -19,  1.71DE3209CDCD9)
 131 ic(S4, -2.5050225177523807003E-8  , -26, -1.AE5C0E319A4EF)
 132 ic(S5,  1.5868926979889205164E-10 , -33,  1.5CF61DF672B13)
 133
 134 #ifdef vccast
 135 #define S0      vccast(S0)
 136 #define S1      vccast(S1)
 137 #define S2      vccast(S2)
 138 #define S3      vccast(S3)
 139 #define S4      vccast(S4)
 140 #define S5      vccast(S5)
 141 #define S6      vccast(S6)
 142 #endif
 143
 144 #if defined(vax)||defined(tahoe)
 145 #  define       sin__S(z)       (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*(S5+z*S6)))))))
 146 #else   /* defined(vax)||defined(tahoe) */
 147 #  define       sin__S(z)       (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*S5))))))
 148 #endif  /* defined(vax)||defined(tahoe) */
 149
 150 /* cos__C(x*x) ... re-implemented as a macro
 151  * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS)
 152  * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X)
 153  * CODED IN C BY K.C. NG, 1/21/85;
 154  * REVISED BY K.C. NG on 8/13/85.
 155  *
 156  *                          x*x
 157  * RETURN   cos(k*x) - 1 + ----- on [-PI/4,PI/4],  where k = pi/PI,
 158  *                           2
 159  * PI is the rounded value of pi in machine precision :
 160  *
 161  *      Decimal:
 162  *              pi = 3.141592653589793 23846264338327 .....
 163  *    53 bits   PI = 3.141592653589793 115997963 ..... ,
 164  *    56 bits   PI = 3.141592653589793 227020265 ..... ,
 165  *
 166  *      Hexadecimal:
 167  *              pi = 3.243F6A8885A308D313198A2E....
 168  *    53 bits   PI = 3.243F6A8885A30  =  2 * 1.921FB54442D18
 169  *    56 bits   PI = 3.243F6A8885A308 =  4 * .C90FDAA22168C2
 170  *
 171  *
 172  * Method:
 173  *      1. Let z=x*x. Create a polynomial approximation to
 174  *          cos(k*x)-1+z/2  =  z*z*(C0 + C1*z^1 + ... + C5*z^5)
 175  *      then
 176  *      cos__C(z) =  z*z*(C0 + C1*z^1 + ... + C5*z^5)
 177  *
 178  *      The coefficient C's are obtained by a special Remez algorithm.
 179  *
 180  * Accuracy:
 181  *      In the absence of rounding error, the approximation has absolute error
 182  *      less than 2**(-64) for VAX D FORMAT, 2**(-58.3) for IEEE DOUBLE.
 183  *
 184  *
 185  * Constants:
 186  * The hexadecimal values are the intended ones for the following constants.
 187  * The decimal values may be used, provided that the compiler will convert
 188  * from decimal to binary accurately enough to produce the hexadecimal values
 189  * shown.
 190  */
 191
 192 vc(C0,  4.1666666666666504759E-2  ,aaaa,3e2a,a9f0,aaaa,  -4,  .AAAAAAAAAAA9F0)
 193 vc(C1, -1.3888888888865302059E-3  ,0b60,bbb6,0cca,b60a,  -9, -.B60B60B60A0CCA)
 194 vc(C2,  2.4801587285601038265E-5  ,0d00,38d0,098f,cdcd, -15,  .D00D00CDCD098F)
 195 vc(C3, -2.7557313470902390219E-7  ,f27b,b593,e805,b593, -21, -.93F27BB593E805)
 196 vc(C4,  2.0875623401082232009E-9  ,74c8,320f,3ff0,fa1e, -28,  .8F74C8FA1E3FF0)
 197 vc(C5, -1.1355178117642986178E-11 ,c32d,ae47,5a63,0a5c, -36, -.C7C32D0A5C5A63)
 198
 199 ic(C0,  4.1666666666666504759E-2  ,  -5,  1.555555555553E)
 200 ic(C1, -1.3888888888865301516E-3  , -10, -1.6C16C16C14199)
 201 ic(C2,  2.4801587269650015769E-5  , -16,  1.A01A01971CAEB)
 202 ic(C3, -2.7557304623183959811E-7  , -22, -1.27E4F1314AD1A)
 203 ic(C4,  2.0873958177697780076E-9  , -29,  1.1EE3B60DDDC8C)
 204 ic(C5, -1.1250289076471311557E-11 , -37, -1.8BD5986B2A52E)
 205
 206 #ifdef vccast
 207 #define C0      vccast(C0)
 208 #define C1      vccast(C1)
 209 #define C2      vccast(C2)
 210 #define C3      vccast(C3)
 211 #define C4      vccast(C4)
 212 #define C5      vccast(C5)
 213 #endif
 214
 215 #define cos__C(z)       (z*z*(C0+z*(C1+z*(C2+z*(C3+z*(C4+z*C5))))))