lib/libm/noieee_src/trig.h

   1 /*      $NetBSD: trig.h,v 1.6 2003/08/07 16:44:53 agc Exp $     */
   2 /*
   3  * Copyright (c) 1987, 1993
   4  *      The Regents of the University of California.  All rights reserved.
   5  *
   6  * Redistribution and use in source and binary forms, with or without
   7  * modification, are permitted provided that the following conditions
   8  * are met:
   9  * 1. Redistributions of source code must retain the above copyright
  10  *    notice, this list of conditions and the following disclaimer.
  11  * 2. Redistributions in binary form must reproduce the above copyright
  12  *    notice, this list of conditions and the following disclaimer in the
  13  *    documentation and/or other materials provided with the distribution.
  14  * 3. Neither the name of the University nor the names of its contributors
  15  *    may be used to endorse or promote products derived from this software
  16  *    without specific prior written permission.
  17  *
  18  * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
  19  * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  20  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
  21  * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
  22  * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
  23  * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
  24  * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  25  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
  26  * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
  27  * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
  28  * SUCH DAMAGE.
  29  *
  30  *      @(#)trig.h      8.1 (Berkeley) 6/4/93
  31  */
  32
  33 vc(thresh, 2.6117239648121182150E-1 ,b863,3f85,6ea0,6b02, -1, .85B8636B026EA0)
  34 vc(PIo4,   7.8539816339744830676E-1 ,0fda,4049,68c2,a221,  0, .C90FDAA22168C2)
  35 vc(PIo2,   1.5707963267948966135E0  ,0fda,40c9,68c2,a221,  1, .C90FDAA22168C2)
  36 vc(PI3o4,  2.3561944901923449203E0  ,cbe3,4116,0e92,f999,  2, .96CBE3F9990E92)
  37 vc(PI,     3.1415926535897932270E0  ,0fda,4149,68c2,a221,  2, .C90FDAA22168C2)
  38 vc(PI2,    6.2831853071795864540E0  ,0fda,41c9,68c2,a221,  3, .C90FDAA22168C2)
  39
  40 ic(thresh, 2.6117239648121182150E-1 , -2, 1.0B70C6D604DD4)
  41 ic(PIo4,   7.8539816339744827900E-1 , -1, 1.921FB54442D18)
  42 ic(PIo2,   1.5707963267948965580E0  ,  0, 1.921FB54442D18)
  43 ic(PI3o4,  2.3561944901923448370E0  ,  1, 1.2D97C7F3321D2)
  44 ic(PI,     3.1415926535897931160E0  ,  1, 1.921FB54442D18)
  45 ic(PI2,    6.2831853071795862320E0  ,  2, 1.921FB54442D18)
  46
  47 #ifdef vccast
  48 #define thresh  vccast(thresh)
  49 #define PIo4    vccast(PIo4)
  50 #define PIo2    vccast(PIo2)
  51 #define PI3o4   vccast(PI3o4)
  52 #define PI      vccast(PI)
  53 #define PI2     vccast(PI2)
  54 #endif
  55
  56 #ifdef national
  57 static long fmaxx[]     = { 0xffffffff, 0x7fefffff};
  58 #define   fmax    (*(double*)fmaxx)
  59 #endif  /* national */
  60
  61 #ifdef _LIBM_DECLARE
  62 const double
  63         __zero = 0,
  64         __one = 1,
  65         __negone = -1,
  66         __half = 1.0/2.0,
  67 #ifdef __vax__
  68         __small = 1E-9, /* 1+small**2 == 1; better values for small:
  69                           *             small   = 1.5E-9 for VAX D
  70                           *                     = 1.2E-8 for IEEE Double
  71                           *                     = 2.8E-10 for IEEE Extended
  72                           */
  73         __big = 1E18;   /* big := 1/(small**2) */
  74 #else
  75         __small = 1E-10, /* 1+small**2 == 1; better values for small:
  76                           *             small   = 1.5E-9 for VAX D
  77                           *                     = 1.2E-8 for IEEE Double
  78                           *                     = 2.8E-10 for IEEE Extended
  79                           */
  80         __big = 1E20;   /* big := 1/(small**2) */
  81 #endif
  82 #else
  83 extern const double __zero, __one, __negone, __half, __small, __big;
  84 #endif
  85
  86 /* sin__S(x*x) ... re-implemented as a macro
  87  * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
  88  * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X)
  89  * CODED IN C BY K.C. NG, 1/21/85;
  90  * REVISED BY K.C. NG on 8/13/85.
  91  *
  92  *          sin(x*k) - x
  93  * RETURN  --------------- on [-PI/4,PI/4] , where k=pi/PI, PI is the rounded
  94  *                  x
  95  * value of pi in machine precision:
  96  *
  97  *      Decimal:
  98  *              pi = 3.141592653589793 23846264338327 .....
  99  *    53 bits   PI = 3.141592653589793 115997963 ..... ,
 100  *    56 bits   PI = 3.141592653589793 227020265 ..... ,
 101  *
 102  *      Hexadecimal:
 103  *              pi = 3.243F6A8885A308D313198A2E....
 104  *    53 bits   PI = 3.243F6A8885A30  =  2 * 1.921FB54442D18
 105  *    56 bits   PI = 3.243F6A8885A308 =  4 * .C90FDAA22168C2
 106  *
 107  * Method:
 108  *      1. Let z=x*x. Create a polynomial approximation to
 109  *          (sin(k*x)-x)/x  =  z*(S0 + S1*z^1 + ... + S5*z^5).
 110  *      Then
 111  *      sin__S(x*x) = z*(S0 + S1*z^1 + ... + S5*z^5)
 112  *
 113  *      The coefficient S's are obtained by a special Remez algorithm.
 114  *
 115  * Accuracy:
 116  *      In the absence of rounding error, the approximation has absolute error
 117  *      less than 2**(-61.11) for VAX D FORMAT, 2**(-57.45) for IEEE DOUBLE.
 118  *
 119  * Constants:
 120  * The hexadecimal values are the intended ones for the following constants.
 121  * The decimal values may be used, provided that the compiler will convert
 122  * from decimal to binary accurately enough to produce the hexadecimal values
 123  * shown.
 124  *
 125  */
 126
 127 vc(S0, -1.6666666666666646660E-1  ,aaaa,bf2a,aa71,aaaa,  -2, -.AAAAAAAAAAAA71)
 128 vc(S1,  8.3333333333297230413E-3  ,8888,3d08,477f,8888,  -6,  .8888888888477F)
 129 vc(S2, -1.9841269838362403710E-4  ,0d00,ba50,1057,cf8a, -12, -.D00D00CF8A1057)
 130 vc(S3,  2.7557318019967078930E-6  ,ef1c,3738,bedc,a326, -18,  .B8EF1CA326BEDC)
 131 vc(S4, -2.5051841873876551398E-8  ,3195,b3d7,e1d3,374c, -25, -.D73195374CE1D3)
 132 vc(S5,  1.6028995389845827653E-10 ,3d9c,3030,cccc,6d26, -32,  .B03D9C6D26CCCC)
 133 vc(S6, -6.2723499671769283121E-13 ,8d0b,ac30,ea82,7561, -40, -.B08D0B7561EA82)
 134
 135 ic(S0, -1.6666666666666463126E-1  ,  -3, -1.555555555550C)
 136 ic(S1,  8.3333333332992771264E-3  ,  -7,  1.111111110C461)
 137 ic(S2, -1.9841269816180999116E-4  , -13, -1.A01A019746345)
 138 ic(S3,  2.7557309793219876880E-6  , -19,  1.71DE3209CDCD9)
 139 ic(S4, -2.5050225177523807003E-8  , -26, -1.AE5C0E319A4EF)
 140 ic(S5,  1.5868926979889205164E-10 , -33,  1.5CF61DF672B13)
 141
 142 #ifdef vccast
 143 #define S0      vccast(S0)
 144 #define S1      vccast(S1)
 145 #define S2      vccast(S2)
 146 #define S3      vccast(S3)
 147 #define S4      vccast(S4)
 148 #define S5      vccast(S5)
 149 #define S6      vccast(S6)
 150 #endif
 151
 152 #if defined(__vax__)||defined(tahoe)
 153 #  define       sin__S(z)       (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*(S5+z*S6)))))))
 154 #else   /* defined(__vax__)||defined(tahoe) */
 155 #  define       sin__S(z)       (z*(S0+z*(S1+z*(S2+z*(S3+z*(S4+z*S5))))))
 156 #endif  /* defined(__vax__)||defined(tahoe) */
 157
 158 /* cos__C(x*x) ... re-implemented as a macro
 159  * DOUBLE PRECISION (VAX D FORMAT 56 BITS, IEEE DOUBLE 53 BITS)
 160  * STATIC KERNEL FUNCTION OF SIN(X), COS(X), AND TAN(X)
 161  * CODED IN C BY K.C. NG, 1/21/85;
 162  * REVISED BY K.C. NG on 8/13/85.
 163  *
 164  *                          x*x
 165  * RETURN   cos(k*x) - 1 + ----- on [-PI/4,PI/4],  where k = pi/PI,
 166  *                           2
 167  * PI is the rounded value of pi in machine precision :
 168  *
 169  *      Decimal:
 170  *              pi = 3.141592653589793 23846264338327 .....
 171  *    53 bits   PI = 3.141592653589793 115997963 ..... ,
 172  *    56 bits   PI = 3.141592653589793 227020265 ..... ,
 173  *
 174  *      Hexadecimal:
 175  *              pi = 3.243F6A8885A308D313198A2E....
 176  *    53 bits   PI = 3.243F6A8885A30  =  2 * 1.921FB54442D18
 177  *    56 bits   PI = 3.243F6A8885A308 =  4 * .C90FDAA22168C2
 178  *
 179  *
 180  * Method:
 181  *      1. Let z=x*x. Create a polynomial approximation to
 182  *          cos(k*x)-1+z/2  =  z*z*(C0 + C1*z^1 + ... + C5*z^5)
 183  *      then
 184  *      cos__C(z) =  z*z*(C0 + C1*z^1 + ... + C5*z^5)
 185  *
 186  *      The coefficient C's are obtained by a special Remez algorithm.
 187  *
 188  * Accuracy:
 189  *      In the absence of rounding error, the approximation has absolute error
 190  *      less than 2**(-64) for VAX D FORMAT, 2**(-58.3) for IEEE DOUBLE.
 191  *
 192  *
 193  * Constants:
 194  * The hexadecimal values are the intended ones for the following constants.
 195  * The decimal values may be used, provided that the compiler will convert
 196  * from decimal to binary accurately enough to produce the hexadecimal values
 197  * shown.
 198  */
 199
 200 vc(C0,  4.1666666666666504759E-2  ,aaaa,3e2a,a9f0,aaaa,  -4,  .AAAAAAAAAAA9F0)
 201 vc(C1, -1.3888888888865302059E-3  ,0b60,bbb6,0cca,b60a,  -9, -.B60B60B60A0CCA)
 202 vc(C2,  2.4801587285601038265E-5  ,0d00,38d0,098f,cdcd, -15,  .D00D00CDCD098F)
 203 vc(C3, -2.7557313470902390219E-7  ,f27b,b593,e805,b593, -21, -.93F27BB593E805)
 204 vc(C4,  2.0875623401082232009E-9  ,74c8,320f,3ff0,fa1e, -28,  .8F74C8FA1E3FF0)
 205 vc(C5, -1.1355178117642986178E-11 ,c32d,ae47,5a63,0a5c, -36, -.C7C32D0A5C5A63)
 206
 207 ic(C0,  4.1666666666666504759E-2  ,  -5,  1.555555555553E)
 208 ic(C1, -1.3888888888865301516E-3  , -10, -1.6C16C16C14199)
 209 ic(C2,  2.4801587269650015769E-5  , -16,  1.A01A01971CAEB)
 210 ic(C3, -2.7557304623183959811E-7  , -22, -1.27E4F1314AD1A)
 211 ic(C4,  2.0873958177697780076E-9  , -29,  1.1EE3B60DDDC8C)
 212 ic(C5, -1.1250289076471311557E-11 , -37, -1.8BD5986B2A52E)
 213
 214 #ifdef vccast
 215 #define C0      vccast(C0)
 216 #define C1      vccast(C1)
 217 #define C2      vccast(C2)
 218 #define C3      vccast(C3)
 219 #define C4      vccast(C4)
 220 #define C5      vccast(C5)
 221 #endif
 222
 223 #define cos__C(z)       (z*z*(C0+z*(C1+z*(C2+z*(C3+z*(C4+z*C5))))))