1*24604Szliu /* 2*24604Szliu * Copyright (c) 1985 Regents of the University of California. 3*24604Szliu * 4*24604Szliu * Use and reproduction of this software are granted in accordance with 5*24604Szliu * the terms and conditions specified in the Berkeley Software License 6*24604Szliu * Agreement (in particular, this entails acknowledgement of the programs' 7*24604Szliu * source, and inclusion of this notice) with the additional understanding 8*24604Szliu * that all recipients should regard themselves as participants in an 9*24604Szliu * ongoing research project and hence should feel obligated to report 10*24604Szliu * their experiences (good or bad) with these elementary function codes, 11*24604Szliu * using "sendbug 4bsd-bugs@BERKELEY", to the authors. 12*24604Szliu */ 13*24604Szliu 14*24604Szliu #ifndef lint 15*24604Szliu static char sccsid[] = "@(#)log__L.c 1.1 (ELEFUNT) 09/06/85"; 16*24604Szliu #endif not lint 17*24604Szliu 18*24604Szliu /* log__L(Z) 19*24604Szliu * LOG(1+X) - 2S X 20*24604Szliu * RETURN --------------- WHERE Z = S*S, S = ------- , 0 <= Z <= .0294... 21*24604Szliu * S 2 + X 22*24604Szliu * 23*24604Szliu * DOUBLE PRECISION (VAX D FORMAT 56 bits or IEEE DOUBLE 53 BITS) 24*24604Szliu * KERNEL FUNCTION FOR LOG; TO BE USED IN LOG1P, LOG, AND POW FUNCTIONS 25*24604Szliu * CODED IN C BY K.C. NG, 1/19/85; 26*24604Szliu * REVISED BY K.C. Ng, 2/3/85, 4/16/85. 27*24604Szliu * 28*24604Szliu * Method : 29*24604Szliu * 1. Polynomial approximation: let s = x/(2+x). 30*24604Szliu * Based on log(1+x) = log(1+s) - log(1-s) 31*24604Szliu * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 32*24604Szliu * 33*24604Szliu * (log(1+x) - 2s)/s is computed by 34*24604Szliu * 35*24604Szliu * z*(L1 + z*(L2 + z*(... (L7 + z*L8)...))) 36*24604Szliu * 37*24604Szliu * where z=s*s. (See the listing below for Lk's values.) The 38*24604Szliu * coefficients are obtained by a special Remez algorithm. 39*24604Szliu * 40*24604Szliu * Accuracy: 41*24604Szliu * Assuming no rounding error, the maximum magnitude of the approximation 42*24604Szliu * error (absolute) is 2**(-58.49) for IEEE double, and 2**(-63.63) 43*24604Szliu * for VAX D format. 44*24604Szliu * 45*24604Szliu * Constants: 46*24604Szliu * The hexadecimal values are the intended ones for the following constants. 47*24604Szliu * The decimal values may be used, provided that the compiler will convert 48*24604Szliu * from decimal to binary accurately enough to produce the hexadecimal values 49*24604Szliu * shown. 50*24604Szliu */ 51*24604Szliu 52*24604Szliu #ifdef VAX /* VAX D format (56 bits) */ 53*24604Szliu /* static double */ 54*24604Szliu /* L1 = 6.6666666666666703212E-1 , Hex 2^ 0 * .AAAAAAAAAAAAC5 */ 55*24604Szliu /* L2 = 3.9999999999970461961E-1 , Hex 2^ -1 * .CCCCCCCCCC2684 */ 56*24604Szliu /* L3 = 2.8571428579395698188E-1 , Hex 2^ -1 * .92492492F85782 */ 57*24604Szliu /* L4 = 2.2222221233634724402E-1 , Hex 2^ -2 * .E38E3839B7AF2C */ 58*24604Szliu /* L5 = 1.8181879517064680057E-1 , Hex 2^ -2 * .BA2EB4CC39655E */ 59*24604Szliu /* L6 = 1.5382888777946145467E-1 , Hex 2^ -2 * .9D8551E8C5781D */ 60*24604Szliu /* L7 = 1.3338356561139403517E-1 , Hex 2^ -2 * .8895B3907FCD92 */ 61*24604Szliu /* L8 = 1.2500000000000000000E-1 , Hex 2^ -2 * .80000000000000 */ 62*24604Szliu static long L1x[] = { 0xaaaa402a, 0xaac5aaaa}; 63*24604Szliu static long L2x[] = { 0xcccc3fcc, 0x2684cccc}; 64*24604Szliu static long L3x[] = { 0x49243f92, 0x578292f8}; 65*24604Szliu static long L4x[] = { 0x8e383f63, 0xaf2c39b7}; 66*24604Szliu static long L5x[] = { 0x2eb43f3a, 0x655ecc39}; 67*24604Szliu static long L6x[] = { 0x85513f1d, 0x781de8c5}; 68*24604Szliu static long L7x[] = { 0x95b33f08, 0xcd92907f}; 69*24604Szliu static long L8x[] = { 0x00003f00, 0x00000000}; 70*24604Szliu #define L1 (*(double*)L1x) 71*24604Szliu #define L2 (*(double*)L2x) 72*24604Szliu #define L3 (*(double*)L3x) 73*24604Szliu #define L4 (*(double*)L4x) 74*24604Szliu #define L5 (*(double*)L5x) 75*24604Szliu #define L6 (*(double*)L6x) 76*24604Szliu #define L7 (*(double*)L7x) 77*24604Szliu #define L8 (*(double*)L8x) 78*24604Szliu #else /* IEEE double */ 79*24604Szliu static double 80*24604Szliu L1 = 6.6666666666667340202E-1 , /*Hex 2^ -1 * 1.5555555555592 */ 81*24604Szliu L2 = 3.9999999999416702146E-1 , /*Hex 2^ -2 * 1.999999997FF24 */ 82*24604Szliu L3 = 2.8571428742008753154E-1 , /*Hex 2^ -2 * 1.24924941E07B4 */ 83*24604Szliu L4 = 2.2222198607186277597E-1 , /*Hex 2^ -3 * 1.C71C52150BEA6 */ 84*24604Szliu L5 = 1.8183562745289935658E-1 , /*Hex 2^ -3 * 1.74663CC94342F */ 85*24604Szliu L6 = 1.5314087275331442206E-1 , /*Hex 2^ -3 * 1.39A1EC014045B */ 86*24604Szliu L7 = 1.4795612545334174692E-1 ; /*Hex 2^ -3 * 1.2F039F0085122 */ 87*24604Szliu #endif 88*24604Szliu 89*24604Szliu double log__L(z) 90*24604Szliu double z; 91*24604Szliu { 92*24604Szliu #ifdef VAX 93*24604Szliu return(z*(L1+z*(L2+z*(L3+z*(L4+z*(L5+z*(L6+z*(L7+z*L8)))))))); 94*24604Szliu #else /* IEEE double */ 95*24604Szliu return(z*(L1+z*(L2+z*(L3+z*(L4+z*(L5+z*(L6+z*L7))))))); 96*24604Szliu #endif 97*24604Szliu } 98