1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * 4 * Use and reproduction of this software are granted in accordance with 5 * the terms and conditions specified in the Berkeley Software License 6 * Agreement (in particular, this entails acknowledgement of the programs' 7 * source, and inclusion of this notice) with the additional understanding 8 * that all recipients should regard themselves as participants in an 9 * ongoing research project and hence should feel obligated to report 10 * their experiences (good or bad) with these elementary function codes, 11 * using "sendbug 4bsd-bugs@BERKELEY", to the authors. 12 */ 13 14 #ifndef lint 15 static char sccsid[] = 16 "@(#)log__L.c 1.2 (Berkeley) 8/21/85; 1.2 (ucb.elefunt) 09/11/85"; 17 #endif not lint 18 19 /* log__L(Z) 20 * LOG(1+X) - 2S X 21 * RETURN --------------- WHERE Z = S*S, S = ------- , 0 <= Z <= .0294... 22 * S 2 + X 23 * 24 * DOUBLE PRECISION (VAX D FORMAT 56 bits or IEEE DOUBLE 53 BITS) 25 * KERNEL FUNCTION FOR LOG; TO BE USED IN LOG1P, LOG, AND POW FUNCTIONS 26 * CODED IN C BY K.C. NG, 1/19/85; 27 * REVISED BY K.C. Ng, 2/3/85, 4/16/85. 28 * 29 * Method : 30 * 1. Polynomial approximation: let s = x/(2+x). 31 * Based on log(1+x) = log(1+s) - log(1-s) 32 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 33 * 34 * (log(1+x) - 2s)/s is computed by 35 * 36 * z*(L1 + z*(L2 + z*(... (L7 + z*L8)...))) 37 * 38 * where z=s*s. (See the listing below for Lk's values.) The 39 * coefficients are obtained by a special Remez algorithm. 40 * 41 * Accuracy: 42 * Assuming no rounding error, the maximum magnitude of the approximation 43 * error (absolute) is 2**(-58.49) for IEEE double, and 2**(-63.63) 44 * for VAX D format. 45 * 46 * Constants: 47 * The hexadecimal values are the intended ones for the following constants. 48 * The decimal values may be used, provided that the compiler will convert 49 * from decimal to binary accurately enough to produce the hexadecimal values 50 * shown. 51 */ 52 53 #ifdef VAX /* VAX D format (56 bits) */ 54 /* static double */ 55 /* L1 = 6.6666666666666703212E-1 , Hex 2^ 0 * .AAAAAAAAAAAAC5 */ 56 /* L2 = 3.9999999999970461961E-1 , Hex 2^ -1 * .CCCCCCCCCC2684 */ 57 /* L3 = 2.8571428579395698188E-1 , Hex 2^ -1 * .92492492F85782 */ 58 /* L4 = 2.2222221233634724402E-1 , Hex 2^ -2 * .E38E3839B7AF2C */ 59 /* L5 = 1.8181879517064680057E-1 , Hex 2^ -2 * .BA2EB4CC39655E */ 60 /* L6 = 1.5382888777946145467E-1 , Hex 2^ -2 * .9D8551E8C5781D */ 61 /* L7 = 1.3338356561139403517E-1 , Hex 2^ -2 * .8895B3907FCD92 */ 62 /* L8 = 1.2500000000000000000E-1 , Hex 2^ -2 * .80000000000000 */ 63 static long L1x[] = { 0xaaaa402a, 0xaac5aaaa}; 64 static long L2x[] = { 0xcccc3fcc, 0x2684cccc}; 65 static long L3x[] = { 0x49243f92, 0x578292f8}; 66 static long L4x[] = { 0x8e383f63, 0xaf2c39b7}; 67 static long L5x[] = { 0x2eb43f3a, 0x655ecc39}; 68 static long L6x[] = { 0x85513f1d, 0x781de8c5}; 69 static long L7x[] = { 0x95b33f08, 0xcd92907f}; 70 static long L8x[] = { 0x00003f00, 0x00000000}; 71 #define L1 (*(double*)L1x) 72 #define L2 (*(double*)L2x) 73 #define L3 (*(double*)L3x) 74 #define L4 (*(double*)L4x) 75 #define L5 (*(double*)L5x) 76 #define L6 (*(double*)L6x) 77 #define L7 (*(double*)L7x) 78 #define L8 (*(double*)L8x) 79 #else /* IEEE double */ 80 static double 81 L1 = 6.6666666666667340202E-1 , /*Hex 2^ -1 * 1.5555555555592 */ 82 L2 = 3.9999999999416702146E-1 , /*Hex 2^ -2 * 1.999999997FF24 */ 83 L3 = 2.8571428742008753154E-1 , /*Hex 2^ -2 * 1.24924941E07B4 */ 84 L4 = 2.2222198607186277597E-1 , /*Hex 2^ -3 * 1.C71C52150BEA6 */ 85 L5 = 1.8183562745289935658E-1 , /*Hex 2^ -3 * 1.74663CC94342F */ 86 L6 = 1.5314087275331442206E-1 , /*Hex 2^ -3 * 1.39A1EC014045B */ 87 L7 = 1.4795612545334174692E-1 ; /*Hex 2^ -3 * 1.2F039F0085122 */ 88 #endif 89 90 double log__L(z) 91 double z; 92 { 93 #ifdef VAX 94 return(z*(L1+z*(L2+z*(L3+z*(L4+z*(L5+z*(L6+z*(L7+z*L8)))))))); 95 #else /* IEEE double */ 96 return(z*(L1+z*(L2+z*(L3+z*(L4+z*(L5+z*(L6+z*L7))))))); 97 #endif 98 } 99