1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * All rights reserved. 4 * 5 * %sccs.include.redist.c% 6 * 7 * All recipients should regard themselves as participants in an ongoing 8 * research project and hence should feel obligated to report their 9 * experiences (good or bad) with these elementary function codes, using 10 * the sendbug(8) program, to the authors. 11 */ 12 13 #ifndef lint 14 static char sccsid[] = "@(#)log__L.c 5.5 (Berkeley) 06/01/90"; 15 #endif /* not lint */ 16 17 /* log__L(Z) 18 * LOG(1+X) - 2S X 19 * RETURN --------------- WHERE Z = S*S, S = ------- , 0 <= Z <= .0294... 20 * S 2 + X 21 * 22 * DOUBLE PRECISION (VAX D FORMAT 56 bits or IEEE DOUBLE 53 BITS) 23 * KERNEL FUNCTION FOR LOG; TO BE USED IN LOG1P, LOG, AND POW FUNCTIONS 24 * CODED IN C BY K.C. NG, 1/19/85; 25 * REVISED BY K.C. Ng, 2/3/85, 4/16/85. 26 * 27 * Method : 28 * 1. Polynomial approximation: let s = x/(2+x). 29 * Based on log(1+x) = log(1+s) - log(1-s) 30 * = 2s + 2/3 s**3 + 2/5 s**5 + ....., 31 * 32 * (log(1+x) - 2s)/s is computed by 33 * 34 * z*(L1 + z*(L2 + z*(... (L7 + z*L8)...))) 35 * 36 * where z=s*s. (See the listing below for Lk's values.) The 37 * coefficients are obtained by a special Remez algorithm. 38 * 39 * Accuracy: 40 * Assuming no rounding error, the maximum magnitude of the approximation 41 * error (absolute) is 2**(-58.49) for IEEE double, and 2**(-63.63) 42 * for VAX D format. 43 * 44 * Constants: 45 * The hexadecimal values are the intended ones for the following constants. 46 * The decimal values may be used, provided that the compiler will convert 47 * from decimal to binary accurately enough to produce the hexadecimal values 48 * shown. 49 */ 50 51 #include "mathimpl.h" 52 53 vc(L1, 6.6666666666666703212E-1 ,aaaa,402a,aac5,aaaa, 0, .AAAAAAAAAAAAC5) 54 vc(L2, 3.9999999999970461961E-1 ,cccc,3fcc,2684,cccc, -1, .CCCCCCCCCC2684) 55 vc(L3, 2.8571428579395698188E-1 ,4924,3f92,5782,92f8, -1, .92492492F85782) 56 vc(L4, 2.2222221233634724402E-1 ,8e38,3f63,af2c,39b7, -2, .E38E3839B7AF2C) 57 vc(L5, 1.8181879517064680057E-1 ,2eb4,3f3a,655e,cc39, -2, .BA2EB4CC39655E) 58 vc(L6, 1.5382888777946145467E-1 ,8551,3f1d,781d,e8c5, -2, .9D8551E8C5781D) 59 vc(L7, 1.3338356561139403517E-1 ,95b3,3f08,cd92,907f, -2, .8895B3907FCD92) 60 vc(L8, 1.2500000000000000000E-1 ,0000,3f00,0000,0000, -2, .80000000000000) 61 62 ic(L1, 6.6666666666667340202E-1, -1, 1.5555555555592) 63 ic(L2, 3.9999999999416702146E-1, -2, 1.999999997FF24) 64 ic(L3, 2.8571428742008753154E-1, -2, 1.24924941E07B4) 65 ic(L4, 2.2222198607186277597E-1, -3, 1.C71C52150BEA6) 66 ic(L5, 1.8183562745289935658E-1, -3, 1.74663CC94342F) 67 ic(L6, 1.5314087275331442206E-1, -3, 1.39A1EC014045B) 68 ic(L7, 1.4795612545334174692E-1, -3, 1.2F039F0085122) 69 70 #ifdef vccast 71 #define L1 vccast(L1) 72 #define L2 vccast(L2) 73 #define L3 vccast(L3) 74 #define L4 vccast(L4) 75 #define L5 vccast(L5) 76 #define L6 vccast(L6) 77 #define L7 vccast(L7) 78 #define L8 vccast(L8) 79 #endif 80 81 double log__L(z) 82 double z; 83 { 84 #if defined(vax)||defined(tahoe) 85 return(z*(L1+z*(L2+z*(L3+z*(L4+z*(L5+z*(L6+z*(L7+z*L8)))))))); 86 #else /* defined(vax)||defined(tahoe) */ 87 return(z*(L1+z*(L2+z*(L3+z*(L4+z*(L5+z*(L6+z*L7))))))); 88 #endif /* defined(vax)||defined(tahoe) */ 89 } 90