sysdeps/libm-ieee754/e_log.c

   1 /* @(#)e_log.c 5.1 93/09/24 */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunPro, a Sun Microsystems, Inc. business.
   7  * Permission to use, copy, modify, and distribute this
   8  * software is freely granted, provided that this notice
   9  * is preserved.
  10  * ====================================================
  11  */
  12 /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/25,
  13    for performance improvement on pipelined processors.
  14 */
  15
  16 #if defined(LIBM_SCCS) && !defined(lint)
  17 static char rcsid[] = "$NetBSD: e_log.c,v 1.8 1995/05/10 20:45:49 jtc Exp $";
  18 #endif
  19
  20 /* __ieee754_log(x)
  21  * Return the logarithm of x
  22  *
  23  * Method :
  24  *   1. Argument Reduction: find k and f such that
  25  *                      x = 2^k * (1+f),
  26  *         where  sqrt(2)/2 < 1+f < sqrt(2) .
  27  *
  28  *   2. Approximation of log(1+f).
  29  *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  30  *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  31  *               = 2s + s*R
  32  *      We use a special Reme algorithm on [0,0.1716] to generate
  33  *      a polynomial of degree 14 to approximate R The maximum error
  34  *      of this polynomial approximation is bounded by 2**-58.45. In
  35  *      other words,
  36  *                      2      4      6      8      10      12      14
  37  *          R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
  38  *      (the values of Lg1 to Lg7 are listed in the program)
  39  *      and
  40  *          |      2          14          |     -58.45
  41  *          | Lg1*s +...+Lg7*s    -  R(z) | <= 2
  42  *          |                             |
  43  *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  44  *      In order to guarantee error in log below 1ulp, we compute log
  45  *      by
  46  *              log(1+f) = f - s*(f - R)        (if f is not too large)
  47  *              log(1+f) = f - (hfsq - s*(hfsq+R)).     (better accuracy)
  48  *
  49  *      3. Finally,  log(x) = k*ln2 + log(1+f).
  50  *                          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
  51  *         Here ln2 is split into two floating point number:
  52  *                      ln2_hi + ln2_lo,
  53  *         where n*ln2_hi is always exact for |n| < 2000.
  54  *
  55  * Special cases:
  56  *      log(x) is NaN with signal if x < 0 (including -INF) ;
  57  *      log(+INF) is +INF; log(0) is -INF with signal;
  58  *      log(NaN) is that NaN with no signal.
  59  *
  60  * Accuracy:
  61  *      according to an error analysis, the error is always less than
  62  *      1 ulp (unit in the last place).
  63  *
  64  * Constants:
  65  * The hexadecimal values are the intended ones for the following
  66  * constants. The decimal values may be used, provided that the
  67  * compiler will convert from decimal to binary accurately enough
  68  * to produce the hexadecimal values shown.
  69  */
  70
  71 #include "math.h"
  72 #include "math_private.h"
  73 #define half Lg[8]
  74 #define two Lg[9]
  75 #ifdef __STDC__
  76 static const double
  77 #else
  78 static double
  79 #endif
  80 ln2_hi  =  6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
  81 ln2_lo  =  1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
  82 two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
  83  Lg[] = {0.0,
  84  6.666666666666735130e-01,  /* 3FE55555 55555593 */
  85  3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
  86  2.857142874366239149e-01,  /* 3FD24924 94229359 */
  87  2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
  88  1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
  89  1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
  90  1.479819860511658591e-01,  /* 3FC2F112 DF3E5244 */
  91  0.5,
  92  2.0};
  93 #ifdef __STDC__
  94 static const double zero   =  0.0;
  95 #else
  96 static double zero   =  0.0;
  97 #endif
  98
  99 #ifdef __STDC__
 100         double __ieee754_log(double x)
 101 #else
 102         double __ieee754_log(x)
 103         double x;
 104 #endif
 105 {
 106         double hfsq,f,s,z,R,w,dk,t11,t12,t21,t22,w2,zw2;
 107 #ifdef DO_NOT_USE_THIS
 108         double t1,t2;
 109 #endif
 110         int32_t k,hx,i,j;
 111         u_int32_t lx;
 112
 113         EXTRACT_WORDS(hx,lx,x);
 114
 115         k=0;
 116         if (hx < 0x00100000) {                  /* x < 2**-1022  */
 117             if (((hx&0x7fffffff)|lx)==0)
 118                 return -two54/(x-x);            /* log(+-0)=-inf */
 119             if (hx<0) return (x-x)/(x-x);       /* log(-#) = NaN */
 120             k -= 54; x *= two54; /* subnormal number, scale up x */
 121             GET_HIGH_WORD(hx,x);
 122         }
 123         if (hx >= 0x7ff00000) return x+x;
 124         k += (hx>>20)-1023;
 125         hx &= 0x000fffff;
 126         i = (hx+0x95f64)&0x100000;
 127         SET_HIGH_WORD(x,hx|(i^0x3ff00000));     /* normalize x or x/2 */
 128         k += (i>>20);
 129         f = x-1.0;
 130         if((0x000fffff&(2+hx))<3) {     /* |f| < 2**-20 */
 131             if(f==zero) {
 132               if(k==0) return zero;  else {dk=(double)k;
 133                                            return dk*ln2_hi+dk*ln2_lo;}
 134             }
 135             R = f*f*(half-0.33333333333333333*f);
 136             if(k==0) return f-R; else {dk=(double)k;
 137                      return dk*ln2_hi-((R-dk*ln2_lo)-f);}
 138         }
 139         s = f/(two+f);
 140         dk = (double)k;
 141         z = s*s;
 142         i = hx-0x6147a;
 143         w = z*z;
 144         j = 0x6b851-hx;
 145 #ifdef DO_NOT_USE_THIS
 146         t1= w*(Lg2+w*(Lg4+w*Lg6));
 147         t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
 148         R = t2+t1;
 149 #else
 150         t21 = Lg[5]+w*Lg[7]; w2=w*w;
 151         t22 = Lg[1]+w*Lg[3]; zw2=z*w2;
 152         t11 = Lg[4]+w*Lg[6];
 153         t12 = w*Lg[2];
 154         R = t12 + w2*t11 + z*t22 + zw2*t21;
 155 #endif
 156         i |= j;
 157         if(i>0) {
 158             hfsq=0.5*f*f;
 159             if(k==0) return f-(hfsq-s*(hfsq+R)); else
 160                      return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
 161         } else {
 162             if(k==0) return f-s*(f-R); else
 163                      return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
 164         }
 165 }