StdLib/LibC/Math/e_log.c

   1 /** @file
   2   Compute the logrithm of x.
   3
   4   Copyright (c) 2010 - 2011, Intel Corporation. All rights reserved.<BR>
   5   This program and the accompanying materials are licensed and made available under
   6   the terms and conditions of the BSD License that accompanies this distribution.
   7   The full text of the license may be found at
   8   http://opensource.org/licenses/bsd-license.
   9
  10   THE PROGRAM IS DISTRIBUTED UNDER THE BSD LICENSE ON AN "AS IS" BASIS,
  11   WITHOUT WARRANTIES OR REPRESENTATIONS OF ANY KIND, EITHER EXPRESS OR IMPLIED.
  12
  13  * ====================================================
  14  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  15  *
  16  * Developed at SunPro, a Sun Microsystems, Inc. business.
  17  * Permission to use, copy, modify, and distribute this
  18  * software is freely granted, provided that this notice
  19  * is preserved.
  20  * ====================================================
  21
  22   e_log.c 5.1 93/09/24
  23   NetBSD: e_log.c,v 1.12 2002/05/26 22:01:51 wiz Exp
  24 **/
  25 #include  <LibConfig.h>
  26 #include  <sys/EfiCdefs.h>
  27
  28 #if defined(_MSC_VER)           /* Handle Microsoft VC++ compiler specifics. */
  29   // potential divide by 0 -- near line 118, (x-x)/zero is on purpose
  30   #pragma warning ( disable : 4723 )
  31 #endif
  32
  33 /* __ieee754_log(x)
  34  * Return the logrithm of x
  35  *
  36  * Method :
  37  *   1. Argument Reduction: find k and f such that
  38  *      x = 2^k * (1+f),
  39  *     where  sqrt(2)/2 < 1+f < sqrt(2) .
  40  *
  41  *   2. Approximation of log(1+f).
  42  *  Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  43  *     = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  44  *         = 2s + s*R
  45  *      We use a special Reme algorithm on [0,0.1716] to generate
  46  *  a polynomial of degree 14 to approximate R The maximum error
  47  *  of this polynomial approximation is bounded by 2**-58.45. In
  48  *  other words,
  49  *            2      4      6      8      10      12      14
  50  *      R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
  51  *    (the values of Lg1 to Lg7 are listed in the program)
  52  *  and
  53  *      |      2          14          |     -58.45
  54  *      | Lg1*s +...+Lg7*s    -  R(z) | <= 2
  55  *      |                             |
  56  *  Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  57  *  In order to guarantee error in log below 1ulp, we compute log
  58  *  by
  59  *    log(1+f) = f - s*(f - R)  (if f is not too large)
  60  *    log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
  61  *
  62  *  3. Finally,  log(x) = k*ln2 + log(1+f).
  63  *          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
  64  *     Here ln2 is split into two floating point number:
  65  *      ln2_hi + ln2_lo,
  66  *     where n*ln2_hi is always exact for |n| < 2000.
  67  *
  68  * Special cases:
  69  *  log(x) is NaN with signal if x < 0 (including -INF) ;
  70  *  log(+INF) is +INF; log(0) is -INF with signal;
  71  *  log(NaN) is that NaN with no signal.
  72  *
  73  * Accuracy:
  74  *  according to an error analysis, the error is always less than
  75  *  1 ulp (unit in the last place).
  76  *
  77  * Constants:
  78  * The hexadecimal values are the intended ones for the following
  79  * constants. The decimal values may be used, provided that the
  80  * compiler will convert from decimal to binary accurately enough
  81  * to produce the hexadecimal values shown.
  82  */
  83
  84 #include "math.h"
  85 #include "math_private.h"
  86 #include  <errno.h>
  87
  88 static const double
  89 ln2_hi  =  6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
  90 ln2_lo  =  1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
  91 two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
  92 Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
  93 Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
  94 Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
  95 Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
  96 Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
  97 Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
  98 Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
  99
 100 static const double zero   =  0.0;
 101
 102 double
 103 __ieee754_log(double x)
 104 {
 105   double hfsq,f,s,z,R,w,t1,t2,dk;
 106   int32_t k,hx,i,j;
 107   u_int32_t lx;
 108
 109   EXTRACT_WORDS(hx,lx,x);
 110
 111   k=0;
 112   if (hx < 0x00100000) {      /* x < 2**-1022  */
 113     if (((hx&0x7fffffff)|lx)==0)
 114       return -two54/zero;     /* log(+-0)=-inf */
 115     if (hx<0) {
 116       errno = EDOM;
 117       return (x-x)/zero;      /* log(-#) = NaN */
 118     }
 119       k -= 54; x *= two54;    /* subnormal number, scale up x */
 120       GET_HIGH_WORD(hx,x);
 121   }
 122   if (hx >= 0x7ff00000) return x+x;
 123   k += (hx>>20)-1023;
 124   hx &= 0x000fffff;
 125   i = (hx+0x95f64)&0x100000;
 126   SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */
 127   k += (i>>20);
 128   f = x-1.0;
 129   if((0x000fffff&(2+hx))<3) { /* |f| < 2**-20 */
 130     if(f==zero) { if(k==0) return zero;  else {dk=(double)k;
 131          return dk*ln2_hi+dk*ln2_lo;}
 132     }
 133     R = f*f*(0.5-0.33333333333333333*f);
 134     if(k==0) return f-R; else {dk=(double)k;
 135            return dk*ln2_hi-((R-dk*ln2_lo)-f);}
 136   }
 137   s = f/(2.0+f);
 138   dk = (double)k;
 139   z = s*s;
 140   i = hx-0x6147a;
 141   w = z*z;
 142   j = 0x6b851-hx;
 143   t1= w*(Lg2+w*(Lg4+w*Lg6));
 144   t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
 145   i |= j;
 146   R = t2+t1;
 147   if(i>0) {
 148       hfsq=0.5*f*f;
 149       if(k==0) return f-(hfsq-s*(hfsq+R)); else
 150          return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
 151   } else {
 152       if(k==0) return f-s*(f-R); else
 153          return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
 154   }
 155 }