libm/e_log.c

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