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  1.  
  2. /* @(#)e_log.c 5.1 93/09/24 */
  3. /*
  4.  * ====================================================
  5.  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  6.  *
  7.  * Developed at SunPro, a Sun Microsystems, Inc. business.
  8.  * Permission to use, copy, modify, and distribute this
  9.  * software is freely granted, provided that this notice
  10.  * is preserved.
  11.  * ====================================================
  12.  */
  13.  
  14. /* __ieee754_log(x)
  15.  * Return the logrithm of x
  16.  *
  17.  * Method :                  
  18.  *   1. Argument Reduction: find k and f such that
  19.  *                      x = 2^k * (1+f),
  20.  *         where  sqrt(2)/2 < 1+f < sqrt(2) .
  21.  *
  22.  *   2. Approximation of log(1+f).
  23.  *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  24.  *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  25.  *               = 2s + s*R
  26.  *      We use a special Reme algorithm on [0,0.1716] to generate
  27.  *      a polynomial of degree 14 to approximate R The maximum error
  28.  *      of this polynomial approximation is bounded by 2**-58.45. In
  29.  *      other words,
  30.  *                      2      4      6      8      10      12      14
  31.  *          R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
  32.  *      (the values of Lg1 to Lg7 are listed in the program)
  33.  *      and
  34.  *          |      2          14          |     -58.45
  35.  *          | Lg1*s +...+Lg7*s    -  R(z) | <= 2
  36.  *          |                             |
  37.  *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  38.  *      In order to guarantee error in log below 1ulp, we compute log
  39.  *      by
  40.  *              log(1+f) = f - s*(f - R)        (if f is not too large)
  41.  *              log(1+f) = f - (hfsq - s*(hfsq+R)).     (better accuracy)
  42.  *     
  43.  *      3. Finally,  log(x) = k*ln2 + log(1+f).  
  44.  *                          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
  45.  *         Here ln2 is split into two floating point number:
  46.  *                      ln2_hi + ln2_lo,
  47.  *         where n*ln2_hi is always exact for |n| < 2000.
  48.  *
  49.  * Special cases:
  50.  *      log(x) is NaN with signal if x < 0 (including -INF) ;
  51.  *      log(+INF) is +INF; log(0) is -INF with signal;
  52.  *      log(NaN) is that NaN with no signal.
  53.  *
  54.  * Accuracy:
  55.  *      according to an error analysis, the error is always less than
  56.  *      1 ulp (unit in the last place).
  57.  *
  58.  * Constants:
  59.  * The hexadecimal values are the intended ones for the following
  60.  * constants. The decimal values may be used, provided that the
  61.  * compiler will convert from decimal to binary accurately enough
  62.  * to produce the hexadecimal values shown.
  63.  */
  64.  
  65. #include "fdlibm.h"
  66.  
  67. #ifndef _DOUBLE_IS_32BITS
  68.  
  69. #ifdef __STDC__
  70. static const double
  71. #else
  72. static double
  73. #endif
  74. ln2_hi  =  6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
  75. ln2_lo  =  1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
  76. two54   =  1.80143985094819840000e+16,  /* 43500000 00000000 */
  77. Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
  78. Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
  79. Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
  80. Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
  81. Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
  82. Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
  83. Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
  84.  
  85. #ifdef __STDC__
  86. static const double zero   =  0.0;
  87. #else
  88. static double zero   =  0.0;
  89. #endif
  90.  
  91. #ifdef __STDC__
  92.         double __ieee754_log(double x)
  93. #else
  94.         double __ieee754_log(x)
  95.         double x;
  96. #endif
  97. {
  98.         double hfsq,f,s,z,R,w,t1,t2,dk;
  99.         __int32_t k,hx,i,j;
  100.         __uint32_t lx;
  101.  
  102.         EXTRACT_WORDS(hx,lx,x);
  103.  
  104.         k=0;
  105.         if (hx < 0x00100000) {                  /* x < 2**-1022  */
  106.             if (((hx&0x7fffffff)|lx)==0)
  107.                 return -two54/zero;             /* log(+-0)=-inf */
  108.             if (hx<0) return (x-x)/zero;        /* log(-#) = NaN */
  109.             k -= 54; x *= two54; /* subnormal number, scale up x */
  110.             GET_HIGH_WORD(hx,x);
  111.         }
  112.         if (hx >= 0x7ff00000) return x+x;
  113.         k += (hx>>20)-1023;
  114.         hx &= 0x000fffff;
  115.         i = (hx+0x95f64)&0x100000;
  116.         SET_HIGH_WORD(x,hx|(i^0x3ff00000));     /* normalize x or x/2 */
  117.         k += (i>>20);
  118.         f = x-1.0;
  119.         if((0x000fffff&(2+hx))<3) {     /* |f| < 2**-20 */
  120.           if(f==zero) { if(k==0) return zero;  else {dk=(double)k;
  121.                                return dk*ln2_hi+dk*ln2_lo;}}
  122.             R = f*f*(0.5-0.33333333333333333*f);
  123.             if(k==0) return f-R; else {dk=(double)k;
  124.                      return dk*ln2_hi-((R-dk*ln2_lo)-f);}
  125.         }
  126.         s = f/(2.0+f);
  127.         dk = (double)k;
  128.         z = s*s;
  129.         i = hx-0x6147a;
  130.         w = z*z;
  131.         j = 0x6b851-hx;
  132.         t1= w*(Lg2+w*(Lg4+w*Lg6));
  133.         t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
  134.         i |= j;
  135.         R = t2+t1;
  136.         if(i>0) {
  137.             hfsq=0.5*f*f;
  138.             if(k==0) return f-(hfsq-s*(hfsq+R)); else
  139.                      return dk*ln2_hi-((hfsq-(s*(hfsq+R)+dk*ln2_lo))-f);
  140.         } else {
  141.             if(k==0) return f-s*(f-R); else
  142.                      return dk*ln2_hi-((s*(f-R)-dk*ln2_lo)-f);
  143.         }
  144. }
  145.  
  146. #endif /* defined(_DOUBLE_IS_32BITS) */
  147.