github.com/afumu/libc@v0.0.6/musl/src/math/log1p.c (about)

     1  /* origin: FreeBSD /usr/src/lib/msun/src/s_log1p.c */
     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  /* double log1p(double x)
    13   * Return the natural logarithm of 1+x.
    14   *
    15   * Method :
    16   *   1. Argument Reduction: find k and f such that
    17   *                      1+x = 2^k * (1+f),
    18   *         where  sqrt(2)/2 < 1+f < sqrt(2) .
    19   *
    20   *      Note. If k=0, then f=x is exact. However, if k!=0, then f
    21   *      may not be representable exactly. In that case, a correction
    22   *      term is need. Let u=1+x rounded. Let c = (1+x)-u, then
    23   *      log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
    24   *      and add back the correction term c/u.
    25   *      (Note: when x > 2**53, one can simply return log(x))
    26   *
    27   *   2. Approximation of log(1+f): See log.c
    28   *
    29   *   3. Finally, log1p(x) = k*ln2 + log(1+f) + c/u. See log.c
    30   *
    31   * Special cases:
    32   *      log1p(x) is NaN with signal if x < -1 (including -INF) ;
    33   *      log1p(+INF) is +INF; log1p(-1) is -INF with signal;
    34   *      log1p(NaN) is that NaN with no signal.
    35   *
    36   * Accuracy:
    37   *      according to an error analysis, the error is always less than
    38   *      1 ulp (unit in the last place).
    39   *
    40   * Constants:
    41   * The hexadecimal values are the intended ones for the following
    42   * constants. The decimal values may be used, provided that the
    43   * compiler will convert from decimal to binary accurately enough
    44   * to produce the hexadecimal values shown.
    45   *
    46   * Note: Assuming log() return accurate answer, the following
    47   *       algorithm can be used to compute log1p(x) to within a few ULP:
    48   *
    49   *              u = 1+x;
    50   *              if(u==1.0) return x ; else
    51   *                         return log(u)*(x/(u-1.0));
    52   *
    53   *       See HP-15C Advanced Functions Handbook, p.193.
    54   */
    55  
    56  #include "libm.h"
    57  
    58  static const double
    59  ln2_hi = 6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
    60  ln2_lo = 1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
    61  Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
    62  Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
    63  Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
    64  Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
    65  Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
    66  Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
    67  Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
    68  
    69  double log1p(double x)
    70  {
    71  	union {double f; uint64_t i;} u = {x};
    72  	double_t hfsq,f,c,s,z,R,w,t1,t2,dk;
    73  	uint32_t hx,hu;
    74  	int k;
    75  
    76  	hx = u.i>>32;
    77  	k = 1;
    78  	if (hx < 0x3fda827a || hx>>31) {  /* 1+x < sqrt(2)+ */
    79  		if (hx >= 0xbff00000) {  /* x <= -1.0 */
    80  			if (x == -1)
    81  				return x/0.0; /* log1p(-1) = -inf */
    82  			return (x-x)/0.0;     /* log1p(x<-1) = NaN */
    83  		}
    84  		if (hx<<1 < 0x3ca00000<<1) {  /* |x| < 2**-53 */
    85  			/* underflow if subnormal */
    86  			if ((hx&0x7ff00000) == 0)
    87  				FORCE_EVAL((float)x);
    88  			return x;
    89  		}
    90  		if (hx <= 0xbfd2bec4) {  /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
    91  			k = 0;
    92  			c = 0;
    93  			f = x;
    94  		}
    95  	} else if (hx >= 0x7ff00000)
    96  		return x;
    97  	if (k) {
    98  		u.f = 1 + x;
    99  		hu = u.i>>32;
   100  		hu += 0x3ff00000 - 0x3fe6a09e;
   101  		k = (int)(hu>>20) - 0x3ff;
   102  		/* correction term ~ log(1+x)-log(u), avoid underflow in c/u */
   103  		if (k < 54) {
   104  			c = k >= 2 ? 1-(u.f-x) : x-(u.f-1);
   105  			c /= u.f;
   106  		} else
   107  			c = 0;
   108  		/* reduce u into [sqrt(2)/2, sqrt(2)] */
   109  		hu = (hu&0x000fffff) + 0x3fe6a09e;
   110  		u.i = (uint64_t)hu<<32 | (u.i&0xffffffff);
   111  		f = u.f - 1;
   112  	}
   113  	hfsq = 0.5*f*f;
   114  	s = f/(2.0+f);
   115  	z = s*s;
   116  	w = z*z;
   117  	t1 = w*(Lg2+w*(Lg4+w*Lg6));
   118  	t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
   119  	R = t2 + t1;
   120  	dk = k;
   121  	return s*(hfsq+R) + (dk*ln2_lo+c) - hfsq + f + dk*ln2_hi;
   122  }