s_log1p.cs

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/* double log1p(double x)
 *
 * Method :
 *   1. Argument Reduction: find k and f such that
 *                      1+x = 2^k * (1+f),
 *         where  sqrt(2)/2 < 1+f < sqrt(2) .
 *
 *      Note. If k=0, then f=x is exact. However, if k!=0, then f
 *      may not be representable exactly. In that case, a correction
 *      term is need. Let u=1+x rounded. Let c = (1+x)-u, then
 *      log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
 *      and add back the correction term c/u.
 *      (Note: when x > 2**53, one can simply return log(x))
 *
 *   2. Approximation of log1p(f).
 *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
 *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
 *               = 2s + s*R
 *      We use a special Reme algorithm on [0,0.1716] to generate
 *      a polynomial of degree 14 to approximate R The maximum error
 *      of this polynomial approximation is bounded by 2**-58.45. In
 *      other words,
 *                      2      4      6      8      10      12      14
 *          R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
 *      (the values of Lp1 to Lp7 are listed in the program)
 *      and
 *          |      2          14          |     -58.45
 *          | Lp1*s +...+Lp7*s    -  R(z) | <= 2
 *          |                             |
 *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
 *      In order to guarantee error in log below 1ulp, we compute log
 *      by
 *              log1p(f) = f - (hfsq - s*(hfsq+R)).
 *
 *      3. Finally, log1p(x) = k*ln2 + log1p(f).
 *                           = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
 *         Here ln2 is split into two floating point number:
 *                      ln2_hi + ln2_lo,
 *         where n*ln2_hi is always exact for |n| < 2000.
 *
 * Special cases:
 *      log1p(x) is NaN with signal if x < -1 (including -INF) ;
 *      log1p(+INF) is +INF; log1p(-1) is -INF with signal;
 *      log1p(NaN) is that NaN with no signal.
 *
 * Accuracy:
 *      according to an error analysis, the error is always less than
 *      1 ulp (unit in the last place).
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 *
 * Note: Assuming log() return accurate answer, the following
 *       algorithm can be used to compute log1p(x) to within a few ULP:
 *
 *              u = 1+x;
 *              if(u==1.0) return x ; else
 *                         return log(u)*(x/(u-1.0));
 *
 *       See HP-15C Advanced Functions Handbook, p.193.
 */
 
namespace IKVM.Runtime.Util.Java.Lang
{
    static partial class fdlibm
    {
        internal static double log1p(double x)
        {
            const double
    ln2_hi = 6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
    ln2_lo = 1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
    two54 = 1.80143985094819840000e+16,  /* 43500000 00000000 */
    Lp1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
    Lp2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
    Lp3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
    Lp4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
    Lp5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
    Lp6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
    Lp7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
 
            const double zero = 0.0;
 
            double hfsq, f = 0, c = 0, s, z, R, u;
            int k, hx, hu = 0, ax;
 
            hx = __HI(x);           /* high word of x */
            ax = hx & 0x7fffffff;
 
            k = 1;
            if (hx < 0x3FDA827A)
            {                  /* x < 0.41422  */
                if (ax >= 0x3ff00000)
                {                /* x <= -1.0 */
                    /*
                     * Added redundant test against hx to work around VC++
                     * code generation problem.
                     */
                    if (x == -1.0 && (hx == unchecked((int)0xbff00000))) /* log1p(-1)=-inf */
                        return -two54 / zero;
                    else
                        return (x - x) / (x - x);           /* log1p(x<-1)=NaN */
                }
                if (ax < 0x3e200000)
                {                 /* |x| < 2**-29 */
                    if (two54 + x > zero                 /* raise inexact */
                        && ax < 0x3c900000)            /* |x| < 2**-54 */
                        return x;
                    else
                        return x - x * x * 0.5;
                }
                if (hx > 0 || hx <= (unchecked((int)0xbfd2bec3)))
                {
                    k = 0; f = x; hu = 1;
                }  /* -0.2929<x<0.41422 */
            }
            if (hx >= 0x7ff00000) return x + x;
            if (k != 0)
            {
                if (hx < 0x43400000)
                {
                    u = 1.0 + x;
                    hu = __HI(u);           /* high word of u */
                    k = (hu >> 20) - 1023;
                    c = (k > 0) ? 1.0 - (u - x) : x - (u - 1.0);/* correction term */
                    c /= u;
                }
                else
                {
                    u = x;
                    hu = __HI(u);           /* high word of u */
                    k = (hu >> 20) - 1023;
                    c = 0;
                }
                hu &= 0x000fffff;
                if (hu < 0x6a09e)
                {
                    u = __HI(u, hu | 0x3ff00000);        /* normalize u */
                }
                else
                {
                    k += 1;
                    u = __HI(u, hu | 0x3fe00000);        /* normalize u/2 */
                    hu = (0x00100000 - hu) >> 2;
                }
                f = u - 1.0;
            }
            hfsq = 0.5 * f * f;
            if (hu == 0)
            {     /* |f| < 2**-20 */
                if (f == zero)
                {
                    if (k == 0) return zero;
                    else { c += k * ln2_lo; return k * ln2_hi + c; }
                }
                R = hfsq * (1.0 - 0.66666666666666666 * f);
                if (k == 0) return f - R;
                else
                    return k * ln2_hi - ((R - (k * ln2_lo + c)) - f);
            }
            s = f / (2.0 + f);
            z = s * s;
            R = z * (Lp1 + z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 + z * (Lp6 + z * Lp7))))));
            if (k == 0) return f - (hfsq - s * (hfsq + R));
            else
                return k * ln2_hi - ((hfsq - (s * (hfsq + R) + (k * ln2_lo + c))) - f);
        }
    }
}