25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * CDDL HEADER START
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25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Copyright 2011 Nexenta Systems, Inc. All rights reserved.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Copyright 2005 Sun Microsystems, Inc. All rights reserved.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Use is subject to license terms.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* INDENT OFF */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 1. Argument Reduction: find k and f such that
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 1+x = 2^k * (1+f),
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * where sqrt(2)/2 < 1+f < sqrt(2) .
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Note. If k=0, then f=x is exact. However, if k != 0, then f
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * may not be representable exactly. In that case, a correction
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * term is need. Let u=1+x rounded. Let c = (1+x)-u, then
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * and add back the correction term c/u.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (Note: when x > 2**53, one can simply return log(x))
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 2. Approximation of log1p(f).
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = 2s + 2/3 s**3 + 2/5 s**5 + .....,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * We use a special Reme algorithm on [0,0.1716] to generate
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * a polynomial of degree 14 to approximate R The maximum error
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * of this polynomial approximation is bounded by 2**-58.45. In
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * other words,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 2 4 6 8 10 12 14
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * (the values of Lp1 to Lp7 are listed in the program)
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * | 2 14 | -58.45
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * | Lp1*s +...+Lp7*s - R(z) | <= 2
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * In order to guarantee error in log below 1ulp, we compute log
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * log1p(f) = f - (hfsq - s*(hfsq+R)).
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 3. Finally, log1p(x) = k*ln2 + log1p(f).
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Here ln2 is splitted into two floating point number:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * ln2_hi + ln2_lo,
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * where n*ln2_hi is always exact for |n| < 2000.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Special cases:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * log1p(x) is NaN with signal if x < -1 (including -INF) ;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * log1p(+INF) is +INF; log1p(-1) is -INF with signal;
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * log1p(NaN) is that NaN with no signal.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * according to an error analysis, the error is always less than
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * 1 ulp (unit in the last place).
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * The hexadecimal values are the intended ones for the following
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * constants. The decimal values may be used, provided that the
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * compiler will convert from decimal to binary accurately enough
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * to produce the hexadecimal values shown.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * Note: Assuming log() return accurate answer, the following
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * algorithm can be used to compute log1p(x) to within a few ULP:
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * if (u == 1.0) return x ; else
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * return log(u)*(x/(u-1.0));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * See HP-15C Advanced Functions Handbook, p.193.
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* INDENT ON */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* ln2_hi */ 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* ln2_lo */ 1.90821492927058770002e-10, /* 3dea39ef 35793c76 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* two54 */ 1.80143985094819840000e+16, /* 43500000 00000000 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* Lp1 */ 6.666666666666735130e-01, /* 3FE55555 55555593 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* Lp2 */ 3.999999999940941908e-01, /* 3FD99999 9997FA04 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* Lp3 */ 2.857142874366239149e-01, /* 3FD24924 94229359 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* Lp4 */ 2.222219843214978396e-01, /* 3FCC71C5 1D8E78AF */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* Lp5 */ 1.818357216161805012e-01, /* 3FC74664 96CB03DE */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* Lp6 */ 1.531383769920937332e-01, /* 3FC39A09 D078C69F */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis/* Lp7 */ 1.479819860511658591e-01, /* 3FC2F112 DF3E5244 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis hx = ((int *)&x)[HIWORD]; /* high word of x */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if (ax >= 0x7ff00000) { /* x is inf or nan */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if (((hx - 0xfff00000) | ((int *)&x)[LOWORD]) == 0) /* -inf */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis return (x * x);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis return (_SVID_libm_err(x, x, x == -1.0 ? 43 : 44));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if (two54 + x > zero && /* raise inexact */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis return (x - x * x * 0.5);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis if (hx > 0 || hx <= (int)0xbfd2bec3) { /* -0.2929<x<0.41422 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* We will initialize 'c' here. */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis hu = ((int *)&u)[HIWORD]; /* high word of u */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis * correction term
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis c = k > 0 ? 1.0 - (u - x) : x - (u - 1.0);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis hu = ((int *)&u)[HIWORD]; /* high word of u */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis } else { /* normalize u/2 */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis /* We already initialized 'c' before, when (k != 0) */
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis return (k * ln2_hi + c);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis R = hfsq * (1.0 - 0.66666666666666666 * f);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis return (f - R);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis return (k * ln2_hi - ((R - (k * ln2_lo + c)) - f));
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis s = f / (2.0 + f);
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis R = z * (Lp1 + z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 +
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis return (k * ln2_hi - ((hfsq - (s * (hfsq + R) +
25c28e83beb90e7c80452a7c818c5e6f73a07dc8Piotr Jasiukajtis (k * ln2_lo + c))) - f));