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| //
// ====================================================
// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
//
// Developed at SunSoft, a Sun Microsystems, Inc. business.
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
//
// 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.
//
var ln2_hi = 6.93147180369123816490e-01; // 3fe62e42 fee00000
var ln2_lo = 1.90821492927058770002e-10; // 3dea39ef 35793c76
var two54 = 1.80143985094819840000e+16; // 43500000 00000000
var Lp1 = 6.666666666666735130e-01; // 3FE55555 55555593
var Lp2 = 3.999999999940941908e-01; // 3FD99999 9997FA04
var Lp3 = 2.857142874366239149e-01; // 3FD24924 94229359
var Lp4 = 2.222219843214978396e-01; // 3FCC71C5 1D8E78AF
var Lp5 = 1.818357216161805012e-01; // 3FC74664 96CB03DE
var Lp6 = 1.531383769920937332e-01; // 3FC39A09 D078C69F
var Lp7 = 1.479819860511658591e-01; // 3FC2F112 DF3E5244
function log1p (x) {
var hx = _DoubleHi(x);
var ax = hx & 0x7fffffff;
var k = 1;
var f = 0;
var hu = 1;
var c = 0;
var u = 0;
if (hx < 0x3fda827a) {
// x < 0.41422
if (ax >= 0x3ff00000) {
// x <= -1: log1p(-1) = inf, log1p(x<-1) = NaN
if (x == -1) {
return -two54/0;
} else {
return (x - x) / (x - x);
}
}
if (ax < 0x3e200000) {
// |x| < 2^-29
if ((two54 + x > 0) && (ax < 0x3c900000)) {
// |x| < 2^-54, so just return x
return x;
} else {
return x - x*x*0.5;
}
}
// (int) 0xbfd2bec3 = -0x402d413d
if ((hx > 0) || (hx <= -0x402D413D)) {
// -.2929 < x < 0.41422
k = 0;
f = x;
hu = 1;
}
}
if (hx >= 0x7ff00000)
return x + x;
if (k != 0) {
if (hx < 0x43400000) {
// x < 9.007199254740992d15
u = 1 + x;
hu = _DoubleHi(u);
k = (hu >> 20) - 1023;
c = (k > 0) ? 1 - (u - x) : x - (u - 1);
c = c / u;
/* istanbul ignore if */
//if (verbose > 0)
console.log("x, u, hu, k, c = " + x + ", " + u + ", " + hu + ", " + k + ", " + c);
} else {
u = x;
hu = _DoubleHi(u);
k = (hu >> 20) - 1023;
c = 0;
}
hu = hu & 0xfffff;
if (hu < 0x6a09e) {
// Normalize u
u = _ConstructDouble(hu | 0x3ff00000, _DoubleLo(u));
} else {
++k;
u = _ConstructDouble(hu | 0x3fe00000, _DoubleLo(u));
hu = (0x00100000 - hu) >> 2;
}
f = u - 1;
}
var hfsq = 0.5 * f * f;
/* istanbul ignore if */
//if (verbose > 0)
console.log("hu, f = " + hu + ", " + f);
if (hu == 0) {
// |f| < 2^-20;
console.log("k = " + k);
if (f == 0) {
Iif (k == 0) {
return 0.0;
} else {
return k*ln2_hi + (c + k*ln2_lo);
}
}
var R = hfsq * (1 - (2/3)*f);
Iif (k == 0) {
return f - R;
} else {
return k*ln2_hi-((R-(k*ln2_lo+c))-f);
}
}
var s = f/(2.0+f);
var z = s*s;
var R = z*(Lp1+z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))));
/* istanbul ignore if */
Iif (verbose > 0)
console.log("hfsq, s, z, r = " + hfsq + ", " + s + ", " + z + ", " + R);
if (k==0) {
return f-(hfsq-s*(hfsq+R));
} else {
return k*ln2_hi-((hfsq-(s*(hfsq+R)+(k*ln2_lo+c)))-f);
}
}
|