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| //
// ====================================================
// Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
//
// Permission to use, copy, modify, and distribute this
// software is freely granted, provided that this notice
// is preserved.
// ====================================================
//
// Translated to Javascript by Raymond Toy (rtoy@google.com)
// expm1(x)
// Returns exp(x)-1, the exponential of x minus 1.
//
// Method
// 1. Argument reduction:
// Given x, find r and integer k such that
//
// x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
//
// Here a correction term c will be computed to compensate
// the error in r when rounded to a floating-point number.
//
// 2. Approximating expm1(r) by a special rational function on
// the interval [0,0.34658]:
// Since
// r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
// we define R1(r*r) by
// r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
// That is,
// R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
// = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
// = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
// We use a special Remes algorithm on [0,0.347] to generate
// a polynomial of degree 5 in r*r to approximate R1. The
// maximum error of this polynomial approximation is bounded
// by 2**-61. In other words,
// R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
// where Q1 = -1.6666666666666567384E-2,
// Q2 = 3.9682539681370365873E-4,
// Q3 = -9.9206344733435987357E-6,
// Q4 = 2.5051361420808517002E-7,
// Q5 = -6.2843505682382617102E-9;
// (where z=r*r, and the values of Q1 to Q5 are listed below)
// with error bounded by
// | 5 | -61
// | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
// | |
//
// expm1(r) = exp(r)-1 is then computed by the following
// specific way which minimize the accumulation rounding error:
// 2 3
// r r [ 3 - (R1 + R1*r/2) ]
// expm1(r) = r + --- + --- * [--------------------]
// 2 2 [ 6 - r*(3 - R1*r/2) ]
//
// To compensate the error in the argument reduction, we use
// expm1(r+c) = expm1(r) + c + expm1(r)*c
// ~ expm1(r) + c + r*c
// Thus c+r*c will be added in as the correction terms for
// expm1(r+c). Now rearrange the term to avoid optimization
// screw up:
// ( 2 2 )
// ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
// expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
// ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
// ( )
//
// = r - E
// 3. Scale back to obtain expm1(x):
// From step 1, we have
// expm1(x) = either 2^k*[expm1(r)+1] - 1
// = or 2^k*[expm1(r) + (1-2^-k)]
// 4. Implementation notes:
// (A). To save one multiplication, we scale the coefficient Qi
// to Qi*2^i, and replace z by (x^2)/2.
// (B). To achieve maximum accuracy, we compute expm1(x) by
// (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
// (ii) if k=0, return r-E
// (iii) if k=-1, return 0.5*(r-E)-0.5
// (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
// else return 1.0+2.0*(r-E);
// (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
// (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
// (vii) return 2^k(1-((E+2^-k)-r))
//
// Special cases:
// expm1(INF) is INF, expm1(NaN) is NaN;
// expm1(-INF) is -1, and
// for finite argument, only expm1(0)=0 is exact.
//
// Accuracy:
// according to an error analysis, the error is always less than
// 1 ulp (unit in the last place).
//
// Misc. info.
// For IEEE double
// if x > 7.09782712893383973096e+02 then expm1(x) overflow
//
// 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.
var huge = 1e300;
var tiny = 1e-300;
var o_threshold = 7.09782712893383973096e+02; // 0x40862E42, 0xFEFA39EF
var ln2_hi = 6.93147180369123816490e-01; // 0x3fe62e42, 0xfee00000
var ln2_lo = 1.90821492927058770002e-10; // 0x3dea39ef, 0x35793c76
var invln2 = 1.44269504088896338700e+00; // 0x3ff71547, 0x652b82fe
// scaled coefficients related to expm1
var Q1 = -3.33333333333331316428e-02; // BFA11111 111110F4
var Q2 = 1.58730158725481460165e-03; // 3F5A01A0 19FE5585
var Q3 = -7.93650757867487942473e-05; // BF14CE19 9EAADBB7
var Q4 = 4.00821782732936239552e-06; // 3ED0CFCA 86E65239
var Q5 = -2.01099218183624371326e-07; // BE8AFDB7 6E09C32D
function expm1(x)
{
var y;
var hi;
var lo;
var k;
var t;
var c;
var hx = _DoubleHi(x);
var xsb = hx & 0x80000000; // sign bit of x
// y = |x|
if (xsb == 0)
y = x;
else
y = -x;
hx &= 0x7fffffff; // High word of |x|
// Filter out huge and non-finite argument
/* istanbul ignore if */
Iif (verbose > 0)
console.log("hx = " + hx);
if (hx >= 0x4043687a) { // if |x| ~=> 56*ln2
if (hx >= 0x40862e42) { // if |x| >= 709.78
if (hx >= 0x7ff00000) {
if (((hx & 0xfffff) | _DoubleLo(x)) != 0) {
// NaN
return x + x;
} else {
// expm1(inf) = inf; expm1(-inf) = -1
return xsb == 0 ? x : -1.0;
}
}
if (x > o_threshold) {
// Overflow
return huge * huge;
}
}
if (xsb != 0) {
// x < -56*ln2 so return -1, with inexact. For Javascript,
// we can probably skip the stuff to set the inexact flag
// and just return -1.
// I don't think the else case is ever reachable because x
// < -38 and tiny = 1e300 so x + tiny < 0 always. JS
// doesn't have an inexact, so maybe just delete the if
// test?
/* istanbul ignore else */
Eif (x + tiny < 0) // This raises inexact.
return tiny - 1;
}
}
// Argument reduction
/* istanbul ignore if */
Iif (verbose > 0)
console.log("Arg reduction");
if (hx > 0x3fd62e42) { // if |x| > 0.5 * ln2
/* istanbul ignore if */
Iif (verbose > 0)
console.log("hx > 0.5*log2");
if (hx < 0x3ff0a2b2) { // and |x| < 1.5 * ln2
/* istanbul ignore if */
Iif (verbose > 0)
console.log("hx < 1.5 * log2");
if (xsb == 0) {
hi = x - ln2_hi;
lo = ln2_lo;
k = 1;
} else {
hi = x + ln2_hi;
lo = -ln2_lo;
k = -1;
}
} else {
// k = (invln2 * x + ((xsb == 0) ? 0.5 : -0.5));
k = (invln2 * x + ((xsb == 0) ? 0.5 : -0.5)) | 0;
/* istanbul ignore if */
Iif (verbose > 0)
console.log("invln2*x, k = " + (invln2*x) + ", " + k);
t = k;
// t*ln2_hi is exact here.
hi = x - t * ln2_hi;
lo = t * ln2_lo;
}
x = hi - lo;
c = (hi - x) - lo;
} else if (hx < 0x3c900000) {
// When |x| < 2^-54, we can // return x, setting inexact flags
// when x != 0. The inexact stuff is probably not needed in
// Javascript.
/* istanbul ignore if */
Iif (verbose > 0)
console.log("hx < 0x3c900000");
t = huge + x;
return x - (t - (huge + x));
} else {
/* istanbul ignore if */
Iif (verbose > 0)
console.log("Fall through");
k = 0;
}
// x is now in primary range
/* istanbul ignore if */
Iif (verbose > 0)
console.log("In range, k = " + k);
var hfx = 0.5 * x;
var hxs = x * hfx;
var r1 = 1 + hxs * (Q1 + hxs * (Q2 + hxs * (Q3 + hxs * (Q4 + hxs * Q5))));
t = 3 - r1 * hfx;
var e = hxs * ((r1 - t) / (6 - x*t));
if (k == 0) {
// c is 0
return x - (x*e - hxs);
} else {
e = (x * (e - c) - c);
e -= hxs;
if (k == -1)
return 0.5 * (x - e) - 0.5;
if (k == 1) {
if (x < -0.25)
return -2 * (e - (x + 0.5));
else
return 1 + 2 * (x - e);
}
if (k <= -2 || k > 56) {
/* istanbul ignore if */
Iif (verbose > 0)
console.log("k <= -2 || k > 56: k = " + k);
// suffice to return exp(x)+1
y = 1 - (e - x);
// Add k to y's exponent
y = _ConstructDouble(_DoubleHi(y) + (k << 20), _DoubleLo(y));
/* istanbul ignore if */
Iif (verbose > 0)
console.log("New y = " + y + ", result = " + (y - 1));
return y - 1;
}
if (k < 20) {
// t = 1 - 2^k
t = _ConstructDouble(0x3ff00000 - (0x200000 >> k), 0);
y = t - (e - x);
// Add k to y's exponent
y = _ConstructDouble(_DoubleHi(y) + (k << 20), _DoubleLo(y));
} else {
// t = 2^-k
t = _ConstructDouble((0x3ff - k) << 20, 0);
y = x - (e + t);
y += 1;
// Add k to y's exponent
y = _ConstructDouble(_DoubleHi(y) + (k << 20), _DoubleLo(y));
}
}
return y;
}
|