mirror of
https://github.com/openjdk/jdk.git
synced 2025-08-27 23:04:50 +02:00
8301396: Port fdlibm expm1 to Java
Reviewed-by: bpb
This commit is contained in:
Joe Darcy
parent
3be5317b59
commit
34493248c0
4 changed files with 504 additions and 19 deletions
|
|
@ -985,4 +985,213 @@ class FdLibm {
|
|||
}
|
||||
}
|
||||
}
|
||||
|
||||
/* 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 Reme 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.
|
||||
*/
|
||||
static class Expm1 {
|
||||
private static final double one = 1.0;
|
||||
private static final double huge = 1.0e+300;
|
||||
private static final double tiny = 1.0e-300;
|
||||
private static final double o_threshold = 0x1.62e42fefa39efp9; // 7.09782712893383973096e+02
|
||||
private static final double ln2_hi = 0x1.62e42feep-1; // 6.93147180369123816490e-01
|
||||
private static final double ln2_lo = 0x1.a39ef35793c76p-33; // 1.90821492927058770002e-10
|
||||
private static final double invln2 = 0x1.71547652b82fep0; // 1.44269504088896338700e+00
|
||||
// scaled coefficients related to expm1
|
||||
private static final double Q1 = -0x1.11111111110f4p-5; // -3.33333333333331316428e-02
|
||||
private static final double Q2 = 0x1.a01a019fe5585p-10; // 1.58730158725481460165e-03
|
||||
private static final double Q3 = -0x1.4ce199eaadbb7p-14; // -7.93650757867487942473e-05
|
||||
private static final double Q4 = 0x1.0cfca86e65239p-18; // 4.00821782732936239552e-06
|
||||
private static final double Q5 = -0x1.afdb76e09c32dp-23; // -2.01099218183624371326e-07
|
||||
|
||||
static double compute(double x) {
|
||||
double y, hi, lo, c=0, t, e, hxs, hfx, r1;
|
||||
int k, xsb;
|
||||
/*unsigned*/ int hx;
|
||||
|
||||
hx = __HI(x); // high word of x
|
||||
xsb = hx & 0x8000_0000; // sign bit of x
|
||||
y = Math.abs(x);
|
||||
hx &= 0x7fff_ffff; // high word of |x|
|
||||
|
||||
// filter out huge and non-finite argument
|
||||
if (hx >= 0x4043_687A) { // if |x| >= 56*ln2
|
||||
if (hx >= 0x4086_2E42) { // if |x| >= 709.78...
|
||||
if (hx >= 0x7ff_00000) {
|
||||
if (((hx & 0xf_ffff) | __LO(x)) != 0) {
|
||||
return x + x; // NaN
|
||||
} else {
|
||||
return (xsb == 0)? x : -1.0; // exp(+-inf)={inf,-1}
|
||||
}
|
||||
}
|
||||
if (x > o_threshold) {
|
||||
return huge*huge; // overflow
|
||||
}
|
||||
}
|
||||
if (xsb != 0) { // x < -56*ln2, return -1.0 with inexact
|
||||
if (x + tiny < 0.0) { // raise inexact
|
||||
return tiny - one; // return -1
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
// argument reduction
|
||||
if (hx > 0x3fd6_2e42) { // if |x| > 0.5 ln2
|
||||
if (hx < 0x3FF0_A2B2) { // and |x| < 1.5 ln2
|
||||
if (xsb == 0) {
|
||||
hi = x - ln2_hi;
|
||||
lo = ln2_lo;
|
||||
k = 1;
|
||||
} else {
|
||||
hi = x + ln2_hi;
|
||||
lo = -ln2_lo;
|
||||
k = -1;
|
||||
}
|
||||
} else {
|
||||
k = (int)(invln2*x + ((xsb == 0) ? 0.5 : -0.5));
|
||||
t = k;
|
||||
hi = x - t*ln2_hi; // t*ln2_hi is exact here
|
||||
lo = t*ln2_lo;
|
||||
}
|
||||
x = hi - lo;
|
||||
c = (hi - x) - lo;
|
||||
} else if (hx < 0x3c90_0000) { // when |x| < 2**-54, return x
|
||||
t = huge + x; // return x with inexact flags when x != 0
|
||||
return x - (t - (huge + x));
|
||||
} else {
|
||||
k = 0;
|
||||
}
|
||||
|
||||
// x is now in primary range
|
||||
hfx = 0.5*x;
|
||||
hxs = x*hfx;
|
||||
r1 = one + hxs*(Q1 + hxs*(Q2 + hxs*(Q3 + hxs*(Q4 + hxs*Q5))));
|
||||
t = 3.0 - r1*hfx;
|
||||
e = hxs *((r1 - t)/(6.0 - x*t));
|
||||
if (k == 0) {
|
||||
return x - (x*e - hxs); // c is 0
|
||||
} 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 -2.0*(e - (x + 0.5));
|
||||
} else {
|
||||
return one + 2.0*(x - e);
|
||||
}
|
||||
}
|
||||
if (k <= -2 || k > 56) { // suffice to return exp(x) - 1
|
||||
y = one - (e - x);
|
||||
y = __HI(y, __HI(y) + (k << 20)); // add k to y's exponent
|
||||
return y - one;
|
||||
}
|
||||
t = one;
|
||||
if (k < 20) {
|
||||
t = __HI(t, 0x3ff0_0000 - (0x2_00000 >> k)); // t = 1-2^-k
|
||||
y = t - ( e - x);
|
||||
y = __HI(y, __HI(y) + (k << 20)); // add k to y's exponent
|
||||
} else {
|
||||
t = __HI(t, ((0x3ff - k) << 20)); // 2^-k
|
||||
y = x - (e + t);
|
||||
y += one;
|
||||
y = __HI(y, __HI(y) + (k << 20)); // add k to y's exponent
|
||||
}
|
||||
}
|
||||
return y;
|
||||
}
|
||||
}
|
||||
}
|
||||
|
|
|
|||
|
|
@ -2093,7 +2093,9 @@ public final class StrictMath {
|
|||
* @return the value <i>e</i><sup>{@code x}</sup> - 1.
|
||||
* @since 1.5
|
||||
*/
|
||||
public static native double expm1(double x);
|
||||
public static double expm1(double x) {
|
||||
return FdLibm.Expm1.compute(x);
|
||||
}
|
||||
|
||||
/**
|
||||
* Returns the natural logarithm of the sum of the argument and 1.
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue