8139688: Port fdlibm exp to Java

Reviewed-by: bpb, nadezhin
This commit is contained in:
Joe Darcy 2016-12-16 21:43:29 -08:00
parent 2e27b2e68a
commit be91309965
6 changed files with 441 additions and 15 deletions

View file

@ -1,5 +1,5 @@
/*
* Copyright (c) 1998, 2015, Oracle and/or its affiliates. All rights reserved.
* Copyright (c) 1998, 2016, Oracle and/or its affiliates. All rights reserved.
* DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
*
* This code is free software; you can redistribute it and/or modify it
@ -79,7 +79,8 @@ class FdLibm {
*/
private static double __LO(double x, int low) {
long transX = Double.doubleToRawLongBits(x);
return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L)|low );
return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L) |
(low & 0x0000_0000_FFFF_FFFFL));
}
/**
@ -96,7 +97,8 @@ class FdLibm {
*/
private static double __HI(double x, int high) {
long transX = Double.doubleToRawLongBits(x);
return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL)|( ((long)high)) << 32 );
return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL) |
( ((long)high)) << 32 );
}
/**
@ -580,4 +582,152 @@ class FdLibm {
return s * z;
}
}
/**
* Returns the exponential of x.
*
* Method
* 1. Argument reduction:
* Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
* Given x, find r and integer k such that
*
* x = k*ln2 + r, |r| <= 0.5*ln2.
*
* Here r will be represented as r = hi-lo for better
* accuracy.
*
* 2. Approximation of exp(r) by a special rational function on
* the interval [0,0.34658]:
* Write
* R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
* We use a special Reme algorithm on [0,0.34658] to generate
* a polynomial of degree 5 to approximate R. The maximum error
* of this polynomial approximation is bounded by 2**-59. In
* other words,
* R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
* (where z=r*r, and the values of P1 to P5 are listed below)
* and
* | 5 | -59
* | 2.0+P1*z+...+P5*z - R(z) | <= 2
* | |
* The computation of exp(r) thus becomes
* 2*r
* exp(r) = 1 + -------
* R - r
* r*R1(r)
* = 1 + r + ----------- (for better accuracy)
* 2 - R1(r)
* where
* 2 4 10
* R1(r) = r - (P1*r + P2*r + ... + P5*r ).
*
* 3. Scale back to obtain exp(x):
* From step 1, we have
* exp(x) = 2^k * exp(r)
*
* Special cases:
* exp(INF) is INF, exp(NaN) is NaN;
* exp(-INF) is 0, and
* for finite argument, only exp(0)=1 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 exp(x) overflow
* if x < -7.45133219101941108420e+02 then exp(x) underflow
*
* 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 Exp {
private static final double one = 1.0;
private static final double[] half = {0.5, -0.5,};
private static final double huge = 1.0e+300;
private static final double twom1000= 0x1.0p-1000; // 9.33263618503218878990e-302 = 2^-1000
private static final double o_threshold= 0x1.62e42fefa39efp9; // 7.09782712893383973096e+02
private static final double u_threshold= -0x1.74910d52d3051p9; // -7.45133219101941108420e+02;
private static final double[] ln2HI ={ 0x1.62e42feep-1, // 6.93147180369123816490e-01
-0x1.62e42feep-1}; // -6.93147180369123816490e-01
private static final double[] ln2LO ={ 0x1.a39ef35793c76p-33, // 1.90821492927058770002e-10
-0x1.a39ef35793c76p-33}; // -1.90821492927058770002e-10
private static final double invln2 = 0x1.71547652b82fep0; // 1.44269504088896338700e+00
private static final double P1 = 0x1.555555555553ep-3; // 1.66666666666666019037e-01
private static final double P2 = -0x1.6c16c16bebd93p-9; // -2.77777777770155933842e-03
private static final double P3 = 0x1.1566aaf25de2cp-14; // 6.61375632143793436117e-05
private static final double P4 = -0x1.bbd41c5d26bf1p-20; // -1.65339022054652515390e-06
private static final double P5 = 0x1.6376972bea4d0p-25; // 4.13813679705723846039e-08
// should be able to forgo strictfp due to controlled over/underflow
public static strictfp double compute(double x) {
double y;
double hi = 0.0;
double lo = 0.0;
double c;
double t;
int k = 0;
int xsb;
/*unsigned*/ int hx;
hx = __HI(x); /* high word of x */
xsb = (hx >> 31) & 1; /* sign bit of x */
hx &= 0x7fffffff; /* high word of |x| */
/* filter out non-finite argument */
if (hx >= 0x40862E42) { /* if |x| >= 709.78... */
if (hx >= 0x7ff00000) {
if (((hx & 0xfffff) | __LO(x)) != 0)
return x + x; /* NaN */
else
return (xsb == 0) ? x : 0.0; /* exp(+-inf) = {inf, 0} */
}
if (x > o_threshold)
return huge * huge; /* overflow */
if (x < u_threshold) // unsigned compare needed here?
return twom1000 * twom1000; /* underflow */
}
/* argument reduction */
if (hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
hi = x - ln2HI[xsb];
lo=ln2LO[xsb];
k = 1 - xsb - xsb;
} else {
k = (int)(invln2 * x + half[xsb]);
t = k;
hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
lo = t*ln2LO[0];
}
x = hi - lo;
} else if (hx < 0x3e300000) { /* when |x|<2**-28 */
if (huge + x > one)
return one + x; /* trigger inexact */
} else {
k = 0;
}
/* x is now in primary range */
t = x * x;
c = x - t*(P1 + t*(P2 + t*(P3 + t*(P4 + t*P5))));
if (k == 0)
return one - ((x*c)/(c - 2.0) - x);
else
y = one - ((lo - (x*c)/(2.0 - c)) - hi);
if(k >= -1021) {
y = __HI(y, __HI(y) + (k << 20)); /* add k to y's exponent */
return y;
} else {
y = __HI(y, __HI(y) + ((k + 1000) << 20)); /* add k to y's exponent */
return y * twom1000;
}
}
}
}