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src/java.base/share/classes/java/lang/FdLibm.java
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@@ -1,7 +1,7 @@
/*
- * 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
* under the terms of the GNU General Public License version 2 only, as
* published by the Free Software Foundation. Oracle designates this
@@ -77,11 +77,12 @@
* Return a double with its low-order bits of the second argument
* and the high-order bits of the first argument..
*/
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));
}
/**
* Return the high-order 32 bits of the double argument as an int.
*/
@@ -94,11 +95,12 @@
* Return a double with its high-order bits of the second argument
* and the low-order bits of the first argument..
*/
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 );
}
/**
* cbrt(x)
* Return cube root of x
@@ -578,6 +580,154 @@
z = __HI(z, z_hi);
}
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;
+ }
+ }
+ }
}
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