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src/java.base/share/classes/java/lang/FdLibm.java
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@@ -1,8 +1,7 @@
-
/*
- * Copyright (c) 1998, 2001, Oracle and/or its affiliates. All rights reserved.
+ * Copyright (c) 1998, 2015, 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
@@ -22,78 +21,564 @@
* Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
* or visit www.oracle.com if you need additional information or have any
* questions.
*/
-#include "fdlibm.h"
+package java.lang;
+
+/**
+ * Port of the "Freely Distributable Math Library", version 5.3, from
+ * C to Java.
+ *
+ * <p>The C version of fdlibm relied on the idiom of pointer aliasing
+ * a 64-bit double floating-point value as a two-element array of
+ * 32-bit integers and reading and writing the two halves of the
+ * double independently. This coding pattern was problematic to C
+ * optimizers and not directly expressible in Java. Therefore, rather
+ * than a memory level overlay, if portions of a double need to be
+ * operated on as integer values, the standard library methods for
+ * bitwise floating-point to integer conversion,
+ * Double.longBitsToDouble and Double.doubleToRawLongBits, are directly
+ * or indirectly used.
+ *
+ * <p>The C version of fdlibm also took some pains to signal the
+ * correct IEEE 754 exceptional conditions divide by zero, invalid,
+ * overflow and underflow. For example, overflow would be signaled by
+ * {@code huge * huge} where {@code huge} was a large constant that
+ * would overflow when squared. Since IEEE floating-point exceptional
+ * handling is not supported natively in the JVM, such coding patterns
+ * have been omitted from this port. For example, rather than {@code
+ * return huge * huge}, this port will use {@code return INFINITY}.
+ *
+ * <p>Various comparison and arithmetic operations in fdlibm could be
+ * done either based on the integer view of a value or directly on the
+ * floating-point representation. Which idiom is faster may depend on
+ * platform specific factors. However, for code clarity if no other
+ * reason, this port will favor expressing the semantics of those
+ * operations in terms of floating-point operations when convenient to
+ * do so.
+ */
+class FdLibm {
+ // Constants used by multiple algorithms
+ private static final double INFINITY = Double.POSITIVE_INFINITY;
+
+ private FdLibm() {
+ throw new UnsupportedOperationException("No FdLibm instances for you.");
+ }
-/* cbrt(x)
+ /**
+ * Return the low-order 32 bits of the double argument as an int.
+ */
+ private static int __LO(double x) {
+ long transducer = Double.doubleToRawLongBits(x);
+ return (int)transducer;
+ }
+
+ /**
+ * 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 the high-order 32 bits of the double argument as an int.
+ */
+ private static int __HI(double x) {
+ long transducer = Double.doubleToRawLongBits(x);
+ return (int)(transducer >> 32);
+ }
+
+ /**
+ * 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 );
+ }
+
+ /**
+ * cbrt(x)
* Return cube root of x
*/
-#ifdef __STDC__
-static const unsigned
-#else
-static unsigned
-#endif
- B1 = 715094163, /* B1 = (682-0.03306235651)*2**20 */
- B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */
-
-#ifdef __STDC__
-static const double
-#else
-static double
-#endif
-C = 5.42857142857142815906e-01, /* 19/35 = 0x3FE15F15, 0xF15F15F1 */
-D = -7.05306122448979611050e-01, /* -864/1225 = 0xBFE691DE, 0x2532C834 */
-E = 1.41428571428571436819e+00, /* 99/70 = 0x3FF6A0EA, 0x0EA0EA0F */
-F = 1.60714285714285720630e+00, /* 45/28 = 0x3FF9B6DB, 0x6DB6DB6E */
-G = 3.57142857142857150787e-01; /* 5/14 = 0x3FD6DB6D, 0xB6DB6DB7 */
-
-#ifdef __STDC__
- double cbrt(double x)
-#else
- double cbrt(x)
- double x;
-#endif
-{
+ public static class Cbrt {
+ // unsigned
+ private static final int B1 = 715094163; /* B1 = (682-0.03306235651)*2**20 */
+ private static final int B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */
+
+ private static final double C = 5.42857142857142815906e-01; /* 19/35 = 0x3FE15F15, 0xF15F15F1 */
+ private static final double D = -7.05306122448979611050e-01; /* -864/1225 = 0xBFE691DE, 0x2532C834 */
+ private static final double E = 1.41428571428571436819e+00; /* 99/70 = 0x3FF6A0EA, 0x0EA0EA0F */
+ private static final double F = 1.60714285714285720630e+00; /* 45/28 = 0x3FF9B6DB, 0x6DB6DB6E */
+ private static final double G = 3.57142857142857150787e-01; /* 5/14 = 0x3FD6DB6D, 0xB6DB6DB7 */
+
+ public static strictfp double compute(double x) {
int hx;
- double r,s,t=0.0,w;
- unsigned sign;
+ double t = 0.0;
+ double sign;
+
+ if (!Double.isFinite(x) || x == 0.0)
+ return x; // Handles signed zeros properly
+ sign = (x < 0.0) ? -1.0: 1.0;
- hx = __HI(x); /* high word of x */
- sign=hx&0x80000000; /* sign= sign(x) */
- hx ^=sign;
- if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */
- if((hx|__LO(x))==0)
- return(x); /* cbrt(0) is itself */
-
- __HI(x) = hx; /* x <- |x| */
- /* rough cbrt to 5 bits */
- if(hx<0x00100000) /* subnormal number */
- {__HI(t)=0x43500000; /* set t= 2**54 */
- t*=x; __HI(t)=__HI(t)/3+B2;
+ x = Math.abs(x); // x <- |x|
+ hx = __HI(x); // high word of x
+
+ // Rough cbrt to 5 bits
+ if (hx < 0x00100000) { // subnormal number
+ t = __HI(t, 0x43500000); // set t= 2**54
+ t *= x;
+ t = __HI(t, __HI(t)/3 + B2);
+ } else {
+ t = __HI(t, hx/3 + B1);
}
+
+ // New cbrt to 23 bits, may be implemented in single precision
+ double r, s, w;
+ r = t * t/x;
+ s = C + r*t;
+ t *= G + F/(s + E + D/s);
+
+ // Chopped to 20 bits and make it larger than cbrt(x)
+ t = __LO(t, 0);
+ t = __HI(t, __HI(t) + 0x00000001);
+
+ // One step newton iteration to 53 bits with error less than 0.667 ulps
+ s = t * t; // t*t is exact
+ r = x / s;
+ w = t + t;
+ r = (r - t)/(w + r); // r-s is exact
+ t = t + t*r;
+
+ // Restore the original sign bit
+ return sign * t;
+ }
+ }
+
+ /**
+ * hypot(x,y)
+ *
+ * Method :
+ * If (assume round-to-nearest) z = x*x + y*y
+ * has error less than sqrt(2)/2 ulp, than
+ * sqrt(z) has error less than 1 ulp (exercise).
+ *
+ * So, compute sqrt(x*x + y*y) with some care as
+ * follows to get the error below 1 ulp:
+ *
+ * Assume x > y > 0;
+ * (if possible, set rounding to round-to-nearest)
+ * 1. if x > 2y use
+ * x1*x1 + (y*y + (x2*(x + x1))) for x*x + y*y
+ * where x1 = x with lower 32 bits cleared, x2 = x - x1; else
+ * 2. if x <= 2y use
+ * t1*y1 + ((x-y) * (x-y) + (t1*y2 + t2*y))
+ * where t1 = 2x with lower 32 bits cleared, t2 = 2x - t1,
+ * y1= y with lower 32 bits chopped, y2 = y - y1.
+ *
+ * NOTE: scaling may be necessary if some argument is too
+ * large or too tiny
+ *
+ * Special cases:
+ * hypot(x,y) is INF if x or y is +INF or -INF; else
+ * hypot(x,y) is NAN if x or y is NAN.
+ *
+ * Accuracy:
+ * hypot(x,y) returns sqrt(x^2 + y^2) with error less
+ * than 1 ulp (unit in the last place)
+ */
+ public static class Hypot {
+ public static final double TWO_MINUS_600 = 0x1.0p-600;
+ public static final double TWO_PLUS_600 = 0x1.0p+600;
+
+ public static strictfp double compute(double x, double y) {
+ double a = Math.abs(x);
+ double b = Math.abs(y);
+
+ if (!Double.isFinite(a) || !Double.isFinite(b)) {
+ if (a == INFINITY || b == INFINITY)
+ return INFINITY;
else
- __HI(t)=hx/3+B1;
+ return a + b; // Propagate NaN significand bits
+ }
+ if (b > a) {
+ double tmp = a;
+ a = b;
+ b = tmp;
+ }
+ assert a >= b;
- /* new cbrt to 23 bits, may be implemented in single precision */
- r=t*t/x;
- s=C+r*t;
- t*=G+F/(s+E+D/s);
+ // Doing bitwise conversion after screening for NaN allows
+ // the code to not worry about the possibility of
+ // "negative" NaN values.
+
+ // Note: the ha and hb variables are the high-order
+ // 32-bits of a and b stored as integer values. The ha and
+ // hb values are used first for a rough magnitude
+ // comparison of a and b and second for simulating higher
+ // precision by allowing a and b, respectively, to be
+ // decomposed into non-overlapping portions. Both of these
+ // uses could be eliminated. The magnitude comparison
+ // could be eliminated by extracting and comparing the
+ // exponents of a and b or just be performing a
+ // floating-point divide. Splitting a floating-point
+ // number into non-overlapping portions can be
+ // accomplished by judicious use of multiplies and
+ // additions. For details see T. J. Dekker, A Floating
+ // Point Technique for Extending the Available Precision ,
+ // Numerische Mathematik, vol. 18, 1971, pp.224-242 and
+ // subsequent work.
- /* chopped to 20 bits and make it larger than cbrt(x) */
- __LO(t)=0; __HI(t)+=0x00000001;
+ int ha = __HI(a); // high word of a
+ int hb = __HI(b); // high word of b
+ if ((ha - hb) > 0x3c00000) {
+ return a + b; // x / y > 2**60
+ }
- /* one step newton iteration to 53 bits with error less than 0.667 ulps */
- s=t*t; /* t*t is exact */
- r=x/s;
- w=t+t;
- r=(r-t)/(w+r); /* r-s is exact */
- t=t+t*r;
+ int k = 0;
+ if (a > 0x1.00000_ffff_ffffp500) { // a > ~2**500
+ // scale a and b by 2**-600
+ ha -= 0x25800000;
+ hb -= 0x25800000;
+ a = a * TWO_MINUS_600;
+ b = b * TWO_MINUS_600;
+ k += 600;
+ }
+ double t1, t2;
+ if (b < 0x1.0p-500) { // b < 2**-500
+ if (b < Double.MIN_NORMAL) { // subnormal b or 0 */
+ if (b == 0.0)
+ return a;
+ t1 = 0x1.0p1022; // t1 = 2^1022
+ b *= t1;
+ a *= t1;
+ k -= 1022;
+ } else { // scale a and b by 2^600
+ ha += 0x25800000; // a *= 2^600
+ hb += 0x25800000; // b *= 2^600
+ a = a * TWO_PLUS_600;
+ b = b * TWO_PLUS_600;
+ k -= 600;
+ }
+ }
+ // medium size a and b
+ double w = a - b;
+ if (w > b) {
+ t1 = 0;
+ t1 = __HI(t1, ha);
+ t2 = a - t1;
+ w = Math.sqrt(t1*t1 - (b*(-b) - t2 * (a + t1)));
+ } else {
+ double y1, y2;
+ a = a + a;
+ y1 = 0;
+ y1 = __HI(y1, hb);
+ y2 = b - y1;
+ t1 = 0;
+ t1 = __HI(t1, ha + 0x00100000);
+ t2 = a - t1;
+ w = Math.sqrt(t1*y1 - (w*(-w) - (t1*y2 + t2*b)));
+ }
+ if (k != 0) {
+ return Math.powerOfTwoD(k) * w;
+ } else
+ return w;
+ }
+ }
+
+ /**
+ * Compute x**y
+ * n
+ * Method: Let x = 2 * (1+f)
+ * 1. Compute and return log2(x) in two pieces:
+ * log2(x) = w1 + w2,
+ * where w1 has 53 - 24 = 29 bit trailing zeros.
+ * 2. Perform y*log2(x) = n+y' by simulating multi-precision
+ * arithmetic, where |y'| <= 0.5.
+ * 3. Return x**y = 2**n*exp(y'*log2)
+ *
+ * Special cases:
+ * 1. (anything) ** 0 is 1
+ * 2. (anything) ** 1 is itself
+ * 3. (anything) ** NAN is NAN
+ * 4. NAN ** (anything except 0) is NAN
+ * 5. +-(|x| > 1) ** +INF is +INF
+ * 6. +-(|x| > 1) ** -INF is +0
+ * 7. +-(|x| < 1) ** +INF is +0
+ * 8. +-(|x| < 1) ** -INF is +INF
+ * 9. +-1 ** +-INF is NAN
+ * 10. +0 ** (+anything except 0, NAN) is +0
+ * 11. -0 ** (+anything except 0, NAN, odd integer) is +0
+ * 12. +0 ** (-anything except 0, NAN) is +INF
+ * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
+ * 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
+ * 15. +INF ** (+anything except 0,NAN) is +INF
+ * 16. +INF ** (-anything except 0,NAN) is +0
+ * 17. -INF ** (anything) = -0 ** (-anything)
+ * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
+ * 19. (-anything except 0 and inf) ** (non-integer) is NAN
+ *
+ * Accuracy:
+ * pow(x,y) returns x**y nearly rounded. In particular
+ * pow(integer,integer)
+ * always returns the correct integer provided it is
+ * representable.
+ */
+ public static class Pow {
+ public static strictfp double compute(final double x, final double y) {
+ double z;
+ double r, s, t, u, v, w;
+ int i, j, k, n;
+
+ // y == zero: x**0 = 1
+ if (y == 0.0)
+ return 1.0;
+
+ // +/-NaN return x + y to propagate NaN significands
+ if (Double.isNaN(x) || Double.isNaN(y))
+ return x + y;
+
+ final double y_abs = Math.abs(y);
+ double x_abs = Math.abs(x);
+ // Special values of y
+ if (y == 2.0) {
+ return x * x;
+ } else if (y == 0.5) {
+ if (x >= -Double.MAX_VALUE) // Handle x == -infinity later
+ return Math.sqrt(x + 0.0); // Add 0.0 to properly handle x == -0.0
+ } else if (y_abs == 1.0) { // y is +/-1
+ return (y == 1.0) ? x : 1.0 / x;
+ } else if (y_abs == INFINITY) { // y is +/-infinity
+ if (x_abs == 1.0)
+ return y - y; // inf**+/-1 is NaN
+ else if (x_abs > 1.0) // (|x| > 1)**+/-inf = inf, 0
+ return (y >= 0) ? y : 0.0;
+ else // (|x| < 1)**-/+inf = inf, 0
+ return (y < 0) ? -y : 0.0;
+ }
+
+ final int hx = __HI(x);
+ int ix = hx & 0x7fffffff;
+
+ /*
+ * When x < 0, determine if y is an odd integer:
+ * y_is_int = 0 ... y is not an integer
+ * y_is_int = 1 ... y is an odd int
+ * y_is_int = 2 ... y is an even int
+ */
+ int y_is_int = 0;
+ if (hx < 0) {
+ if (y_abs >= 0x1.0p53) // |y| >= 2^53 = 9.007199254740992E15
+ y_is_int = 2; // y is an even integer since ulp(2^53) = 2.0
+ else if (y_abs >= 1.0) { // |y| >= 1.0
+ long y_abs_as_long = (long) y_abs;
+ if ( ((double) y_abs_as_long) == y_abs) {
+ y_is_int = 2 - (int)(y_abs_as_long & 0x1L);
+ }
+ }
+ }
+
+ // Special value of x
+ if (x_abs == 0.0 ||
+ x_abs == INFINITY ||
+ x_abs == 1.0) {
+ z = x_abs; // x is +/-0, +/-inf, +/-1
+ if (y < 0.0)
+ z = 1.0/z; // z = (1/|x|)
+ if (hx < 0) {
+ if (((ix - 0x3ff00000) | y_is_int) == 0) {
+ z = (z-z)/(z-z); // (-1)**non-int is NaN
+ } else if (y_is_int == 1)
+ z = -1.0 * z; // (x < 0)**odd = -(|x|**odd)
+ }
+ return z;
+ }
+
+ n = (hx >> 31) + 1;
+
+ // (x < 0)**(non-int) is NaN
+ if ((n | y_is_int) == 0)
+ return (x-x)/(x-x);
+
+ s = 1.0; // s (sign of result -ve**odd) = -1 else = 1
+ if ( (n | (y_is_int - 1)) == 0)
+ s = -1.0; // (-ve)**(odd int)
+
+ double p_h, p_l, t1, t2;
+ // |y| is huge
+ if (y_abs > 0x1.00000_ffff_ffffp31) { // if |y| > ~2**31
+ final double INV_LN2 = 0x1.7154_7652_b82fep0; // 1.44269504088896338700e+00 = 1/ln2
+ final double INV_LN2_H = 0x1.715476p0; // 1.44269502162933349609e+00 = 24 bits of 1/ln2
+ final double INV_LN2_L = 0x1.4ae0_bf85_ddf44p-26; // 1.92596299112661746887e-08 = 1/ln2 tail
+
+ // Over/underflow if x is not close to one
+ if (x_abs < 0x1.fffff_0000_0000p-1) // |x| < ~0.9999995231628418
+ return (y < 0.0) ? s * INFINITY : s * 0.0;
+ if (x_abs > 0x1.00000_ffff_ffffp0) // |x| > ~1.0
+ return (y > 0.0) ? s * INFINITY : s * 0.0;
+ /*
+ * now |1-x| is tiny <= 2**-20, sufficient to compute
+ * log(x) by x - x^2/2 + x^3/3 - x^4/4
+ */
+ t = x_abs - 1.0; // t has 20 trailing zeros
+ w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
+ u = INV_LN2_H * t; // INV_LN2_H has 21 sig. bits
+ v = t * INV_LN2_L - w * INV_LN2;
+ t1 = u + v;
+ t1 =__LO(t1, 0);
+ t2 = v - (t1 - u);
+ } else {
+ final double CP = 0x1.ec70_9dc3_a03fdp-1; // 9.61796693925975554329e-01 = 2/(3ln2)
+ final double CP_H = 0x1.ec709ep-1; // 9.61796700954437255859e-01 = (float)cp
+ final double CP_L = -0x1.e2fe_0145_b01f5p-28; // -7.02846165095275826516e-09 = tail of CP_H
+
+ double z_h, z_l, ss, s2, s_h, s_l, t_h, t_l;
+ n = 0;
+ // Take care of subnormal numbers
+ if (ix < 0x00100000) {
+ x_abs *= 0x1.0p53; // 2^53 = 9007199254740992.0
+ n -= 53;
+ ix = __HI(x_abs);
+ }
+ n += ((ix) >> 20) - 0x3ff;
+ j = ix & 0x000fffff;
+ // Determine interval
+ ix = j | 0x3ff00000; // Normalize ix
+ if (j <= 0x3988E)
+ k = 0; // |x| <sqrt(3/2)
+ else if (j < 0xBB67A)
+ k = 1; // |x| <sqrt(3)
+ else {
+ k = 0;
+ n += 1;
+ ix -= 0x00100000;
+ }
+ x_abs = __HI(x_abs, ix);
+
+ // Compute ss = s_h + s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5)
+
+ final double BP[] = {1.0,
+ 1.5};
+ final double DP_H[] = {0.0,
+ 0x1.2b80_34p-1}; // 5.84962487220764160156e-01
+ final double DP_L[] = {0.0,
+ 0x1.cfde_b43c_fd006p-27};// 1.35003920212974897128e-08
+
+ // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3
+ final double L1 = 0x1.3333_3333_33303p-1; // 5.99999999999994648725e-01
+ final double L2 = 0x1.b6db_6db6_fabffp-2; // 4.28571428578550184252e-01
+ final double L3 = 0x1.5555_5518_f264dp-2; // 3.33333329818377432918e-01
+ final double L4 = 0x1.1746_0a91_d4101p-2; // 2.72728123808534006489e-01
+ final double L5 = 0x1.d864_a93c_9db65p-3; // 2.30660745775561754067e-01
+ final double L6 = 0x1.a7e2_84a4_54eefp-3; // 2.06975017800338417784e-01
+ u = x_abs - BP[k]; // BP[0]=1.0, BP[1]=1.5
+ v = 1.0 / (x_abs + BP[k]);
+ ss = u * v;
+ s_h = ss;
+ s_h = __LO(s_h, 0);
+ // t_h=x_abs + BP[k] High
+ t_h = 0.0;
+ t_h = __HI(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18) );
+ t_l = x_abs - (t_h - BP[k]);
+ s_l = v * ((u - s_h * t_h) - s_h * t_l);
+ // Compute log(x_abs)
+ s2 = ss * ss;
+ r = s2 * s2* (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
+ r += s_l * (s_h + ss);
+ s2 = s_h * s_h;
+ t_h = 3.0 + s2 + r;
+ t_h = __LO(t_h, 0);
+ t_l = r - ((t_h - 3.0) - s2);
+ // u+v = ss*(1+...)
+ u = s_h * t_h;
+ v = s_l * t_h + t_l * ss;
+ // 2/(3log2)*(ss + ...)
+ p_h = u + v;
+ p_h = __LO(p_h, 0);
+ p_l = v - (p_h - u);
+ z_h = CP_H * p_h; // CP_H + CP_L = 2/(3*log2)
+ z_l = CP_L * p_h + p_l * CP + DP_L[k];
+ // log2(x_abs) = (ss + ..)*2/(3*log2) = n + DP_H + z_h + z_l
+ t = (double)n;
+ t1 = (((z_h + z_l) + DP_H[k]) + t);
+ t1 = __LO(t1, 0);
+ t2 = z_l - (((t1 - t) - DP_H[k]) - z_h);
+ }
- /* retore the sign bit */
- __HI(t) |= sign;
- return(t);
+ // Split up y into (y1 + y2) and compute (y1 + y2) * (t1 + t2)
+ double y1 = y;
+ y1 = __LO(y1, 0);
+ p_l = (y - y1) * t1 + y * t2;
+ p_h = y1 * t1;
+ z = p_l + p_h;
+ j = __HI(z);
+ i = __LO(z);
+ if (j >= 0x40900000) { // z >= 1024
+ if (((j - 0x40900000) | i)!=0) // if z > 1024
+ return s * INFINITY; // Overflow
+ else {
+ final double OVT = 8.0085662595372944372e-0017; // -(1024-log2(ovfl+.5ulp))
+ if (p_l + OVT > z - p_h)
+ return s * INFINITY; // Overflow
+ }
+ } else if ((j & 0x7fffffff) >= 0x4090cc00 ) { // z <= -1075
+ if (((j - 0xc090cc00) | i)!=0) // z < -1075
+ return s * 0.0; // Underflow
+ else {
+ if (p_l <= z - p_h)
+ return s * 0.0; // Underflow
+ }
+ }
+ /*
+ * Compute 2**(p_h+p_l)
+ */
+ // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3
+ final double P1 = 0x1.5555_5555_5553ep-3; // 1.66666666666666019037e-01
+ final double P2 = -0x1.6c16_c16b_ebd93p-9; // -2.77777777770155933842e-03
+ final double P3 = 0x1.1566_aaf2_5de2cp-14; // 6.61375632143793436117e-05
+ final double P4 = -0x1.bbd4_1c5d_26bf1p-20; // -1.65339022054652515390e-06
+ final double P5 = 0x1.6376_972b_ea4d0p-25; // 4.13813679705723846039e-08
+ final double LG2 = 0x1.62e4_2fef_a39efp-1; // 6.93147180559945286227e-01
+ final double LG2_H = 0x1.62e43p-1; // 6.93147182464599609375e-01
+ final double LG2_L = -0x1.05c6_10ca_86c39p-29; // -1.90465429995776804525e-09
+ i = j & 0x7fffffff;
+ k = (i >> 20) - 0x3ff;
+ n = 0;
+ if (i > 0x3fe00000) { // if |z| > 0.5, set n = [z + 0.5]
+ n = j + (0x00100000 >> (k + 1));
+ k = ((n & 0x7fffffff) >> 20) - 0x3ff; // new k for n
+ t = 0.0;
+ t = __HI(t, (n & ~(0x000fffff >> k)) );
+ n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
+ if (j < 0)
+ n = -n;
+ p_h -= t;
+ }
+ t = p_l + p_h;
+ t = __LO(t, 0);
+ u = t * LG2_H;
+ v = (p_l - (t - p_h)) * LG2 + t * LG2_L;
+ z = u + v;
+ w = v - (z - u);
+ t = z * z;
+ t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
+ r = (z * t1)/(t1 - 2.0) - (w + z * w);
+ z = 1.0 - (r - z);
+ j = __HI(z);
+ j += (n << 20);
+ if ((j >> 20) <= 0)
+ z = Math.scalb(z, n); // subnormal output
+ else {
+ int z_hi = __HI(z);
+ z_hi += (n << 20);
+ z = __HI(z, z_hi);
+ }
+ return s * z;
+ }
+ }
}
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