| /* |
| * Copyright (c) 1998, 2017, 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 |
| * particular file as subject to the "Classpath" exception as provided |
| * by Oracle in the LICENSE file that accompanied this code. |
| * |
| * This code is distributed in the hope that it will be useful, but WITHOUT |
| * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
| * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
| * version 2 for more details (a copy is included in the LICENSE file that |
| * accompanied this code). |
| * |
| * You should have received a copy of the GNU General Public License version |
| * 2 along with this work; if not, write to the Free Software Foundation, |
| * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. |
| * |
| * 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. |
| */ |
| |
| 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."); |
| } |
| |
| /** |
| * 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 & 0x0000_0000_FFFF_FFFFL)); |
| } |
| |
| /** |
| * 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 |
| */ |
| 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 = 0x1.15f15f15f15f1p-1; // 19/35 ~= 5.42857142857142815906e-01 |
| private static final double D = -0x1.691de2532c834p-1; // -864/1225 ~= 7.05306122448979611050e-01 |
| private static final double E = 0x1.6a0ea0ea0ea0fp0; // 99/70 ~= 1.41428571428571436819e+00 |
| private static final double F = 0x1.9b6db6db6db6ep0; // 45/28 ~= 1.60714285714285720630e+00 |
| private static final double G = 0x1.6db6db6db6db7p-2; // 5/14 ~= 3.57142857142857150787e-01 |
| |
| private Cbrt() { |
| throw new UnsupportedOperationException(); |
| } |
| |
| public static strictfp double compute(double x) { |
| double t = 0.0; |
| double sign; |
| |
| if (x == 0.0 || !Double.isFinite(x)) |
| return x; // Handles signed zeros properly |
| |
| sign = (x < 0.0) ? -1.0: 1.0; |
| |
| x = Math.abs(x); // x <- |x| |
| |
| // Rough cbrt to 5 bits |
| if (x < 0x1.0p-1022) { // subnormal number |
| t = 0x1.0p54; // set t= 2**54 |
| t *= x; |
| t = __HI(t, __HI(t)/3 + B2); |
| } else { |
| int hx = __HI(x); // high word of x |
| 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; |
| |
| private Hypot() { |
| throw new UnsupportedOperationException(); |
| } |
| |
| 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 |
| return a + b; // Propagate NaN significand bits |
| } |
| |
| if (b > a) { |
| double tmp = a; |
| a = b; |
| b = tmp; |
| } |
| assert a >= b; |
| |
| // 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. |
| |
| 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 |
| } |
| |
| 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 { |
| private Pow() { |
| throw new UnsupportedOperationException(); |
| } |
| |
| 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); |
| } |
| |
| // 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; |
| } |
| } |
| |
| /** |
| * 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 |
| |
| private Exp() { |
| throw new UnsupportedOperationException(); |
| } |
| |
| // 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; |
| } |
| } |
| } |
| } |
