/* * 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 * 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. */ class FdLibm { /** * Return the low-order 32 bits of the double argument as an int. */ private static int __LO(double x) { long transducer = Double.doubleToLongBits(x); return (int)transducer; } /** * Return the a double with its low-order bits reset. */ private static double __LO(double x, int low) { long transX = Double.doubleToLongBits(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.doubleToLongBits(x); return (int)(transducer >> 32); } /** * Return the a double with its high-order bits reset. */ private static double __HI(double x, int high) { long transX = Double.doubleToLongBits(x); return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL)|( ((long)high)) << 32 ); } /** * 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 muti-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 { static final double bp[] = {1.0, 1.5}; static final double dp_h[] = {0.0, 0x1.2b8034p-1}; // 5.84962487220764160156e-01 static final double dp_l[] = {0.0, 0x1.cfdeb43cfd006p-27};// 1.35003920212974897128e-08 static final double zero = 0.0; static final double one = 1.0; static final double two = 2.0; static final double two53 = 0x1.0p53; // 9007199254740992.0 static final double huge = 1.0e300; static final double tiny = 1.0e-300; // Poly coefs for (3/2)*(log(x)-2s-2/3*s**3 static final double L1 = 0x1.3333333333303p-1; // 5.99999999999994648725e-01 static final double L2 = 0x1.b6db6db6fabffp-2; // 4.28571428578550184252e-01 static final double L3 = 0x1.55555518f264dp-2; // 3.33333329818377432918e-01 static final double L4 = 0x1.17460a91d4101p-2; // 2.72728123808534006489e-01 static final double L5 = 0x1.d864a93c9db65p-3; // 2.30660745775561754067e-01 static final double L6 = 0x1.a7e284a454eefp-3; // 2.06975017800338417784e-01 static final double P1 = 0x1.555555555553ep-3; // 1.66666666666666019037e-01 static final double P2 = -0x1.6c16c16bebd93p-9; // -2.77777777770155933842e-03 static final double P3 = 0x1.1566aaf25de2cp-14; // 6.61375632143793436117e-05 static final double P4 = -0x1.bbd41c5d26bf1p-20; // -1.65339022054652515390e-06 static final double P5 = 0x1.6376972bea4d0p-25; // 4.13813679705723846039e-08 static final double lg2 = 0x1.62e42fefa39efp-1; // 6.93147180559945286227e-01 static final double lg2_h = 0x1.62e43p-1; // 6.93147182464599609375e-01 static final double lg2_l = -0x1.05c610ca86c39p-29; // -1.90465429995776804525e-09 static final double ovt = 8.0085662595372944372e-0017; // -(1024-log2(ovfl+.5ulp)) static final double cp = 0x1.ec709dc3a03fdp-1; // 9.61796693925975554329e-01 = 2/(3ln2) static final double cp_h = 0x1.ec709ep-1; // 9.61796700954437255859e-01 = (float)cp static final double cp_l = -0x1.e2fe0145b01f5p-28; // -7.02846165095275826516e-09 = tail of cp_h static final double ivln2 = 0x1.71547652b82fep0; // 1.44269504088896338700e+00 = 1/ln2 static final double ivln2_h = 0x1.715476p0; // 1.44269502162933349609e+00 = 24 bits of 1/ln2 static final double ivln2_l = 0x1.4ae0bf85ddf44p-26; // 1.92596299112661746887e-08 = 1/ln2 tail public static double pow(final double x, final double y) { double z; double t1, t2, 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 : one / x; } else if (y_abs == Double.POSITIVE_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 : zero; else // (|x| < 1)**-, +inf = inf, 0 return (y < 0) ? -y : zero; } final int hx = __HI(x); // Try to replace with copysign usage // final int lx = __LO(x); // final int hy = __HI(y); final int ly = __LO(y); int ix = hx & 0x7fffffff; final int iy = __HI(y) & 0x7fffffff; // Try to replace with getExponent in yisint /* * When x < 0, determine if y is an odd integer: * yisint = 0 ... y is not an integer * yisint = 1 ... y is an odd int * yisint = 2 ... y is an even int */ { int yisint = 0; if (hx < 0) { if (iy >= 0x43400000) yisint = 2; // even integer y else if (iy >= 0x3ff00000) { k = (iy >> 20) - 0x3ff; // exponent if (k > 20) { j = ly >> (52 - k); if ((j << (52 - k) ) == ly) yisint = 2 - (j & 1); } else if (ly == 0) { j = iy >> (20 - k); if ((j << (20 - k)) == iy) { yisint = 2 - (j & 1); } } } } // Special value of x if (x_abs == 0.0 || x_abs == Double.POSITIVE_INFINITY || x_abs == 1.0) { z = x_abs; // x is +/-0, +/-inf, +/-1 if (y < 0.0) z = one/z; // z = (1/|x|) if (hx < 0) { if (((ix - 0x3ff00000) | yisint) == 0) { z = (z-z)/(z-z); // (-1)**non-int is NaN } else if (yisint == 1) z = -1.0 * z; // (x < 0)**odd = -(|x|**odd) } return z; } n = (hx >> 31) + 1; // (x < 0)**(non-int) is NaN if ((n | yisint) == 0) return (x-x)/(x-x); s = one; // s (sign of result -ve**odd) = -1 else = 1 if ( (n | (yisint - 1)) == 0) s = -one; // (-ve)**(odd int) } double p_h, p_l; // |y| is huge if (y_abs > 0x1.0p31) { // if |y| > 2**31 if (y_abs > 0x1.0p64){ // if |y| > 2**64, must over/underflow if (ix <= 0x3fefffff) return (y < 0.0) ? huge*huge : tiny*tiny; if (ix >= 0x3ff00000) return (y > 0.0) ? huge*huge : tiny*tiny; } // Over/underflow if x is not close to one if (ix < 0x3fefffff) return (y < 0.0) ? s*huge*huge : s*tiny*tiny; if (ix > 0x3ff00000) return (y > 0.0) ? s*huge*huge : s*tiny*tiny; /* * 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 - one; // t has 20 trailing zeros w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25)); u = ivln2_h * t; // ivln2_h has 21 sig. bits v = t * ivln2_l - w * ivln2; t1 = u + v; t1 =__LO(t1, 0); t2 = v - (t1 - u); } else { 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 *= two53; 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| > 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*huge*huge; // Overflow else { if (p_l + ovt > z - p_h) return s*huge*huge; // Overflow } } else if ((j & 0x7fffffff) >= 0x4090cc00 ) { // z <= -1075 if (((j - 0xc090cc00) | i)!=0) // z < -1075 return s*tiny*tiny; // Underflow else { if (p_l <= z - p_h) return s*tiny*tiny; // Underflow } } /* * Compute 2**(p_h+p_l) */ 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 = zero; 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 - two) - (w + z * w); z = one - (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; } } }