Hi Brian, as agreed in [1], here is the first batch of my code. It includes everything needed to replace Double::toString(double), if so wished. It is accompanied by some JUnit tests in math.DoubleToDecimalTest and by straightforward benchmarks in math.D2DBenchmark
The code still has notices attributing the copyright to me under the GPL2 + Classpath exception license because I just copied the code verbatim from my GitHub repo. What is currently *not* included is the drop-in replacement for Float.toString(float). Before I polish this part, which should take a couple of hours, I'll wait for comments about the code delivered so far. Source files are preceded by a line of the form: -------- <filename> They are * LICENSE * module-info.java * math.Natural.java * math.Powers.java * math.MathUtils.java * math.DoubleToDecimal.java * math.DecimalChecker.java * math.DoubleToDecimalTest.java * math.D2DBenchmark.java Let me know if the format I chose, namely dumb copy&paste, is appropriate. Greetings Raffaello [1] http://mail.openjdk.java.net/pipermail/core-libs-dev/2018-April/052681.html -------- LICENSE GNU GENERAL PUBLIC LICENSE Version 2, June 1991 Copyright (C) 1989, 1991 Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. Preamble The licenses for most software are designed to take away your freedom to share and change it. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change free software--to make sure the software is free for all its users. This General Public License applies to most of the Free Software Foundation's software and to any other program whose authors commit to using it. (Some other Free Software Foundation software is covered by the GNU Lesser General Public License instead.) You can apply it to your programs, too. 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As a special exception, the copyright holders of this library give you permission to link this library with independent modules to produce an executable, regardless of the license terms of these independent modules, and to copy and distribute the resulting executable under terms of your choice, provided that you also meet, for each linked independent module, the terms and conditions of the license of that module. An independent module is a module which is not derived from or based on this library. If you modify this library, you may extend this exception to your version of the library, but you are not obligated to do so. If you do not wish to do so, delete this exception statement from your version. -------- module-info.java /* * Copyright (c) 2018, Raffaello Giulietti. 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. * This particular file is subject to the "Classpath" exception as provided * 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. */ module todec { exports math; } -------- math.Natural.java /* * Copyright (c) 2018, Raffaello Giulietti. 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. * This particular file is subject to the "Classpath" exception as provided * 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. */ package math; import java.math.BigInteger; import static java.lang.Math.max; /** * A minimal, limited implementation of non-negative large integers. * * <p>All operations implemented here are needed in other parts of the package, * while many are missing entirely because they are not needed. */ final class Natural { private static final long I_SZ = Integer.SIZE; private static final long I_MASK = (1L << I_SZ) - 1; private static final int I_MSB = -1 << I_SZ - 1; private static final long L_MSB = -1L << Long.SIZE - 1; /* A large natural is represented as a sequence of len B-ary digits, that is, in base B = 2^Integer.SIZE = 2^32 Its value is d[0] + d[1]*B + d[2]*B^2 + ... + d[len-1]*B^(len-1) where each B-ary digit d[i] is interpreted as unsigned int. As usual, an empty sum has value 0, so if len = 0 then the value is 0. The following invariants hold: 0 <= len <= d.length either len = 0 or d[len-1] != 0 */ private final int[] d; private final int len; private Natural(int[] d) { int i = d.length; while (--i >= 0 && d[i] == 0); // empty body intended this.len = i + 1; this.d = d; } /** * Returns a {@link Natural} with the same value as {@code v}, * which is interpreted as an unsigned {@code long}. */ static Natural valueOf(long v) { return new Natural(new int[]{(int) v, (int) (v >>> 32)}); } /** * Returns a {@link Natural} with the value * {@code v} · 2<sup>{@code n}</sup>. * The value {@code v} is interpreted as an unsigned {@code long} and * {@code n} must be non-negative. */ static Natural valueOfShiftLeft(long v, int n) { // I_SZ = 2^5 int q = n >> 5; int r = n & 0x1f; // length is q plus additional 2 for v and 1 for possible overlapping int[] rd = new int[q + 3]; rd[q] = (int) v << r; rd[q + 1] = (int) (v >>> I_SZ - r); // A safe shift by 64 - r, even when r = 0 rd[q + 2] = (int) (v >>> I_SZ >>> I_SZ - r); return new Natural(rd); } /** * Returns -1, 0 or 1, depending on whether {@code this} is * <, = or > {@code y}, respectively. */ int compareTo(Natural y) { if (len < y.len) { return -1; } if (len > y.len) { return 1; } int i = len; while (--i >= 0 && d[i] == y.d[i]); // empty body intended if (i < 0) { return 0; } // Perform an unsigned int comparision if ((d[i] ^ I_MSB) < (y.d[i] ^ I_MSB)) { return -1; } return 1; } /** * Returns -1, 0 or 1, depending on whether {@code this} is closer to * {@code x}, equally close to both {@code x} and {@code y} or closer to * {@code y}, respectively. */ int closerTo(Natural x, Natural y) { /* computes (2 * this - x - y).compareTo(0) without allocating objects for the intermediate results. */ int cmp = 0; long c = 0; int maxLen = max(max(x.len, y.len), len); for (int i = 0; i < maxLen; ++i) { long td = i < len ? d[i] & I_MASK : 0; long xd = i < x.len ? x.d[i] & I_MASK : 0; long yd = i < y.len ? y.d[i] & I_MASK : 0; long s = (td << 1) - xd - yd + c; cmp |= (int) s; c = s >> I_SZ; } if (c < 0) { return -1; } if (cmp != 0) { return 1; } return 0; } /** * Returns {@code this} * {@code y}, where {@code y} is taken as an * unsigned {@code long}. */ Natural multiply(long y) { // Straightforward paper-and-pencil method for multiplication. int[] rd = new int[len + 2]; long y0 = y & I_MASK; long y1 = y >>> I_SZ; long c = 0; long r1 = 0; long q0 = 0; long q1 = 0; long s; int i = 0; for (; i < len; ++i) { long td = d[i] & I_MASK; long p0 = y0 * td; s = (r1 >>> I_SZ) + (q0 >>> I_SZ) + (q1 & I_MASK) + (p0 & I_MASK) + c; rd[i] = (int) s; c = s >>> I_SZ; r1 = q1; q0 = p0; q1 = y1 * td; } s = (r1 >>> I_SZ) + (q0 >>> I_SZ) + (q1 & I_MASK) + c; rd[i] = (int) s; c = s >>> I_SZ; rd[i + 1] = (int) (c + (q1 >>> I_SZ)); return new Natural(rd); } /** * Returns {@code this} - {@code y}, where it is assumed that * {@code this} ≥ {@code y}. */ Natural subtract(Natural y) { int[] rd = new int[len]; long c = 0; int i = 0; for (; i < y.len; ++i) { long s = (d[i] & I_MASK) - (y.d[i] & I_MASK) + c; rd[i] = (int) s; c = s >> I_SZ; } for (; i < len; ++i) { long s = (d[i] & I_MASK) + c; rd[i] = (int) s; c = s >> I_SZ; } return new Natural(rd); } /** * Returns ⎣{@code this} · 2<sup>-{@code n}</sup>⎦, * where it is assumed that {@code n} ≥ 0 and that the result * is an unsigned {@code long}. */ long shiftRight(int n) { int q = n >> 5; int r = n & 0x1f; long d0 = d[q] & I_MASK; long d1 = d[q + 1] & I_MASK; long d2 = q + 2 < len ? d[q + 2] & I_MASK : 0; // The double shift is safe even when r = 0 return d0 >>> r | d1 << I_SZ - r | d2 << I_SZ << I_SZ - r; } /** * Returns {@code this} + {@code y} · 2<sup>{@code n}</sup>, * where it is assumed that {@code n} ≥ 0. */ Natural addShiftLeft(Natural y, int n) { int maxLen = max(len, y.len); int[] rd = new int[maxLen + 1]; long c = 0; long yd = 0; int i = 0; for (; i < maxLen; ++i) { long t0 = i < len ? d[i] & I_MASK : 0; long y0 = i < y.len ? y.d[i] & I_MASK : 0; yd = yd >>> I_SZ | y0 << n; long s = t0 + (yd & I_MASK) + c; rd[i] = (int) s; c = s >>> I_SZ; } rd[i] = (int) ((yd >>> I_SZ) + c); return new Natural(rd); } private BigInteger toBigInteger() { // additional 1 for the "sign" most significant byte at index 0 byte[] b = new byte[1 + (len << 2)]; for (int i = 1; i <= len; ++i) { int d0 = d[len - i]; b[(i << 2) - 3] = (byte) (d0 >>> 24); b[(i << 2) - 2] = (byte) (d0 >>> 16); b[(i << 2) - 1] = (byte) (d0 >>> 8); b[i << 2] = (byte) d0; } return new BigInteger(b); } /* Here for debugging purposes only. There's otherwise no need for it. */ @Override public String toString() { // Quick-and-dirty solution to avoid implementing a special division. return toBigInteger().toString(); } /* Assumes 0 <= n */ private Natural shiftLeft(int n) { int q = n >> 5; int r = n & 0x1f; // Allocates one int more than necessary to simplify the division if (r == 0) { int[] rd = new int[len + q + 1]; for (int i = 0; i < len; ++i) { rd[q + i] = d[i]; } return new Natural(rd); } int[] rd = new int[len + q + 2]; rd[q] = d[0] << r; int i = 1; for (; i < len; ++i) { // safe shift, as 0 < r < I_SZ rd[q + i] = d[i] << r | d[i - 1] >>> I_SZ - r; } rd[q + i] = d[i - 1] >>> I_SZ - r; return new Natural(rd); } /** * Returns ⎣{@code this} / {@code y}⎦. * <p>Assumes that: * <ul> * <li> {@code this} ≥ 2<sup>32</sup>. * <li> {@code y} > 0. * <li> The result is an unsigned {@code long} ≥ 2<sup>32</sup>. * </ul> */ long divide(Natural y) { int r = Integer.numberOfLeadingZeros(y.d[y.len - 1]) - 1 & 0x1f; // Ensure that v.len >= 2 and that vp meets the inequalities below if (y.len == 1) { r += I_SZ; } Natural u = shiftLeft(r); Natural v = y.shiftLeft(r); // by construction, 2^30 <= vp < 2^31: no need for masking long vp = v.d[v.len - 1]; long v_n2 = v.d[v.len - 2] & I_MASK; long q = 0; for (int k = 1; k >= 0; --k) { int n = v.len + k; // this assumes that n <= u.d.length long up = (long) u.d[n] << I_SZ | u.d[n - 1] & I_MASK; long qb = up / vp; if (qb > I_MASK) qb = I_MASK; long rb = up - qb * vp; while (rb <= I_MASK && (qb * v_n2 ^ L_MSB) > ((rb << I_SZ | u.d[n - 2] & I_MASK) ^ L_MSB)) { qb -= 1; rb += vp; } long s = 0; int i = 0; for (; i < v.len; ++i) { long p = qb * (v.d[i] & I_MASK) + s; long t = (u.d[i + k] & I_MASK) - (p & I_MASK); u.d[i + k] = (int) t; s = (p >>> I_SZ) - (t >> I_SZ); } long t = (u.d[i + k] & I_MASK) - (s & I_MASK); u.d[i + k] = (int) t; s = -(t >> I_SZ); if (s > 0) { qb -= 1; s = 0; for (i = 0; i < v.len; ++i) { t = (u.d[i + k] & I_MASK) + (v.d[i] & I_MASK) + s; u.d[i + k] = (int) t; s = t >>> I_SZ; } } q = q << I_SZ | qb; } return q; } } -------- math.Powers.java /* * Copyright (c) 2018, Raffaello Giulietti. 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. * This particular file is subject to the "Classpath" exception as provided * 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. */ package math; import static math.Natural.valueOf; /** * Package-privately exposes * <ul> * <li> integer powers of 5 as unsigned {@code long}s, up to the exponent * {@link #MAX_POW_5_EXP} * <li> integer powers of 10 as unsigned {@code long}s, up to the exponent * {@link #MAX_POW_10_EXP} * <li> integer powers of 5 as {@link Natural}s, up to the exponent * {@link #MAX_POW_5_N_EXP} * </ul> * * <p> * Since this is a package-private class, no checks are made to ensure * that usages are correct. */ final class Powers { /** * The integer <i>e</i> such that * 5<sup><i>e</i></sup> ≤ <i>M</i> < 5<sup><i>e</i>+1</sup>, * where <i>M</i> is the largest unsigned {@code long}, namely * <i>M</i> = 2<sup>{@link Long#SIZE}</sup> - 1. */ static final int MAX_POW_5_EXP = 27; /* The greatest power of 5 fitting in an unsigned {@code long}, namely 5^MAX_POW_5_EXP */ private static final long MAX_POW_5 = 7_450_580_596_923_828_125L; /** * The integer <i>e</i> such that * 10<sup><i>e</i></sup> ≤ <i>M</i> < 10<sup><i>e</i>+1</sup>, * where <i>M</i> is the largest unsigned {@code long}, namely * <i>M</i> = 2<sup>{@link Long#SIZE}</sup> - 1. */ static final int MAX_POW_10_EXP = 19; /** * The greatest exponent for {@link #pow5(int)}. */ /* MAX_POW_5_N_EXP = Double.H - Double.E_MIN_VALUE */ static final int MAX_POW_5_N_EXP = 340; /** * Powers of 5, as unsigned {@code long}s, for exponents between * 0 and {@link #MAX_POW_5_EXP}. */ static final long[] pow5; /** * Powers of 10, as unsigned {@code long}s, for exponents between * 0 and {@link #MAX_POW_10_EXP}. */ static final long[] pow10; /* pow5n is populated lazily. More precisely, values for the exponents between 0 and MAX_POW_5_EXP are initialized upon class loading. Other values are computed upon request (see pow5()). Invariant: e0max is a multiple of MAX_POW_5_EXP and all values for exponents that are multiples of MAX_POW_5_EXP, up to e0max, are already present in the array. */ private static final Natural[] pow5n; private static int e0max = MAX_POW_5_EXP; static { /* Fully initializes the pow5 and pow10 array and partial initializes pow5n, which will be populated lazily, as need arises. */ pow5n = new Natural[MAX_POW_5_N_EXP + 1]; pow5 = new long[MAX_POW_5_EXP + 1]; pow5[0] = 1; for (int k = 1; k < pow5.length; ++k) { pow5[k] = 5 * pow5[k - 1]; pow5n[k] = valueOf(pow5[k]); } pow10 = new long[MAX_POW_10_EXP + 1]; pow10[0] = 1; for (int k = 1; k < pow10.length; ++k) { pow10[k] = 10 * pow10[k - 1]; } } private Powers() { } /** * Powers of 5, for exponents between 0 and {@link #MAX_POW_5_N_EXP}. */ static synchronized Natural pow5(int e) { if (pow5n[e] != null) { return pow5n[e]; } int e0 = e / MAX_POW_5_EXP * MAX_POW_5_EXP; /* Guard for the loop: mathematically not necessary but measurably enhances performance when there's no need to enter the loop. */ if (e0max < e0) { for (; e0max < e0; e0max += MAX_POW_5_EXP) { pow5n[e0max + MAX_POW_5_EXP] = pow5n[e0max].multiply(MAX_POW_5); } } if (e0 < e) { pow5n[e] = pow5n[e0].multiply(pow5[e - e0]); } return pow5n[e]; } } -------- math.MathUtils.java /* * Copyright (c) 2018, Raffaello Giulietti. 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. * This particular file is subject to the "Classpath" exception as provided * 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. */ package math; import static math.DoubleToDecimal.Double.c; import static math.DoubleToDecimal.Double.q; final class MathUtils { private static final int I = Integer.SIZE; private static final long MASK_I = (1L << I) - 1; /* The doubles below are expressed in hex notation to avoid possible anomalies during decimal tokenization. Hex tokenization is assumed to be completely reliable, as it is simpler from a mathematical perspective. */ // The double closest to log10(2), 0.3010299956639812 in decimal private static final double LOG_10_2 = 0x1.34413509F79FFp-2; // The double closest to log2(10), 3.321928094887362 in decimal private static final double LOG_2_10 = 0x1.A934F0979A371p1; /** * The minimum exponent for {@link #floorPow10d(int)} * and {@link #pow10r(int)} */ static final int MIN_EXP = -324; /** * The maximum exponent for {@link #floorPow10d(int)} * and {@link #pow10r(int)} */ static final int MAX_EXP = 324; private MathUtils() { } /* This implementation is simple but is restricted to its usage here, when the assumptions below hold. It assumes v = 0 (thus c = 0) or 0 <= -q < Long.SIZE Also note that for v < 0 floor(v) = -ceil(-v) */ private static int floor(double v) { if (v >= 0) { long bits = java.lang.Double.doubleToRawLongBits(v); return (int) (c(bits) >>> -q(bits)); } long bits = java.lang.Double.doubleToRawLongBits(-v); int q = q(bits); return -(int) (c(bits) + (1L << -q) - 1 >>> -q); } /** * Returns the integer <i>k</i> such that 10<sup><i>k</i>-1</sup> ≤ * 2<sup>{@code e}</sup> < 10<sup><i>k</i></sup>. * <p> * The result is correct when -198'096'464 ≤ {@code e} ≤ * 146'964'307. * Otherwise the result may or may not be correct. */ static int ord10pow2(int e) { return floor(e * LOG_10_2) + 1; } /** * Returns the integer <i>k</i> such that 2<sup><i>k</i>-1</sup> ≤ * 10<sup>{@code e}</sup> < 2<sup><i>k</i></sup>. * <p> * The result is correct when -44'240'664 ≤ {@code e} ≤ * 59'632'977. * Otherwise the result may or may not be correct. */ static int ord2pow10(int e) { return floor(e * LOG_2_10) + 1; } /** * Returns the high {@link Long#SIZE} bits of the full product * {@code x}{@code y}, namely * ⎣{@code x}{@code y} · 2<sup>-{@link Long#SIZE}</sup>⎦. * <p> * Both {@code x} and {@code y} as well as the result are interpreted as * unsigned {@code long}s. */ static long multiplyHighUnsigned(long x, long y) { /* Unfortunately, the plain version of Karatsuba cannot be applied here: the mixed product would overflow with unrecoverable loss of bits. Thus, the plain paper-and-pencil scheme requiring 4 long multiplications is used. This could be a good candidate a for JIT compiler intrinsic. */ final long x1 = x >>> I; final long x0 = x & MASK_I; final long y1 = y >>> I; final long y0 = y & MASK_I; final long x1y0 = x1 * y0; final long x0y1 = x0 * y1; return x1 * y1 + (x0y1 >>> I) + (x1y0 >>> I) + ((x0y1 & MASK_I) + (x1y0 & MASK_I) + (x0 * y0 >>> I) >>> I); } /** * Returns one of two components of an approximation of a power of 10. * * <p>More precisely, let * 10<sup>{@code e}</sup> = <i>d</i> · 2<sup><i>r</i></sup> * for some integer <i>r</i> and real <i>d</i> with * 2<sup>{@link Long#SIZE}-1</sup> ≤ <i>d</i> < * 2<sup>{@link Long#SIZE}</sup>. * * <p>This method returns ⎣<i>d</i>⎦ as an * unsigned {@code long}, while {@link #pow10r(int)} returns <i>r</i>. * * @param e The exponent of the power of 10, bounded by * {@link #MIN_EXP} ≤ {@code e} ≤ {@link #MAX_EXP} * @see #pow10r(int) */ static long floorPow10d(int e) { return floorPow10d[e - MIN_EXP]; } /** * Returns one of two components of an approximation of a power of 10. * * <p>This method returns <i>r</i> from the representation described in * {@link #floorPow10d(int)}. * * @param e The exponent of the power of 10, bounded by * {@link #MIN_EXP} ≤ {@code e} ≤ {@link #MAX_EXP} * @see #floorPow10d(int) */ static int pow10r(int e) { return ord2pow10(e) - Long.SIZE; } /* The array has been computed and checked using full precision. The values are prefixed with a comment indicating the exponent. Contrary to common coding conventions, its definition is located here, at the end of the file, because the length of its source would be distracting for reading the rest. */ private static final long[] floorPow10d = { /* -324 */ 0xCF42_894A_5DCE_35EAL, /* -323 */ 0x8189_95CE_7AA0_E1B2L, /* -322 */ 0xA1EB_FB42_1949_1A1FL, /* -321 */ 0xCA66_FA12_9F9B_60A6L, /* -320 */ 0xFD00_B897_4782_38D0L, /* -319 */ 0x9E20_735E_8CB1_6382L, /* -318 */ 0xC5A8_9036_2FDD_BC62L, /* -317 */ 0xF712_B443_BBD5_2B7BL, /* -316 */ 0x9A6B_B0AA_5565_3B2DL, /* -315 */ 0xC106_9CD4_EABE_89F8L, /* -314 */ 0xF148_440A_256E_2C76L, /* -313 */ 0x96CD_2A86_5764_DBCAL, /* -312 */ 0xBC80_7527_ED3E_12BCL, /* -311 */ 0xEBA0_9271_E88D_976BL, /* -310 */ 0x9344_5B87_3158_7EA3L, /* -309 */ 0xB815_7268_FDAE_9E4CL, /* -308 */ 0xE61A_CF03_3D1A_45DFL, /* -307 */ 0x8FD0_C162_0630_6BABL, /* -306 */ 0xB3C4_F1BA_87BC_8696L, /* -305 */ 0xE0B6_2E29_29AB_A83CL, /* -304 */ 0x8C71_DCD9_BA0B_4925L, /* -303 */ 0xAF8E_5410_288E_1B6FL, /* -302 */ 0xDB71_E914_32B1_A24AL, /* -301 */ 0x8927_31AC_9FAF_056EL, /* -300 */ 0xAB70_FE17_C79A_C6CAL, /* -299 */ 0xD64D_3D9D_B981_787DL, /* -298 */ 0x85F0_4682_93F0_EB4EL, /* -297 */ 0xA76C_5823_38ED_2621L, /* -296 */ 0xD147_6E2C_0728_6FAAL, /* -295 */ 0x82CC_A4DB_8479_45CAL, /* -294 */ 0xA37F_CE12_6597_973CL, /* -293 */ 0xCC5F_C196_FEFD_7D0CL, /* -292 */ 0xFF77_B1FC_BEBC_DC4FL, /* -291 */ 0x9FAA_CF3D_F736_09B1L, /* -290 */ 0xC795_830D_7503_8C1DL, /* -289 */ 0xF97A_E3D0_D244_6F25L, /* -288 */ 0x9BEC_CE62_836A_C577L, /* -287 */ 0xC2E8_01FB_2445_76D5L, /* -286 */ 0xF3A2_0279_ED56_D48AL, /* -285 */ 0x9845_418C_3456_44D6L, /* -284 */ 0xBE56_91EF_416B_D60CL, /* -283 */ 0xEDEC_366B_11C6_CB8FL, /* -282 */ 0x94B3_A202_EB1C_3F39L, /* -281 */ 0xB9E0_8A83_A5E3_4F07L, /* -280 */ 0xE858_AD24_8F5C_22C9L, /* -279 */ 0x9137_6C36_D999_95BEL, /* -278 */ 0xB585_4744_8FFF_FB2DL, /* -277 */ 0xE2E6_9915_B3FF_F9F9L, /* -276 */ 0x8DD0_1FAD_907F_FC3BL, /* -275 */ 0xB144_2798_F49F_FB4AL, /* -274 */ 0xDD95_317F_31C7_FA1DL, /* -273 */ 0x8A7D_3EEF_7F1C_FC52L, /* -272 */ 0xAD1C_8EAB_5EE4_3B66L, /* -271 */ 0xD863_B256_369D_4A40L, /* -270 */ 0x873E_4F75_E222_4E68L, /* -269 */ 0xA90D_E353_5AAA_E202L, /* -268 */ 0xD351_5C28_3155_9A83L, /* -267 */ 0x8412_D999_1ED5_8091L, /* -266 */ 0xA517_8FFF_668A_E0B6L, /* -265 */ 0xCE5D_73FF_402D_98E3L, /* -264 */ 0x80FA_687F_881C_7F8EL, /* -263 */ 0xA139_029F_6A23_9F72L, /* -262 */ 0xC987_4347_44AC_874EL, /* -261 */ 0xFBE9_1419_15D7_A922L, /* -260 */ 0x9D71_AC8F_ADA6_C9B5L, /* -259 */ 0xC4CE_17B3_9910_7C22L, /* -258 */ 0xF601_9DA0_7F54_9B2BL, /* -257 */ 0x99C1_0284_4F94_E0FBL, /* -256 */ 0xC031_4325_637A_1939L, /* -255 */ 0xF03D_93EE_BC58_9F88L, /* -254 */ 0x9626_7C75_35B7_63B5L, /* -253 */ 0xBBB0_1B92_8325_3CA2L, /* -252 */ 0xEA9C_2277_23EE_8BCBL, /* -251 */ 0x92A1_958A_7675_175FL, /* -250 */ 0xB749_FAED_1412_5D36L, /* -249 */ 0xE51C_79A8_5916_F484L, /* -248 */ 0x8F31_CC09_37AE_58D2L, /* -247 */ 0xB2FE_3F0B_8599_EF07L, /* -246 */ 0xDFBD_CECE_6700_6AC9L, /* -245 */ 0x8BD6_A141_0060_42BDL, /* -244 */ 0xAECC_4991_4078_536DL, /* -243 */ 0xDA7F_5BF5_9096_6848L, /* -242 */ 0x888F_9979_7A5E_012DL, /* -241 */ 0xAAB3_7FD7_D8F5_8178L, /* -240 */ 0xD560_5FCD_CF32_E1D6L, /* -239 */ 0x855C_3BE0_A17F_CD26L, /* -238 */ 0xA6B3_4AD8_C9DF_C06FL, /* -237 */ 0xD060_1D8E_FC57_B08BL, /* -236 */ 0x823C_1279_5DB6_CE57L, /* -235 */ 0xA2CB_1717_B524_81EDL, /* -234 */ 0xCB7D_DCDD_A26D_A268L, /* -233 */ 0xFE5D_5415_0B09_0B02L, /* -232 */ 0x9EFA_548D_26E5_A6E1L, /* -231 */ 0xC6B8_E9B0_709F_109AL, /* -230 */ 0xF867_241C_8CC6_D4C0L, /* -229 */ 0x9B40_7691_D7FC_44F8L, /* -228 */ 0xC210_9436_4DFB_5636L, /* -227 */ 0xF294_B943_E17A_2BC4L, /* -226 */ 0x979C_F3CA_6CEC_5B5AL, /* -225 */ 0xBD84_30BD_0827_7231L, /* -224 */ 0xECE5_3CEC_4A31_4EBDL, /* -223 */ 0x940F_4613_AE5E_D136L, /* -222 */ 0xB913_1798_99F6_8584L, /* -221 */ 0xE757_DD7E_C074_26E5L, /* -220 */ 0x9096_EA6F_3848_984FL, /* -219 */ 0xB4BC_A50B_065A_BE63L, /* -218 */ 0xE1EB_CE4D_C7F1_6DFBL, /* -217 */ 0x8D33_60F0_9CF6_E4BDL, /* -216 */ 0xB080_392C_C434_9DECL, /* -215 */ 0xDCA0_4777_F541_C567L, /* -214 */ 0x89E4_2CAA_F949_1B60L, /* -213 */ 0xAC5D_37D5_B79B_6239L, /* -212 */ 0xD774_85CB_2582_3AC7L, /* -211 */ 0x86A8_D39E_F771_64BCL, /* -210 */ 0xA853_0886_B54D_BDEBL, /* -209 */ 0xD267_CAA8_62A1_2D66L, /* -208 */ 0x8380_DEA9_3DA4_BC60L, /* -207 */ 0xA461_1653_8D0D_EB78L, /* -206 */ 0xCD79_5BE8_7051_6656L, /* -205 */ 0x806B_D971_4632_DFF6L, /* -204 */ 0xA086_CFCD_97BF_97F3L, /* -203 */ 0xC8A8_83C0_FDAF_7DF0L, /* -202 */ 0xFAD2_A4B1_3D1B_5D6CL, /* -201 */ 0x9CC3_A6EE_C631_1A63L, /* -200 */ 0xC3F4_90AA_77BD_60FCL, /* -199 */ 0xF4F1_B4D5_15AC_B93BL, /* -198 */ 0x9917_1105_2D8B_F3C5L, /* -197 */ 0xBF5C_D546_78EE_F0B6L, /* -196 */ 0xEF34_0A98_172A_ACE4L, /* -195 */ 0x9580_869F_0E7A_AC0EL, /* -194 */ 0xBAE0_A846_D219_5712L, /* -193 */ 0xE998_D258_869F_ACD7L, /* -192 */ 0x91FF_8377_5423_CC06L, /* -191 */ 0xB67F_6455_292C_BF08L, /* -190 */ 0xE41F_3D6A_7377_EECAL, /* -189 */ 0x8E93_8662_882A_F53EL, /* -188 */ 0xB238_67FB_2A35_B28DL, /* -187 */ 0xDEC6_81F9_F4C3_1F31L, /* -186 */ 0x8B3C_113C_38F9_F37EL, /* -185 */ 0xAE0B_158B_4738_705EL, /* -184 */ 0xD98D_DAEE_1906_8C76L, /* -183 */ 0x87F8_A8D4_CFA4_17C9L, /* -182 */ 0xA9F6_D30A_038D_1DBCL, /* -181 */ 0xD474_87CC_8470_652BL, /* -180 */ 0x84C8_D4DF_D2C6_3F3BL, /* -179 */ 0xA5FB_0A17_C777_CF09L, /* -178 */ 0xCF79_CC9D_B955_C2CCL, /* -177 */ 0x81AC_1FE2_93D5_99BFL, /* -176 */ 0xA217_27DB_38CB_002FL, /* -175 */ 0xCA9C_F1D2_06FD_C03BL, /* -174 */ 0xFD44_2E46_88BD_304AL, /* -173 */ 0x9E4A_9CEC_1576_3E2EL, /* -172 */ 0xC5DD_4427_1AD3_CDBAL, /* -171 */ 0xF754_9530_E188_C128L, /* -170 */ 0x9A94_DD3E_8CF5_78B9L, /* -169 */ 0xC13A_148E_3032_D6E7L, /* -168 */ 0xF188_99B1_BC3F_8CA1L, /* -167 */ 0x96F5_600F_15A7_B7E5L, /* -166 */ 0xBCB2_B812_DB11_A5DEL, /* -165 */ 0xEBDF_6617_91D6_0F56L, /* -164 */ 0x936B_9FCE_BB25_C995L, /* -163 */ 0xB846_87C2_69EF_3BFBL, /* -162 */ 0xE658_29B3_046B_0AFAL, /* -161 */ 0x8FF7_1A0F_E2C2_E6DCL, /* -160 */ 0xB3F4_E093_DB73_A093L, /* -159 */ 0xE0F2_18B8_D250_88B8L, /* -158 */ 0x8C97_4F73_8372_5573L, /* -157 */ 0xAFBD_2350_644E_EACFL, /* -156 */ 0xDBAC_6C24_7D62_A583L, /* -155 */ 0x894B_C396_CE5D_A772L, /* -154 */ 0xAB9E_B47C_81F5_114FL, /* -153 */ 0xD686_619B_A272_55A2L, /* -152 */ 0x8613_FD01_4587_7585L, /* -151 */ 0xA798_FC41_96E9_52E7L, /* -150 */ 0xD17F_3B51_FCA3_A7A0L, /* -149 */ 0x82EF_8513_3DE6_48C4L, /* -148 */ 0xA3AB_6658_0D5F_DAF5L, /* -147 */ 0xCC96_3FEE_10B7_D1B3L, /* -146 */ 0xFFBB_CFE9_94E5_C61FL, /* -145 */ 0x9FD5_61F1_FD0F_9BD3L, /* -144 */ 0xC7CA_BA6E_7C53_82C8L, /* -143 */ 0xF9BD_690A_1B68_637BL, /* -142 */ 0x9C16_61A6_5121_3E2DL, /* -141 */ 0xC31B_FA0F_E569_8DB8L, /* -140 */ 0xF3E2_F893_DEC3_F126L, /* -139 */ 0x986D_DB5C_6B3A_76B7L, /* -138 */ 0xBE89_5233_8609_1465L, /* -137 */ 0xEE2B_A6C0_678B_597FL, /* -136 */ 0x94DB_4838_40B7_17EFL, /* -135 */ 0xBA12_1A46_50E4_DDEBL, /* -134 */ 0xE896_A0D7_E51E_1566L, /* -133 */ 0x915E_2486_EF32_CD60L, /* -132 */ 0xB5B5_ADA8_AAFF_80B8L, /* -131 */ 0xE323_1912_D5BF_60E6L, /* -130 */ 0x8DF5_EFAB_C597_9C8FL, /* -129 */ 0xB173_6B96_B6FD_83B3L, /* -128 */ 0xDDD0_467C_64BC_E4A0L, /* -127 */ 0x8AA2_2C0D_BEF6_0EE4L, /* -126 */ 0xAD4A_B711_2EB3_929DL, /* -125 */ 0xD89D_64D5_7A60_7744L, /* -124 */ 0x8762_5F05_6C7C_4A8BL, /* -123 */ 0xA93A_F6C6_C79B_5D2DL, /* -122 */ 0xD389_B478_7982_3479L, /* -121 */ 0x8436_10CB_4BF1_60CBL, /* -120 */ 0xA543_94FE_1EED_B8FEL, /* -119 */ 0xCE94_7A3D_A6A9_273EL, /* -118 */ 0x811C_CC66_8829_B887L, /* -117 */ 0xA163_FF80_2A34_26A8L, /* -116 */ 0xC9BC_FF60_34C1_3052L, /* -115 */ 0xFC2C_3F38_41F1_7C67L, /* -114 */ 0x9D9B_A783_2936_EDC0L, /* -113 */ 0xC502_9163_F384_A931L, /* -112 */ 0xF643_35BC_F065_D37DL, /* -111 */ 0x99EA_0196_163F_A42EL, /* -110 */ 0xC064_81FB_9BCF_8D39L, /* -109 */ 0xF07D_A27A_82C3_7088L, /* -108 */ 0x964E_858C_91BA_2655L, /* -107 */ 0xBBE2_26EF_B628_AFEAL, /* -106 */ 0xEADA_B0AB_A3B2_DBE5L, /* -105 */ 0x92C8_AE6B_464F_C96FL, /* -104 */ 0xB77A_DA06_17E3_BBCBL, /* -103 */ 0xE559_9087_9DDC_AABDL, /* -102 */ 0x8F57_FA54_C2A9_EAB6L, /* -101 */ 0xB32D_F8E9_F354_6564L, /* -100 */ 0xDFF9_7724_7029_7EBDL, /* -99 */ 0x8BFB_EA76_C619_EF36L, /* -98 */ 0xAEFA_E514_77A0_6B03L, /* -97 */ 0xDAB9_9E59_9588_85C4L, /* -96 */ 0x88B4_02F7_FD75_539BL, /* -95 */ 0xAAE1_03B5_FCD2_A881L, /* -94 */ 0xD599_44A3_7C07_52A2L, /* -93 */ 0x857F_CAE6_2D84_93A5L, /* -92 */ 0xA6DF_BD9F_B8E5_B88EL, /* -91 */ 0xD097_AD07_A71F_26B2L, /* -90 */ 0x825E_CC24_C873_782FL, /* -89 */ 0xA2F6_7F2D_FA90_563BL, /* -88 */ 0xCBB4_1EF9_7934_6BCAL, /* -87 */ 0xFEA1_26B7_D781_86BCL, /* -86 */ 0x9F24_B832_E6B0_F436L, /* -85 */ 0xC6ED_E63F_A05D_3143L, /* -84 */ 0xF8A9_5FCF_8874_7D94L, /* -83 */ 0x9B69_DBE1_B548_CE7CL, /* -82 */ 0xC244_52DA_229B_021BL, /* -81 */ 0xF2D5_6790_AB41_C2A2L, /* -80 */ 0x97C5_60BA_6B09_19A5L, /* -79 */ 0xBDB6_B8E9_05CB_600FL, /* -78 */ 0xED24_6723_473E_3813L, /* -77 */ 0x9436_C076_0C86_E30BL, /* -76 */ 0xB944_7093_8FA8_9BCEL, /* -75 */ 0xE795_8CB8_7392_C2C2L, /* -74 */ 0x90BD_77F3_483B_B9B9L, /* -73 */ 0xB4EC_D5F0_1A4A_A828L, /* -72 */ 0xE228_0B6C_20DD_5232L, /* -71 */ 0x8D59_0723_948A_535FL, /* -70 */ 0xB0AF_48EC_79AC_E837L, /* -69 */ 0xDCDB_1B27_9818_2244L, /* -68 */ 0x8A08_F0F8_BF0F_156BL, /* -67 */ 0xAC8B_2D36_EED2_DAC5L, /* -66 */ 0xD7AD_F884_AA87_9177L, /* -65 */ 0x86CC_BB52_EA94_BAEAL, /* -64 */ 0xA87F_EA27_A539_E9A5L, /* -63 */ 0xD29F_E4B1_8E88_640EL, /* -62 */ 0x83A3_EEEE_F915_3E89L, /* -61 */ 0xA48C_EAAA_B75A_8E2BL, /* -60 */ 0xCDB0_2555_6531_31B6L, /* -59 */ 0x808E_1755_5F3E_BF11L, /* -58 */ 0xA0B1_9D2A_B70E_6ED6L, /* -57 */ 0xC8DE_0475_64D2_0A8BL, /* -56 */ 0xFB15_8592_BE06_8D2EL, /* -55 */ 0x9CED_737B_B6C4_183DL, /* -54 */ 0xC428_D05A_A475_1E4CL, /* -53 */ 0xF533_0471_4D92_65DFL, /* -52 */ 0x993F_E2C6_D07B_7FABL, /* -51 */ 0xBF8F_DB78_849A_5F96L, /* -50 */ 0xEF73_D256_A5C0_F77CL, /* -49 */ 0x95A8_6376_2798_9AADL, /* -48 */ 0xBB12_7C53_B17E_C159L, /* -47 */ 0xE9D7_1B68_9DDE_71AFL, /* -46 */ 0x9226_7121_62AB_070DL, /* -45 */ 0xB6B0_0D69_BB55_C8D1L, /* -44 */ 0xE45C_10C4_2A2B_3B05L, /* -43 */ 0x8EB9_8A7A_9A5B_04E3L, /* -42 */ 0xB267_ED19_40F1_C61CL, /* -41 */ 0xDF01_E85F_912E_37A3L, /* -40 */ 0x8B61_313B_BABC_E2C6L, /* -39 */ 0xAE39_7D8A_A96C_1B77L, /* -38 */ 0xD9C7_DCED_53C7_2255L, /* -37 */ 0x881C_EA14_545C_7575L, /* -36 */ 0xAA24_2499_6973_92D2L, /* -35 */ 0xD4AD_2DBF_C3D0_7787L, /* -34 */ 0x84EC_3C97_DA62_4AB4L, /* -33 */ 0xA627_4BBD_D0FA_DD61L, /* -32 */ 0xCFB1_1EAD_4539_94BAL, /* -31 */ 0x81CE_B32C_4B43_FCF4L, /* -30 */ 0xA242_5FF7_5E14_FC31L, /* -29 */ 0xCAD2_F7F5_359A_3B3EL, /* -28 */ 0xFD87_B5F2_8300_CA0DL, /* -27 */ 0x9E74_D1B7_91E0_7E48L, /* -26 */ 0xC612_0625_7658_9DDAL, /* -25 */ 0xF796_87AE_D3EE_C551L, /* -24 */ 0x9ABE_14CD_4475_3B52L, /* -23 */ 0xC16D_9A00_9592_8A27L, /* -22 */ 0xF1C9_0080_BAF7_2CB1L, /* -21 */ 0x971D_A050_74DA_7BEEL, /* -20 */ 0xBCE5_0864_9211_1AEAL, /* -19 */ 0xEC1E_4A7D_B695_61A5L, /* -18 */ 0x9392_EE8E_921D_5D07L, /* -17 */ 0xB877_AA32_36A4_B449L, /* -16 */ 0xE695_94BE_C44D_E15BL, /* -15 */ 0x901D_7CF7_3AB0_ACD9L, /* -14 */ 0xB424_DC35_095C_D80FL, /* -13 */ 0xE12E_1342_4BB4_0E13L, /* -12 */ 0x8CBC_CC09_6F50_88CBL, /* -11 */ 0xAFEB_FF0B_CB24_AAFEL, /* -10 */ 0xDBE6_FECE_BDED_D5BEL, /* -9 */ 0x8970_5F41_36B4_A597L, /* -8 */ 0xABCC_7711_8461_CEFCL, /* -7 */ 0xD6BF_94D5_E57A_42BCL, /* -6 */ 0x8637_BD05_AF6C_69B5L, /* -5 */ 0xA7C5_AC47_1B47_8423L, /* -4 */ 0xD1B7_1758_E219_652BL, /* -3 */ 0x8312_6E97_8D4F_DF3BL, /* -2 */ 0xA3D7_0A3D_70A3_D70AL, /* -1 */ 0xCCCC_CCCC_CCCC_CCCCL, /* 0 */ 0x8000_0000_0000_0000L, /* 1 */ 0xA000_0000_0000_0000L, /* 2 */ 0xC800_0000_0000_0000L, /* 3 */ 0xFA00_0000_0000_0000L, /* 4 */ 0x9C40_0000_0000_0000L, /* 5 */ 0xC350_0000_0000_0000L, /* 6 */ 0xF424_0000_0000_0000L, /* 7 */ 0x9896_8000_0000_0000L, /* 8 */ 0xBEBC_2000_0000_0000L, /* 9 */ 0xEE6B_2800_0000_0000L, /* 10 */ 0x9502_F900_0000_0000L, /* 11 */ 0xBA43_B740_0000_0000L, /* 12 */ 0xE8D4_A510_0000_0000L, /* 13 */ 0x9184_E72A_0000_0000L, /* 14 */ 0xB5E6_20F4_8000_0000L, /* 15 */ 0xE35F_A931_A000_0000L, /* 16 */ 0x8E1B_C9BF_0400_0000L, /* 17 */ 0xB1A2_BC2E_C500_0000L, /* 18 */ 0xDE0B_6B3A_7640_0000L, /* 19 */ 0x8AC7_2304_89E8_0000L, /* 20 */ 0xAD78_EBC5_AC62_0000L, /* 21 */ 0xD8D7_26B7_177A_8000L, /* 22 */ 0x8786_7832_6EAC_9000L, /* 23 */ 0xA968_163F_0A57_B400L, /* 24 */ 0xD3C2_1BCE_CCED_A100L, /* 25 */ 0x8459_5161_4014_84A0L, /* 26 */ 0xA56F_A5B9_9019_A5C8L, /* 27 */ 0xCECB_8F27_F420_0F3AL, /* 28 */ 0x813F_3978_F894_0984L, /* 29 */ 0xA18F_07D7_36B9_0BE5L, /* 30 */ 0xC9F2_C9CD_0467_4EDEL, /* 31 */ 0xFC6F_7C40_4581_2296L, /* 32 */ 0x9DC5_ADA8_2B70_B59DL, /* 33 */ 0xC537_1912_364C_E305L, /* 34 */ 0xF684_DF56_C3E0_1BC6L, /* 35 */ 0x9A13_0B96_3A6C_115CL, /* 36 */ 0xC097_CE7B_C907_15B3L, /* 37 */ 0xF0BD_C21A_BB48_DB20L, /* 38 */ 0x9676_9950_B50D_88F4L, /* 39 */ 0xBC14_3FA4_E250_EB31L, /* 40 */ 0xEB19_4F8E_1AE5_25FDL, /* 41 */ 0x92EF_D1B8_D0CF_37BEL, /* 42 */ 0xB7AB_C627_0503_05ADL, /* 43 */ 0xE596_B7B0_C643_C719L, /* 44 */ 0x8F7E_32CE_7BEA_5C6FL, /* 45 */ 0xB35D_BF82_1AE4_F38BL, /* 46 */ 0xE035_2F62_A19E_306EL, /* 47 */ 0x8C21_3D9D_A502_DE45L, /* 48 */ 0xAF29_8D05_0E43_95D6L, /* 49 */ 0xDAF3_F046_51D4_7B4CL, /* 50 */ 0x88D8_762B_F324_CD0FL, /* 51 */ 0xAB0E_93B6_EFEE_0053L, /* 52 */ 0xD5D2_38A4_ABE9_8068L, /* 53 */ 0x85A3_6366_EB71_F041L, /* 54 */ 0xA70C_3C40_A64E_6C51L, /* 55 */ 0xD0CF_4B50_CFE2_0765L, /* 56 */ 0x8281_8F12_81ED_449FL, /* 57 */ 0xA321_F2D7_2268_95C7L, /* 58 */ 0xCBEA_6F8C_EB02_BB39L, /* 59 */ 0xFEE5_0B70_25C3_6A08L, /* 60 */ 0x9F4F_2726_179A_2245L, /* 61 */ 0xC722_F0EF_9D80_AAD6L, /* 62 */ 0xF8EB_AD2B_84E0_D58BL, /* 63 */ 0x9B93_4C3B_330C_8577L, /* 64 */ 0xC278_1F49_FFCF_A6D5L, /* 65 */ 0xF316_271C_7FC3_908AL, /* 66 */ 0x97ED_D871_CFDA_3A56L, /* 67 */ 0xBDE9_4E8E_43D0_C8ECL, /* 68 */ 0xED63_A231_D4C4_FB27L, /* 69 */ 0x945E_455F_24FB_1CF8L, /* 70 */ 0xB975_D6B6_EE39_E436L, /* 71 */ 0xE7D3_4C64_A9C8_5D44L, /* 72 */ 0x90E4_0FBE_EA1D_3A4AL, /* 73 */ 0xB51D_13AE_A4A4_88DDL, /* 74 */ 0xE264_589A_4DCD_AB14L, /* 75 */ 0x8D7E_B760_70A0_8AECL, /* 76 */ 0xB0DE_6538_8CC8_ADA8L, /* 77 */ 0xDD15_FE86_AFFA_D912L, /* 78 */ 0x8A2D_BF14_2DFC_C7ABL, /* 79 */ 0xACB9_2ED9_397B_F996L, /* 80 */ 0xD7E7_7A8F_87DA_F7FBL, /* 81 */ 0x86F0_AC99_B4E8_DAFDL, /* 82 */ 0xA8AC_D7C0_2223_11BCL, /* 83 */ 0xD2D8_0DB0_2AAB_D62BL, /* 84 */ 0x83C7_088E_1AAB_65DBL, /* 85 */ 0xA4B8_CAB1_A156_3F52L, /* 86 */ 0xCDE6_FD5E_09AB_CF26L, /* 87 */ 0x80B0_5E5A_C60B_6178L, /* 88 */ 0xA0DC_75F1_778E_39D6L, /* 89 */ 0xC913_936D_D571_C84CL, /* 90 */ 0xFB58_7849_4ACE_3A5FL, /* 91 */ 0x9D17_4B2D_CEC0_E47BL, /* 92 */ 0xC45D_1DF9_4271_1D9AL, /* 93 */ 0xF574_6577_930D_6500L, /* 94 */ 0x9968_BF6A_BBE8_5F20L, /* 95 */ 0xBFC2_EF45_6AE2_76E8L, /* 96 */ 0xEFB3_AB16_C59B_14A2L, /* 97 */ 0x95D0_4AEE_3B80_ECE5L, /* 98 */ 0xBB44_5DA9_CA61_281FL, /* 99 */ 0xEA15_7514_3CF9_7226L, /* 100 */ 0x924D_692C_A61B_E758L, /* 101 */ 0xB6E0_C377_CFA2_E12EL, /* 102 */ 0xE498_F455_C38B_997AL, /* 103 */ 0x8EDF_98B5_9A37_3FECL, /* 104 */ 0xB297_7EE3_00C5_0FE7L, /* 105 */ 0xDF3D_5E9B_C0F6_53E1L, /* 106 */ 0x8B86_5B21_5899_F46CL, /* 107 */ 0xAE67_F1E9_AEC0_7187L, /* 108 */ 0xDA01_EE64_1A70_8DE9L, /* 109 */ 0x8841_34FE_9086_58B2L, /* 110 */ 0xAA51_823E_34A7_EEDEL, /* 111 */ 0xD4E5_E2CD_C1D1_EA96L, /* 112 */ 0x850F_ADC0_9923_329EL, /* 113 */ 0xA653_9930_BF6B_FF45L, /* 114 */ 0xCFE8_7F7C_EF46_FF16L, /* 115 */ 0x81F1_4FAE_158C_5F6EL, /* 116 */ 0xA26D_A399_9AEF_7749L, /* 117 */ 0xCB09_0C80_01AB_551CL, /* 118 */ 0xFDCB_4FA0_0216_2A63L, /* 119 */ 0x9E9F_11C4_014D_DA7EL, /* 120 */ 0xC646_D635_01A1_511DL, /* 121 */ 0xF7D8_8BC2_4209_A565L, /* 122 */ 0x9AE7_5759_6946_075FL, /* 123 */ 0xC1A1_2D2F_C397_8937L, /* 124 */ 0xF209_787B_B47D_6B84L, /* 125 */ 0x9745_EB4D_50CE_6332L, /* 126 */ 0xBD17_6620_A501_FBFFL, /* 127 */ 0xEC5D_3FA8_CE42_7AFFL, /* 128 */ 0x93BA_47C9_80E9_8CDFL, /* 129 */ 0xB8A8_D9BB_E123_F017L, /* 130 */ 0xE6D3_102A_D96C_EC1DL, /* 131 */ 0x9043_EA1A_C7E4_1392L, /* 132 */ 0xB454_E4A1_79DD_1877L, /* 133 */ 0xE16A_1DC9_D854_5E94L, /* 134 */ 0x8CE2_529E_2734_BB1DL, /* 135 */ 0xB01A_E745_B101_E9E4L, /* 136 */ 0xDC21_A117_1D42_645DL, /* 137 */ 0x8995_04AE_7249_7EBAL, /* 138 */ 0xABFA_45DA_0EDB_DE69L, /* 139 */ 0xD6F8_D750_9292_D603L, /* 140 */ 0x865B_8692_5B9B_C5C2L, /* 141 */ 0xA7F2_6836_F282_B732L, /* 142 */ 0xD1EF_0244_AF23_64FFL, /* 143 */ 0x8335_616A_ED76_1F1FL, /* 144 */ 0xA402_B9C5_A8D3_A6E7L, /* 145 */ 0xCD03_6837_1308_90A1L, /* 146 */ 0x8022_2122_6BE5_5A64L, /* 147 */ 0xA02A_A96B_06DE_B0FDL, /* 148 */ 0xC835_53C5_C896_5D3DL, /* 149 */ 0xFA42_A8B7_3ABB_F48CL, /* 150 */ 0x9C69_A972_84B5_78D7L, /* 151 */ 0xC384_13CF_25E2_D70DL, /* 152 */ 0xF465_18C2_EF5B_8CD1L, /* 153 */ 0x98BF_2F79_D599_3802L, /* 154 */ 0xBEEE_FB58_4AFF_8603L, /* 155 */ 0xEEAA_BA2E_5DBF_6784L, /* 156 */ 0x952A_B45C_FA97_A0B2L, /* 157 */ 0xBA75_6174_393D_88DFL, /* 158 */ 0xE912_B9D1_478C_EB17L, /* 159 */ 0x91AB_B422_CCB8_12EEL, /* 160 */ 0xB616_A12B_7FE6_17AAL, /* 161 */ 0xE39C_4976_5FDF_9D94L, /* 162 */ 0x8E41_ADE9_FBEB_C27DL, /* 163 */ 0xB1D2_1964_7AE6_B31CL, /* 164 */ 0xDE46_9FBD_99A0_5FE3L, /* 165 */ 0x8AEC_23D6_8004_3BEEL, /* 166 */ 0xADA7_2CCC_2005_4AE9L, /* 167 */ 0xD910_F7FF_2806_9DA4L, /* 168 */ 0x87AA_9AFF_7904_2286L, /* 169 */ 0xA995_41BF_5745_2B28L, /* 170 */ 0xD3FA_922F_2D16_75F2L, /* 171 */ 0x847C_9B5D_7C2E_09B7L, /* 172 */ 0xA59B_C234_DB39_8C25L, /* 173 */ 0xCF02_B2C2_1207_EF2EL, /* 174 */ 0x8161_AFB9_4B44_F57DL, /* 175 */ 0xA1BA_1BA7_9E16_32DCL, /* 176 */ 0xCA28_A291_859B_BF93L, /* 177 */ 0xFCB2_CB35_E702_AF78L, /* 178 */ 0x9DEF_BF01_B061_ADABL, /* 179 */ 0xC56B_AEC2_1C7A_1916L, /* 180 */ 0xF6C6_9A72_A398_9F5BL, /* 181 */ 0x9A3C_2087_A63F_6399L, /* 182 */ 0xC0CB_28A9_8FCF_3C7FL, /* 183 */ 0xF0FD_F2D3_F3C3_0B9FL, /* 184 */ 0x969E_B7C4_7859_E743L, /* 185 */ 0xBC46_65B5_9670_6114L, /* 186 */ 0xEB57_FF22_FC0C_7959L, /* 187 */ 0x9316_FF75_DD87_CBD8L, /* 188 */ 0xB7DC_BF53_54E9_BECEL, /* 189 */ 0xE5D3_EF28_2A24_2E81L, /* 190 */ 0x8FA4_7579_1A56_9D10L, /* 191 */ 0xB38D_92D7_60EC_4455L, /* 192 */ 0xE070_F78D_3927_556AL, /* 193 */ 0x8C46_9AB8_43B8_9562L, /* 194 */ 0xAF58_4166_54A6_BABBL, /* 195 */ 0xDB2E_51BF_E9D0_696AL, /* 196 */ 0x88FC_F317_F222_41E2L, /* 197 */ 0xAB3C_2FDD_EEAA_D25AL, /* 198 */ 0xD60B_3BD5_6A55_86F1L, /* 199 */ 0x85C7_0565_6275_7456L, /* 200 */ 0xA738_C6BE_BB12_D16CL, /* 201 */ 0xD106_F86E_69D7_85C7L, /* 202 */ 0x82A4_5B45_0226_B39CL, /* 203 */ 0xA34D_7216_42B0_6084L, /* 204 */ 0xCC20_CE9B_D35C_78A5L, /* 205 */ 0xFF29_0242_C833_96CEL, /* 206 */ 0x9F79_A169_BD20_3E41L, /* 207 */ 0xC758_09C4_2C68_4DD1L, /* 208 */ 0xF92E_0C35_3782_6145L, /* 209 */ 0x9BBC_C7A1_42B1_7CCBL, /* 210 */ 0xC2AB_F989_935D_DBFEL, /* 211 */ 0xF356_F7EB_F835_52FEL, /* 212 */ 0x9816_5AF3_7B21_53DEL, /* 213 */ 0xBE1B_F1B0_59E9_A8D6L, /* 214 */ 0xEDA2_EE1C_7064_130CL, /* 215 */ 0x9485_D4D1_C63E_8BE7L, /* 216 */ 0xB9A7_4A06_37CE_2EE1L, /* 217 */ 0xE811_1C87_C5C1_BA99L, /* 218 */ 0x910A_B1D4_DB99_14A0L, /* 219 */ 0xB54D_5E4A_127F_59C8L, /* 220 */ 0xE2A0_B5DC_971F_303AL, /* 221 */ 0x8DA4_71A9_DE73_7E24L, /* 222 */ 0xB10D_8E14_5610_5DADL, /* 223 */ 0xDD50_F199_6B94_7518L, /* 224 */ 0x8A52_96FF_E33C_C92FL, /* 225 */ 0xACE7_3CBF_DC0B_FB7BL, /* 226 */ 0xD821_0BEF_D30E_FA5AL, /* 227 */ 0x8714_A775_E3E9_5C78L, /* 228 */ 0xA8D9_D153_5CE3_B396L, /* 229 */ 0xD310_45A8_341C_A07CL, /* 230 */ 0x83EA_2B89_2091_E44DL, /* 231 */ 0xA4E4_B66B_68B6_5D60L, /* 232 */ 0xCE1D_E406_42E3_F4B9L, /* 233 */ 0x80D2_AE83_E9CE_78F3L, /* 234 */ 0xA107_5A24_E442_1730L, /* 235 */ 0xC949_30AE_1D52_9CFCL, /* 236 */ 0xFB9B_7CD9_A4A7_443CL, /* 237 */ 0x9D41_2E08_06E8_8AA5L, /* 238 */ 0xC491_798A_08A2_AD4EL, /* 239 */ 0xF5B5_D7EC_8ACB_58A2L, /* 240 */ 0x9991_A6F3_D6BF_1765L, /* 241 */ 0xBFF6_10B0_CC6E_DD3FL, /* 242 */ 0xEFF3_94DC_FF8A_948EL, /* 243 */ 0x95F8_3D0A_1FB6_9CD9L, /* 244 */ 0xBB76_4C4C_A7A4_440FL, /* 245 */ 0xEA53_DF5F_D18D_5513L, /* 246 */ 0x9274_6B9B_E2F8_552CL, /* 247 */ 0xB711_8682_DBB6_6A77L, /* 248 */ 0xE4D5_E823_92A4_0515L, /* 249 */ 0x8F05_B116_3BA6_832DL, /* 250 */ 0xB2C7_1D5B_CA90_23F8L, /* 251 */ 0xDF78_E4B2_BD34_2CF6L, /* 252 */ 0x8BAB_8EEF_B640_9C1AL, /* 253 */ 0xAE96_72AB_A3D0_C320L, /* 254 */ 0xDA3C_0F56_8CC4_F3E8L, /* 255 */ 0x8865_8996_17FB_1871L, /* 256 */ 0xAA7E_EBFB_9DF9_DE8DL, /* 257 */ 0xD51E_A6FA_8578_5631L, /* 258 */ 0x8533_285C_936B_35DEL, /* 259 */ 0xA67F_F273_B846_0356L, /* 260 */ 0xD01F_EF10_A657_842CL, /* 261 */ 0x8213_F56A_67F6_B29BL, /* 262 */ 0xA298_F2C5_01F4_5F42L, /* 263 */ 0xCB3F_2F76_4271_7713L, /* 264 */ 0xFE0E_FB53_D30D_D4D7L, /* 265 */ 0x9EC9_5D14_63E8_A506L, /* 266 */ 0xC67B_B459_7CE2_CE48L, /* 267 */ 0xF81A_A16F_DC1B_81DAL, /* 268 */ 0x9B10_A4E5_E991_3128L, /* 269 */ 0xC1D4_CE1F_63F5_7D72L, /* 270 */ 0xF24A_01A7_3CF2_DCCFL, /* 271 */ 0x976E_4108_8617_CA01L, /* 272 */ 0xBD49_D14A_A79D_BC82L, /* 273 */ 0xEC9C_459D_5185_2BA2L, /* 274 */ 0x93E1_AB82_52F3_3B45L, /* 275 */ 0xB8DA_1662_E7B0_0A17L, /* 276 */ 0xE710_9BFB_A19C_0C9DL, /* 277 */ 0x906A_617D_4501_87E2L, /* 278 */ 0xB484_F9DC_9641_E9DAL, /* 279 */ 0xE1A6_3853_BBD2_6451L, /* 280 */ 0x8D07_E334_5563_7EB2L, /* 281 */ 0xB049_DC01_6ABC_5E5FL, /* 282 */ 0xDC5C_5301_C56B_75F7L, /* 283 */ 0x89B9_B3E1_1B63_29BAL, /* 284 */ 0xAC28_20D9_623B_F429L, /* 285 */ 0xD732_290F_BACA_F133L, /* 286 */ 0x867F_59A9_D4BE_D6C0L, /* 287 */ 0xA81F_3014_49EE_8C70L, /* 288 */ 0xD226_FC19_5C6A_2F8CL, /* 289 */ 0x8358_5D8F_D9C2_5DB7L, /* 290 */ 0xA42E_74F3_D032_F525L, /* 291 */ 0xCD3A_1230_C43F_B26FL, /* 292 */ 0x8044_4B5E_7AA7_CF85L, /* 293 */ 0xA055_5E36_1951_C366L, /* 294 */ 0xC86A_B5C3_9FA6_3440L, /* 295 */ 0xFA85_6334_878F_C150L, /* 296 */ 0x9C93_5E00_D4B9_D8D2L, /* 297 */ 0xC3B8_3581_09E8_4F07L, /* 298 */ 0xF4A6_42E1_4C62_62C8L, /* 299 */ 0x98E7_E9CC_CFBD_7DBDL, /* 300 */ 0xBF21_E440_03AC_DD2CL, /* 301 */ 0xEEEA_5D50_0498_1478L, /* 302 */ 0x9552_7A52_02DF_0CCBL, /* 303 */ 0xBAA7_18E6_8396_CFFDL, /* 304 */ 0xE950_DF20_247C_83FDL, /* 305 */ 0x91D2_8B74_16CD_D27EL, /* 306 */ 0xB647_2E51_1C81_471DL, /* 307 */ 0xE3D8_F9E5_63A1_98E5L, /* 308 */ 0x8E67_9C2F_5E44_FF8FL, /* 309 */ 0xB201_833B_35D6_3F73L, /* 310 */ 0xDE81_E40A_034B_CF4FL, /* 311 */ 0x8B11_2E86_420F_6191L, /* 312 */ 0xADD5_7A27_D293_39F6L, /* 313 */ 0xD94A_D8B1_C738_0874L, /* 314 */ 0x87CE_C76F_1C83_0548L, /* 315 */ 0xA9C2_794A_E3A3_C69AL, /* 316 */ 0xD433_179D_9C8C_B841L, /* 317 */ 0x849F_EEC2_81D7_F328L, /* 318 */ 0xA5C7_EA73_224D_EFF3L, /* 319 */ 0xCF39_E50F_EAE1_6BEFL, /* 320 */ 0x8184_2F29_F2CC_E375L, /* 321 */ 0xA1E5_3AF4_6F80_1C53L, /* 322 */ 0xCA5E_89B1_8B60_2368L, /* 323 */ 0xFCF6_2C1D_EE38_2C42L, /* 324 */ 0x9E19_DB92_B4E3_1BA9L, }; } -------- math.DoubleToDecimal.java /* * Copyright (c) 2018, Raffaello Giulietti. 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. * This particular file is subject to the "Classpath" exception as provided * 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. */ package math; import static java.lang.Double.*; import static java.lang.Math.max; import static java.lang.Long.numberOfLeadingZeros; import static math.MathUtils.*; import static math.DoubleToDecimal.Double.*; import static math.Natural.valueOfShiftLeft; import static math.Powers.*; /** * This class exposes a method to render a {@code double} as a string. */ final public class DoubleToDecimal { /* For full details of the logic in this and the other supporting classes, search the web for d6b9e38fbe27f199d27e19f25acc26452e7e2ece and check that the title reads "Rendering doubles in Java" */ /** * Exposes some constants related to the IEEE 754-2008 breakdown of * {@code double}s and some extractors suited for finite positive values. * * <p>A finite positive {@code double} <i>v</i> has the form * <i>v</i> = <i>c</i>·2<sup><i>q</i></sup>, * where integers <i>c</i>, <i>q</i> meet * <ul> * <li> either 2<sup>{@link #P}-1</sup> ≤ <i>c</i> < * 2<sup>{@link #P}</sup> and {@link #Q_MIN} ≤ <i>q</i> ≤ * {@link #Q_MAX} (normal <i>v</i>) * <li> or 0 < <i>c</i> < 2<sup>{@link #P}-1</sup> and * <i>c</i> = {@link #Q_MIN} (subnormal <i>v</i>) * </ul> */ static final class Double { /** * Precision of normal values in bits. */ static final int P = 53; /** * Length in bits of the exponent field. */ static final int W = (java.lang.Double.SIZE - 1) - (P - 1); /** * Minimum value of the exponent. */ static final int Q_MIN = (-1 << W - 1) - P + 3; /** * Maximum value of the exponent. */ static final int Q_MAX = (1 << W - 1) - P; /** * Minimum value of the coefficient of a normal value. */ static final long C_MIN = 1L << P - 1; /** * Maximum value of the coefficient of a normal value. */ static final long C_MAX = (1L << P) - 1; /** * H = min {n integer | 10^(n-1) > 2^P} */ static final int H = 17; /** * G = max {n integer | 2^(P-1) > 10^n} */ static final int G = 15; /** * The integer <i>e</i> such that * 10<sup><i>e</i>-1</sup> ≤ {@link java.lang.Double#MIN_VALUE} * < 10<sup><i>e</i></sup>. */ static final int E_MIN_VALUE = -323; /** * The integer <i>e</i> such that * 10<sup><i>e</i>-1</sup> ≤ {@link java.lang.Double#MIN_NORMAL} * < 10<sup><i>e</i></sup>. */ static final int E_MIN_NORMAL = -307; /** * The integer <i>e</i> such that * 10<sup><i>e</i>-1</sup> ≤ {@link java.lang.Double#MAX_VALUE} * < 10<sup><i>e</i></sup>. */ static final int E_MAX_VALUE = 309; // Mask to extract the IEEE 754-2008 biased exponent. private static final int BQ_MASK = (1 << W) - 1; // Mask to extract the IEEE 754-2008 fraction bits. private static final long T_MASK = (1L << P - 1) - 1; // Constants for the computation of roundCeilPow10() private static final int D = Long.SIZE - P; private static final long CEIL_EPS = (1L << D) - 1; private static final int ORD_2_MIN_NORMAL = Q_MIN + P; private Double() { } private static int bq(long bits) { return (int) (bits >>> P - 1) & BQ_MASK; } /** * Given the {@code bits} of a finite positive {@code double}, * returns <i>q</i> described in {@link java.lang.Double}. */ static int q(long bits) { int bq = bq(bits); if (bq > 0) { return Q_MIN - 1 + bq; } return Q_MIN; } /** * Given the {@code bits} of a finite positive {@code double}, * returns <i>c</i> described in {@link java.lang.Double}. */ static long c(long bits) { int bq = bq(bits); long t = bits & T_MASK; if (bq > 0) { return C_MIN | t; } return t; } private static int ord2(int q, long c) { // Fast path for the normal case. if (c >= C_MIN) { return P + q; } return Q_MIN + Long.SIZE - numberOfLeadingZeros(c); } /** * For finite positive {@code v}, returns the integer <i>e</i> such that * 2<sup><i>e</i>-1</sup> ≤ {@code v} < * 2<sup><i>e</i></sup>. */ private static int ord2(double v) { long bits = java.lang.Double.doubleToRawLongBits(v); return ord2(q(bits), c(bits)); } /** * For finite positive {@code v}, returns the integer <i>e</i> such that * 10<sup><i>e</i>-1</sup> ≤ {@code v} < * 10<sup><i>e</i></sup>. */ static int ord10(double v) { int ep = ord10pow2(ord2(v)) - 1; if (v < roundCeilPow10(ep)) return ep; return ep + 1; } // Returns the smallest double v such that 10^e <= v. private static double roundCeilPow10(int e) { int e2 = ord2pow10(e); if (e2 >= ORD_2_MIN_NORMAL) { long c = (floorPow10d(e) + CEIL_EPS) >>> D; int q = e2 - P; long bits = (long) (q - (Q_MIN - 1)) << P - 1 | c & T_MASK; return java.lang.Double.longBitsToDouble(bits); } int d = ORD_2_MIN_NORMAL + D - e2; if (d < Long.SIZE) { long c = (floorPow10d(e) + (1L << d) - 1) >>> d; return java.lang.Double.longBitsToDouble(c); } return java.lang.Double.MIN_VALUE; } } // used in the left-to-right extraction of the digits private static final int LTR = 28; private static final int MASK_LTR = (1 << LTR) - 1; // MAX_SIGNIFICAND = 10^H private static final long MAX_SIGNIFICAND = 100_000_000_000_000_000L; // The additional precision, used in reduced() private static final int D = Long.SIZE - P; // for thread-safety, each thread gets its own instance of this class private static final ThreadLocal<DoubleToDecimal> threadLocal = ThreadLocal.withInitial(DoubleToDecimal::new); /* Given finite positive double v, there are two breakdowns: v = c * 2^q, as described in DoubleToDecimal.Double v = f * 10^e, with 0.1 <= f < 1 e, q, and c are kept in the following fields. */ private int e; private int q; private long c; /* For the default IEEE round-to-closest rounding, lout = rout always holds. However, two fields are kept for possible future extensions. Possible values are 0, if the boundary of the rounding interval is included 1, if the boundary of the rounding interval is excluded */ private int lout; // left (closer to 0) boundary private int rout; // right (farther from 0) boundary /* Room for H digits, 3 exponent digits, 2 '-', 1 '.', 1 'E' = H + 7 or for "-0.00" + H digits = H + 5 */ private final char[] buf = new char[H + 7]; private int index; // index of rightmost valid character private DoubleToDecimal() { } /** * Returns a string rendering of the {@code double} argument. * * <p>The characters of the result are all drawn from the ASCII set. * <ul> * <li> Any NaN, whether quiet or signaling, is rendered symbolically * as {@code "NaN"}, regardless of the sign bit. * <li> The infinities +∞ and -∞ are rendered as * {@code "Infinity"} and {@code "-Infinity"}, respectively. * <li> The zeroes +0.0 and -0.0 are rendered as * {@code "0.0"} and {@code "-0.0"}, respectively. * <li> Otherwise {@code v} is finite and non-zero. * It is rendered in two stages: * <ul> * <li> Selection of a decimal: A well-specified non-zero decimal * <i>d</i> is selected to represent {@code v}. * <li> Formatting as a string: The decimal <i>d</i> is formatted * as a string, either in plain or in computerized scientific * notation, depending on its value. * </ul> * </ul> * * <p>A decimal <i>d</i> is said to have length <i>i</i> if it has * the form <i>d</i> = <i>c</i> · 10<sup><i>q</i></sup> * for some integers <i>c</i> and <i>q</i> and if the decimal expansion of * <i>c</i> consists of <i>i</i> digits. Note that if <i>d</i> has some * length, then it has any other greater length as well: grow <i>c</i> by * appending trailing zeroes and decrease <i>q</i> accordingly. * * <p>Abstractly, the unique decimal <i>d</i> to represent {@code v} * is selected as follows: * <ul> * <li>First, all decimals that round to {@code v} according to the * usual round-to-closest rule of IEEE 754 floating-point arithmetic * are tentatively selected, while the other are discarded. * There is never the need to go beyond a length of 17. * <li>Among these, only the ones that have the shortest possible * length not less than 2 are selected and the other are discarded. * <li>Finally, among these, only the one closest to {@code v} is * definitely selected: or if two are equally close to {@code v}, the * one whose least significant digit is even is definitely selected. * </ul> * * <p>The selected decimal <i>d</i> is then formatted as a string. * If <i>d</i> < 0, the first character of the string is the sign * '{@code -}'. Then consider the absolute value and let * |<i>d</i>| = <i>m</i> · 10<sup><i>k</i></sup>, for some unique * real <i>m</i> meeting 1 ≤ <i>m</i> < 10 and integer <i>k</i>. * Further, let the decimal expansion of <i>m</i> be * <i>m</i><sub>1</sub>.<i>m</i><sub>2</sub>…<!-- * --><i>m</i><sub><i>i</i></sub>, * with <i>i</i> ≥ 1 and <i>m</i><sub><i>i</i></sub> ≠ 0. * <ul> * <li>Case -3 ≤ k < 0: |<i>d</i>| is formatted as * 0.0…0<i>m</i><sub>1</sub>…<!-- * --><i>m</i><sub><i>i</i></sub>, * where there are exactly -<i>k</i> leading zeroes before * <i>m</i><sub>1</sub>, including the zero before the decimal point. * For example, {@code "0.01234"}. * <li>Case 0 ≤ <i>k</i> < 7: * <ul> * <li>Subcase <i>i</i> < <i>k</i> + 2: * |<i>d</i>| is formatted as * <i>m</i><sub>1</sub>…<!-- * --><i>m</i><sub><i>i</i></sub>0…0.0, * where there are exactly <i>k</i> + 2 - <i>i</i> trailing zeroes * after <i>m</i><sub><i>i</i></sub>, including the zero after * the decimal point. * For example, {@code "1200.0"}. * <li>Subcase <i>i</i> ≥ <i>k</i> + 2: * |<i>d</i>| is formatted as * <i>m</i><sub>1</sub>…<i>m</i><sub><i>k</i>+1</sub>.<!-- * --><i>m</i><sub><i>k</i>+2</sub>…<!-- * --><i>m</i><sub><i>i</i></sub>. * For example, {@code "1234.567"}. * </ul> * <li>Case <i>k</i> < -3 or <i>k</i> ≥ 7: * computerized scientific notation is used to format |<i>d</i>|, * by combining <i>m</i> and <i>k</i> separated by the exponent * indicator '{@code E}'. * <ul> * <li>Subcase <i>i</i> = 1: * |<i>d</i>| is formatted as * <i>m</i><sub>1</sub>.0E<i>k</i>. * For example, {@code "2.0E23"}. * <li>Subcase <i>i</i> > 1: * |<i>d</i>| is formatted as * <i>m</i><sub>1</sub>.<i>m</i><sub>2</sub>…<!-- * --><i>m</i><sub><i>i</i></sub>E<i>k</i>. * For example, {@code "1.2345E-67"}. * </ul> * The exponent <i>k</i> is formatted as in * {@link Integer#toString(int)}. * </ul> * * @param v the {@code double} to be rendered. * @return a string rendering of the argument. */ public static String toString(double v) { return threadLocalInstance().toDecimal(v); } private static DoubleToDecimal threadLocalInstance() { return threadLocal.get(); } private String toDecimal(double v) { // Get rid of NaNs, infinities and zeroes right at the beginning if (v != v) { return "NaN"; } if (v == POSITIVE_INFINITY) { return "Infinity"; } if (v == NEGATIVE_INFINITY) { return "-Infinity"; } long bits = doubleToRawLongBits(v); if (bits == 0x0000_0000_0000_0000L) { return "0.0"; } if (bits == 0x8000_0000_0000_0000L) { return "-0.0"; } index = -1; if (bits < 0) { append('-'); v = -v; } e = ord10(v); /* When v is an integer less than 10^9, a common case in practice, use a customized faster method. */ long l = (long) v; if (l == v & l < 1_000_000_000L) { return integer(l); } q = q(bits); c = c(bits); lout = rout = (int) (c) & 0x1; /* The reduced() method assumes v is normal, i.e., has full precision P, and that powers of 2 have unequally distant predecessor and successor. MIN_NORMAL is normal and a power of 2 but its predecessor and its successor are equally close to it, so is excluded from reduced(). Note that reduced() might failover to full(). */ if (v > MIN_NORMAL) { return reduced(); } return full(); } private String integer(long l) { return toChars(l * pow10[H - 8 - e], e); } private String full() { long cb; int qb; long cbr; if (c != C_MIN | q == Q_MIN) { cb = c << 1; qb = q - 1; cbr = cb + 1; } else { cb = c << 2; qb = q - 2; cbr = cb + 2; } if (e <= H) { if (e - qb <= H) { return fullCaseM(qb, cb, cbr); } if (H - 3 <= e) { return fullSubcaseS(qb, cb, cbr); } int p = q > Q_MIN || c > C_MIN ? P : Long.SIZE - numberOfLeadingZeros(c - 1); int i = max(ord10pow2(p - 1) - 1, 2); return fullCaseXS(qb, cb, cbr, i); } if (qb - e <= 8 - H) { return fullSubcaseL(qb, cb, cbr); } return fullCaseXL(qb, cb, cbr); } private String fullCaseXS(int qb, long cb, long cb_r, int i) { Natural m = pow5(H - e); Natural vb = m.multiply(cb); Natural vbl = vb.subtract(m); Natural vbr = m.multiply(cb_r); int p = e - H - qb; long sbH = vb.shiftRight(p); for (int g = H - i; g >= 0; --g) { long di = pow10[g]; long sbi = sbH - sbH % di; Natural ubi = valueOfShiftLeft(sbi, p); Natural wbi = valueOfShiftLeft(sbi + di, p); boolean uin = vbl.compareTo(ubi) + lout <= 0; boolean win = wbi.compareTo(vbr) + rout <= 0; if (uin & !win) { return toChars(sbi, e); } if (!uin & win) { return toChars(sbi + di, e); } if (uin) { int cmp = vb.closerTo(ubi, wbi); if (cmp < 0 || cmp == 0 && (sbi / di & 0x1) == 0) { return toChars(sbi, e); } return toChars(sbi + di, e); } } throw new AssertionError("unreachable code"); } private String fullSubcaseS(int qb, long cb, long cb_r) { long m = pow5[H - e]; long vb = cb * m; long vbl = vb - m; long vbr = cb_r * m; int p = e - H - qb; long sbH = vb >> p; for (int g = H - G; g >= 0; --g) { long di = pow10(g); long sbi = sbH - sbH % di; long ubi = sbi << p; long wbi = sbi + di << p; boolean uin = vbl + lout <= ubi; boolean win = wbi + rout <= vbr; if (uin & !win) { return toChars(sbi, e); } if (!uin & win) { return toChars(sbi + di, e); } if (uin) { int cmp = (int) (2 * vb - ubi - wbi); if (cmp < 0 || cmp == 0 && (sbi / di & 0x1) == 0) { return toChars(sbi, e); } return toChars(sbi + di, e); } } throw new AssertionError("unreachable code"); } private String fullCaseM(int qb, long cb, long cb_r) { long m = pow5[H - e] << H - e + qb; long vb = cb * m; long vbl = vb - m; long vbr = cb_r * m; for (int g = H - G; g > 0; --g) { long di = pow10(g); long sbi = vb - vb % di; long tbi = sbi + di; boolean uin = vbl + lout <= sbi; boolean win = tbi + rout <= vbr; if (uin & !win) { return toChars(sbi, e); } if (!uin & win) { return toChars(tbi, e); } if (uin) { int cmp = (int) (2 * vb - sbi - tbi); if (cmp < 0 || cmp == 0 && (vb / di & 0x1) == 0) { return toChars(sbi, e); } return toChars(tbi, e); } } /* The loop didn't produce a shorter result. The full sb_H = s_H = vb is needed. This is done outside the loop, as there's no need to check tb_H = t_H as well. */ return toChars(vb, e); } private String fullSubcaseL(int qb, long cb, long cb_r) { int p = H - e + qb; long vb = cb << p; long vbl = cb - 1 << p; long vbr = cb_r << p; long m = pow5[e - H]; long sbH = vb / m; for (int g = H - G; g >= 0; --g) { long di = pow10(g); long sbi = sbH - sbH % di; long ubi = sbi * m; long wbi = ubi + (pow5[e - H + g] << g); boolean uin = vbl + lout <= ubi; boolean win = wbi + rout <= vbr; if (uin & !win) { return toChars(sbi, e); } if (!uin & win) { return toChars(sbi + di, e); } if (uin) { int cmp = (int) (2 * vb - ubi - wbi); if (cmp < 0 || cmp == 0 && (sbi / di & 0x1) == 0) { return toChars(sbi, e); } return toChars(sbi + di, e); } } throw new AssertionError("unreachable code"); } private String fullCaseXL(int qb, long cb, long cb_r) { int p = H - e + qb; Natural vb = valueOfShiftLeft(cb, p); Natural vbl = valueOfShiftLeft(cb - 1, p); Natural vbr = valueOfShiftLeft(cb_r, p); Natural m = pow5(e - H); long sbH = vb.divide(m); for (int g = H - G; g >= 0; --g) { long di = pow10(g); long sbi = sbH - sbH % di; Natural ubi = m.multiply(sbi); Natural wbi = ubi.addShiftLeft(pow5(e - H + g), g); boolean uin = vbl.compareTo(ubi) + lout <= 0; boolean win = wbi.compareTo(vbr) + rout <= 0; if (uin & !win) { return toChars(sbi, e); } if (!uin & win) { return toChars(sbi + di, e); } if (uin) { int cmp = vb.closerTo(ubi, wbi); if (cmp < 0 || cmp == 0 && (sbi / di & 0x1) == 0) { return toChars(sbi, e); } return toChars(sbi + di, e); } } throw new AssertionError("unreachable code"); } /* A faster version that succeeds in about 99.5% of the cases. It must be invoked only on values greater than MIN_NORMAL. When it fails, it resorts to the full() version. */ private String reduced() { int p = -P - q - pow10r(H - e); long t = floorPow10d(H - e); long cb = c << D; long vh = multiplyHighUnsigned(cb, t); long cbl = cb - (1L << D - (c != Double.C_MIN ? 1 : 2)); long vhl = multiplyHighUnsigned(cbl, t); long cbr = cb + (1L << D - 1); long vhr = multiplyHighUnsigned(cbr, t); long shH = vh >>> p; long vhu = vh + 2; for (int g = H - G; g >= 0; --g) { long di = pow10(g); long uhi = shH - shH % di << p; long whi = uhi + (di << p); boolean uin = uhi - vhl >= 2; boolean wout = whi - vhr >= 2; if (uin & wout) { return toChars(uhi >>> p, e); } boolean uout = uhi - vhl - lout < 0; boolean win = whi - vhr + rout <= 0; if (uout & win) { return toChars(whi >>> p, e); } if (uin & win) { if (vhu - uhi <= whi - vhu) { return toChars(uhi >>> p, e); } if (whi - vh < vh - uhi) { return toChars(whi >>> p, e); } return full(); } if (!uout & !uin | !wout) { return full(); } } throw new AssertionError("unreachable code"); } /* Limited usage, but does magic during JIT compilation. Note that 0 <= g <= 2 = H - G when invoked, so the default branch is never taken. */ private long pow10(int g) { switch (g) { case 0: return 1; case 1: return 10; case 2: return 100; default: return 0; } } /* f comes from integer(), from full() or from reduced(). In the former case 10^8 <= f < 10^9 and the method formats the number (f * 10^8) * 10^(e-H). Otherwise 10^(H-1) <= f <= 10^H and the method formats the number f * 10^(e-H) Division is avoided, where possible, by replacing it with multiplications and shifts. This makes a noticeable difference in performance, in particular when generating the digits of the exponent. For more in-depth readings, see for example Moeller N, Granlund T, "Improved division by invariant integers" ridiculous_fish, "Labor of Division (Episode III): Faster Unsigned Division by Constants" Also, once the quotient is known, the remainder is computed indirectly. */ private String toChars(long f, int e) { int h; // the 1 most significant int m; // the next 8 most significant digits int l; // the 8 least significant digits if (f != MAX_SIGNIFICAND) { long hm; if (f < 1_000_000_000L) { hm = f; l = 0; } else { hm = f / 100_000_000L; l = (int) (f - 100_000_000L * hm); } h = (int) (hm * 1_441_151_881L >>> 57); // h = hm / 100_000_000 m = (int) (hm - 100_000_000 * h); } else { // This might happen for doubles close or equal to powers of 10 h = 1; m = l = 0; e += 1; } /* The left-to-right digits generation in toChars_* is inspired by Bouvier C, Zimmermann P, "Division-Free Binary-to-Decimal Conversion" */ if (0 < e && e <= 7) { return toChars_1(h, m, l, e); } if (-3 < e && e <= 0) { return toChars_2(h, m, l, e); } return toChars_3(h, m, l, e); } private String toChars_1(int h, int m, int l, int e) { // 0 < e <= 7 appendDigit(h); int y = (int) (((long) (m + 1) << LTR) / 100_000_000L) - 1; int t; int i = 1; for (; i < e; ++i) { t = 10 * y; appendDigit(t >>> LTR); y = t & MASK_LTR; } append('.'); for (; i <= 8; ++i) { t = 10 * y; appendDigit(t >>> LTR); y = t & MASK_LTR; } lowDigits(l); return charsToString(); } private String toChars_2(int h, int m, int l, int e) { // -3 < e <= 0 appendDigit(0); append('.'); for (; e < 0; ++e) { appendDigit(0); } appendDigit(h); append8Digits(m); lowDigits(l); return charsToString(); } private String toChars_3(int h, int m, int l, int e) { // computerized scientific notation appendDigit(h); append('.'); append8Digits(m); lowDigits(l); exponent(e - 1); return charsToString(); } private void lowDigits(int l) { if (l != 0) { append8Digits(l); } removeTrailingZeroes(); } private void append8Digits(int v) { int y = (int) (((long) (v + 1) << LTR) / 100_000_000L) - 1; for (int i = 0; i < 8; ++i) { int t = 10 * y; appendDigit(t >>> LTR); y = t & MASK_LTR; } } private void removeTrailingZeroes() { while (buf[index] == '0') { --index; } if (buf[index] == '.') { ++index; } } private void exponent(int e) { append('E'); if (e < 0) { append('-'); e = -e; } if (e < 10) { appendDigit(e); } else if (e < 100) { int d = e * 205 >>> 11; // d = e / 10 appendDigit(d); appendDigit(e - 10 * d); } else { int d = e * 1_311 >>> 17; // d = e / 100 appendDigit(d); e -= 100 * d; d = e * 205 >>> 11; // d = e / 10 appendDigit(d); appendDigit(e - 10 * d); } } private void append(int c) { buf[++index] = (char) c; } private void appendDigit(int d) { buf[++index] = (char) ('0' + d); } private String charsToString() { return new String(buf, 0, index + 1); } } -------- math.DecimalChecker.java /* * Copyright (c) 2018, Raffaello Giulietti. 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. * This particular file is subject to the "Classpath" exception as provided * 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. */ package math; import java.io.IOException; import java.io.StringReader; import java.math.BigDecimal; class DecimalChecker { /* Returns whether s syntactically meets the expected output of Double.toString(double). It is restricted to finite positive outputs. It is an unusually long method but rather straightforward, too. Many conditionals could be merged, but KISS here. */ private static boolean hasCorrectFormat(String s) { try { // first determine interesting boundaries in the string StringReader r = new StringReader(s); int c = r.read(); int i = 0; while (c == '0') { ++i; c = r.read(); } // i is just after zeroes starting the integer int d = i; while ('0' <= c && c <= '9') { ++d; c = r.read(); } // d is just after digits ending the integer int fz = d; if (c == '.') { ++fz; c = r.read(); } // fz is just after a decimal '.' int f = fz; while (c == '0') { ++f; c = r.read(); } // f is just after zeroes starting the fraction int x = f; while ('0' <= c && c <= '9') { ++x; c = r.read(); } // x is just after digits ending the fraction int g = x; if (c == 'E') { ++g; c = r.read(); } // g is just after an exponent indicator 'E' int ez = g; if (c == '-') { ++ez; c = r.read(); } // ez is just after a '-' sign in the exponent int e = ez; while (c == '0') { ++e; c = r.read(); } // e is just after zeroes starting the exponent int z = e; while ('0' <= c && c <= '9') { ++z; c = r.read(); } // z is just after digits ending the exponent // No other chars after the number if (z != s.length()) { return false; } // The integer must be present if (d == 0) { return false; } // The decimal '.' must be present if (fz == d) { return false; } // The fraction must be present if (x == fz) { return false; } // Plain notation, no exponent if (x == z) { // At most one 0 starting the integer if (i > 1) { return false; } // The integer cannot have more than 7 digits if (d > 7) { return false; } // If the integer is 0, at most 2 zeroes start the fraction if (i == 1 && f - fz > 2) { return false; } // OK for plain notation return true; } // Computerized scientific notation // The integer has exactly one nonzero digit if (i != 0 || d != 1) { return false; } // There must be an exponent indicator if (x == g) { return false; } // There must be an exponent if (ez == z) { return false; } // The exponent must not start with zeroes if (ez != e) { return false; } int exp; // The exponent must parse as an int try { exp = Integer.parseInt(s, g, z, 10); } catch (NumberFormatException ex) { return false; } // The exponent must not lie in [-3, 7) if (-3 <= exp && exp < 7) { return false; } // OK for computerized scientific notation return true; } catch (IOException ex) { // An IOException on a StringReader??? Please... return false; } } /* And KISS even here. */ static boolean isCorrect(double v, String s) { if (v != v) { return s.equals("NaN"); } if (Double.doubleToRawLongBits(v) < 0) { if (s.isEmpty() || s.charAt(0) != '-') { return false; } return isCorrect(-v, s.substring(1)); } if (v == Double.POSITIVE_INFINITY) { return s.equals("Infinity"); } if (v == 0) { return s.equals("0.0"); } if (!hasCorrectFormat(s)) { return false; } // s must of course recover v try { if (v != Double.parseDouble(s)) { return false; } } catch (NumberFormatException e) { return false; } // b = d * 10^r for some integers d, r with d > 0 BigDecimal b = new BigDecimal(s); // d > 0 has at most 17 digits, so must fit in a positive long if (b.unscaledValue().bitLength() >= Long.SIZE) { return false; } long d = b.unscaledValue().longValue(); if (d >= 100_000_000_000_000_000L) { return false; } int r = -b.scale(); // Determine the number of digits in d int len2 = Long.SIZE - Long.numberOfLeadingZeros(d); int len10 = MathUtils.ord10pow2(len2) - 1; if (d >= Powers.pow10[len10]) { len10 += 1; } // ord10 is such that 10^(ord10-1) <= v < 10^ord10 int ord10 = r + len10; // Plain format iff -3 < ord10 <= 7 boolean isPlain = -3 < ord10 && ord10 <= 7; // If plain then len10 > ord10, i.e., r < 0 if (isPlain && r >= 0) { return false; } // If plain, trailing zero in fraction only if r = -1 if (isPlain && d % 10 == 0 && r < -1) { return false; } // If not plain, trailing zero in fraction only if len10 = 2 if (!isPlain && d % 10 == 0 && len10 > 2) { return false; } // Get rid of trailing zeroes while (d % 10 == 0) { d /= 10; r += 1; len10 -= 1; } if (len10 > 1) { // Try with a shorter number less than v long dsd = d / 10; int rsd = r + 1; BigDecimal bsd = BigDecimal.valueOf(dsd, -rsd); if (dsd >= 10 && bsd.doubleValue() == v) { return false; } // ... and with a shorter number greater than v long dsu = d / 10 + 1; int rsu = r + 1; BigDecimal bsu = BigDecimal.valueOf(dsu, -rsu); if (dsu > 10 && bsu.doubleValue() == v) { return false; } } BigDecimal bv = new BigDecimal(v); BigDecimal deltav = b.subtract(bv).abs(); // Check if the decimal predecessor is closer long dsp = d - 1; BigDecimal bsp = BigDecimal.valueOf(dsp, -r); int cmpp = 1; if (bsp.doubleValue() == v) { BigDecimal deltap = bsp.subtract(bv).abs(); cmpp = deltap.compareTo(deltav); if (cmpp < 0) { return false; } } // Check if the decimal successor is closer long dss = d + 1; BigDecimal bss = BigDecimal.valueOf(dss, -r); int cmps = 1; if (bss.doubleValue() == v) { BigDecimal deltas = bss.subtract(bv).abs(); cmps = deltas.compareTo(deltav); if (cmps < 0) { return false; } } if (cmpp == 0 && (d & 0x1) != 0) { return false; } if (cmps == 0 && (d & 0x1) != 0) { return false; } return true; } } -------- math.DoubleToDecimalTest.java /* * Copyright (c) 2018, Raffaello Giulietti. 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. * This particular file is subject to the "Classpath" exception as provided * 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. */ package math; import org.junit.Test; import java.util.Random; import static java.lang.Math.scalb; import static junit.framework.TestCase.*; import static math.DecimalChecker.isCorrect; import static math.DoubleToDecimal.Double.E_MAX_VALUE; import static math.DoubleToDecimal.Double.E_MIN_VALUE; public class DoubleToDecimalTest { private String toDecimal(double v) { String s = DoubleToDecimal.toString(v); assertTrue(isCorrect(v, s)); return s; } @Test public void testExtremeValues() { toDecimal(Double.NEGATIVE_INFINITY); toDecimal(-Double.MAX_VALUE); toDecimal(-Double.MIN_NORMAL); toDecimal(-Double.MIN_VALUE); toDecimal(-0.0); toDecimal(0.0); toDecimal(Double.MIN_VALUE); toDecimal(Double.MIN_NORMAL); toDecimal(Double.MAX_VALUE); toDecimal(Double.POSITIVE_INFINITY); toDecimal(Double.NaN); } /* A few powers of 10 are incorrectly rendered by the JDK. The rendering is either too long or it is not the closest decimal. */ @Test public void testPowersOf10() { for (int e = E_MIN_VALUE; e <= E_MAX_VALUE; ++e) { toDecimal(Double.parseDouble("1e" + e)); } } /* Many powers of 2 are incorrectly rendered by the JDK. The rendering is either too long or it is not the closest decimal. */ @Test public void testPowersOf2() { for (double v = Double.MIN_VALUE; v <= Double.MAX_VALUE; v *= 2.0) { toDecimal(v); } } /* There are tons of doubles that are rendered incorrectly by the JDK. While the renderings correctly round back to the original value, they are longer than needed or are not the closest decimal to the double. Here are just a very few examples. */ private static final String[] Anomalies = { // JDK renders these, and others, with 18 digits! "2.82879384806159E17", "1.387364135037754E18", "1.45800632428665E17", // JDK renders these longer than needed. "1.6E-322", "6.3E-322", "7.3879E20", "2.0E23", "7.0E22", "9.2E22", "9.5E21", "3.1E22", "5.63E21", "8.41E21", // JDK does not render these, and many others, as the closest. "9.9E-324", "9.9E-323", "1.9400994884341945E25", "3.6131332396758635E25", "2.5138990223946153E25", }; @Test public void testSomeAnomalies() { for (String dec : Anomalies) { toDecimal(Double.parseDouble(dec)); } } /* Values are from Paxson V, "A Program for Testing IEEE Decimal–Binary Conversion" */ private static final double[] PaxsonSignificands = { 8_511_030_020_275_656L, 5_201_988_407_066_741L, 6_406_892_948_269_899L, 8_431_154_198_732_492L, 6_475_049_196_144_587L, 8_274_307_542_972_842L, 5_381_065_484_265_332L, 6_761_728_585_499_734L, 7_976_538_478_610_756L, 5_982_403_858_958_067L, 5_536_995_190_630_837L, 7_225_450_889_282_194L, 7_225_450_889_282_194L, 8_703_372_741_147_379L, 8_944_262_675_275_217L, 7_459_803_696_087_692L, 6_080_469_016_670_379L, 8_385_515_147_034_757L, 7_514_216_811_389_786L, 8_397_297_803_260_511L, 6_733_459_239_310_543L, 8_091_450_587_292_794L, 6_567_258_882_077_402L, 6_712_731_423_444_934L, 6_712_731_423_444_934L, 5_298_405_411_573_037L, 5_137_311_167_659_507L, 6_722_280_709_661_868L, 5_344_436_398_034_927L, 8_369_123_604_277_281L, 8_995_822_108_487_663L, 8_942_832_835_564_782L, 8_942_832_835_564_782L, 8_942_832_835_564_782L, 6_965_949_469_487_146L, 6_965_949_469_487_146L, 6_965_949_469_487_146L, 7_487_252_720_986_826L, 5_592_117_679_628_511L, 8_887_055_249_355_788L, 6_994_187_472_632_449L, 8_797_576_579_012_143L, 7_363_326_733_505_337L, 8_549_497_411_294_502L, }; private static final int[] PaxsonExponents = { -342, -824, 237, 72, 99, 726, -456, -57, 376, 377, 93, 710, 709, 117, -1, -707, -381, 721, -828, -345, 202, -473, 952, 535, 534, -957, -144, 363, -169, -853, -780, -383, -384, -385, -249, -250, -251, 548, 164, 665, 690, 588, 272, -448, }; @Test public void testPaxson() { for (int i = 0; i < PaxsonSignificands.length; ++i) { toDecimal(scalb(PaxsonSignificands[i], PaxsonExponents[i])); } } /* Tests all integers of the form yx_xxx_000_000_000_000_000, y != 0. These are all exact doubles. */ @Test public void testLongs() { for (int i = 10_000; i < 100_000; ++i) { String s = toDecimal(i * 1e15); String xp = Integer.toString(i); int j = 5; while (--j >= 2 && xp.charAt(j) == '0') ; // empty body intended xp = xp.substring(0, 1) + "." + xp.substring(1, j + 1) + "E19"; assertEquals(xp, s); } } /* Tests all integers up to 100_000. These are all exact doubles. */ @Test public void testInts() { for (int i = -100_000; i <= 100_000; ++i) { String s = toDecimal(i); String xp = Integer.toString(i) + ".0"; assertEquals(xp, s); } } /* Random doubles over the whole range */ @Test public void testRandom() { Random r = new Random(); for (int i = 0; i < 10_000; ++i) { toDecimal(Double.longBitsToDouble(r.nextLong())); } } /* Random doubles over the integer range [0, 10^15). These integers are all exact doubles. */ @Test public void testRandomUnit() { Random r = new Random(); for (int i = 0; i < 10_000; ++i) { toDecimal(r.nextLong() % 1_000_000_000_000_000L); } } /* Random doubles over the range [0, 10^15) as "multiples" of 1e-3 */ @Test public void testRandomMilli() { Random r = new Random(); for (int i = 0; i < 10_000; ++i) { toDecimal(r.nextLong() % 1_000_000_000_000_000_000L / 1e3); } } /* Random doubles over the range [0, 10^15) as "multiples" of 1e-6 */ @Test public void testRandomMicro() { Random r = new Random(); for (int i = 0; i < 10_000; ++i) { toDecimal(r.nextLong() / 1e6); } } } -------- math.D2DBenchmark.java /* * Copyright (c) 2018, Raffaello Giulietti. 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. * This particular file is subject to the "Classpath" exception as provided * 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. */ package math; import java.text.DecimalFormat; import java.util.Random; /* Some simple benchmarks to evaluate speeds. */ public class D2DBenchmark { private static final int N = 100_000_000; private static final double[] x = new double[N]; private static final DecimalFormat intFormat = new DecimalFormat("#,##0"); private static final DecimalFormat doubleFormat = new DecimalFormat("#,##0.000"); private static final int RUNS_PER_LIB = 3; private static Random r; private static long d2dNs; private static long jdkNs; public static void main(String[] args) { if (args.length == 0) { System.out.println("arguments"); System.out.println(" [ <seed> ]"); System.out.println(); } Long seed = args.length > 0 ? Long.parseLong(args[0]) : null; r = seed != null ? new Random(seed) : new Random(); micro(); milli(); integers(); nonNaNRange(); } private static void benchmark() { d2dNs = jdkNs = 0; for (int i = 1; i <= RUNS_PER_LIB; ++i) { benchmarkD2d(i); } for (int i = 1; i <= RUNS_PER_LIB; ++i) { benchmarkJdk(i); } printSpeedup(); } private static void micro() { prepareMicro(); benchmark(); } private static void milli() { prepareMilli(); benchmark(); } private static void integers() { prepareIntegers(); benchmark(); } private static void nonNaNRange() { prepareNonNaNDoubles(); benchmark(); } private static void prepareIntegers() { System.out.print("generating " + intFormat.format(x.length) + " integer random doubles... "); System.out.flush(); for (int i = 0; i < x.length; ++i) { x[i] = r.nextInt(); } System.out.println("finished"); } private static void prepareMilli() { System.out.print("generating " + intFormat.format(x.length) + " \"milli\" random doubles... "); System.out.flush(); for (int i = 0; i < x.length; ++i) { x[i] = r.nextInt() / 1e3; } System.out.println("finished"); } private static void prepareMicro() { System.out.print("generating " + intFormat.format(x.length) + " \"micro\" random doubles... "); System.out.flush(); for (int i = 0; i < x.length; ++i) { x[i] = r.nextInt() / 1e6; } System.out.println("finished"); } private static void prepareNonNaNDoubles() { System.out.print("generating " + intFormat.format(x.length) + " non NaN random doubles... "); System.out.flush(); int i = 0; while (i < x.length) { double v = Double.longBitsToDouble(r.nextLong()); if (v == v) { x[i++] = v; } } System.out.println("finished"); } private static void benchmarkJdk(int take) { long tot = 0; long begin = System.nanoTime(); for (double v : x) { tot += Double.toString(v).length(); } long ns = System.nanoTime() - begin; jdkNs += ns; print("java.lang.Double",take, ns, tot); } private static void benchmarkD2d(int take) { long tot = 0; long begin = System.nanoTime(); for (double v : x) { tot += DoubleToDecimal.toString(v).length(); } long ns = System.nanoTime() - begin; d2dNs += ns; print("math.DoubleToDecimal", take, ns, tot); } private static void print(String lib, int take, long ns, long tot) { System.out.println(lib + "[" + take + "/" + RUNS_PER_LIB + "]"); System.out.println("--------"); System.out.println("n=" + intFormat.format(x.length)); System.out.println("elapsed=" + intFormat.format(ns) + " ns"); System.out.println(intFormat.format(ns / x.length) + " ns/rendering"); System.out.println("total length of output=" + intFormat.format(tot)); System.out.println(); } private static void printSpeedup() { System.out.println("speedup factor=" + doubleFormat.format((double) jdkNs / (double) d2dNs)); System.out.println(); } }
