I figured out a method that works ~35% faster than your 6bits+Carmack+sqrt code, at least with my CPU (x86) and programming language (C/C++). Your results may vary, especially because I don't know how the Java factor will play out.
My approach is threefold:
- First, filter out obvious answers. This includes negative numbers and looking at the last 4 bits. (I found looking at the last six didn't help.) I also answer yes for 0. (In reading the code below, note that my input is
int64 x
.)
if( x < 0 || (x&2) || ((x & 7) == 5) || ((x & 11) == 8) )
return false;
if( x == 0 )
return true;
- Next, check if it's a square modulo 255 = 3 * 5 * 17. Because that's a product of three distinct primes, only about 1/8 of the residues mod 255 are squares. However, in my experience, calling the modulo operator (%) costs more than the benefit one gets, so I use bit tricks involving 255 = 2^8-1 to compute the residue. (For better or worse, I am not using the trick of reading individual bytes out of a word, only bitwise-and and shifts.)
int64 y = x;
y = (y & 4294967295LL) + (y >> 32);
y = (y & 65535) + (y >> 16);
y = (y & 255) + ((y >> 8) & 255) + (y >> 16);
// At this point, y is between 0 and 511. More code can reduce it farther.
To actually check if the residue is a square, I look up the answer in a precomputed table.
if( bad255[y] )
return false;
// However, I just use a table of size 512
- Finally, try to compute the square root using a method similar to Hensel's lemma. (I don't think it's applicable directly, but it works with some modifications.) Before doing that, I divide out all powers of 2 with a binary search:
if((x & 4294967295LL) == 0)
x >>= 32;
if((x & 65535) == 0)
x >>= 16;
if((x & 255) == 0)
x >>= 8;
if((x & 15) == 0)
x >>= 4;
if((x & 3) == 0)
x >>= 2;
At this point, for our number to be a square, it must be 1 mod 8.
if((x & 7) != 1)
return false;
The basic structure of Hensel's lemma is the following. (Note: untested code; if it doesn't work, try t=2 or 8.)
int64 t = 4, r = 1;
t <<= 1; r += ((x - r * r) & t) >> 1;
t <<= 1; r += ((x - r * r) & t) >> 1;
t <<= 1; r += ((x - r * r) & t) >> 1;
// Repeat until t is 2^33 or so. Use a loop if you want.
The idea is that at each iteration, you add one bit onto r, the "current" square root of x; each square root is accurate modulo a larger and larger power of 2, namely t/2. At the end, r and t/2-r will be square roots of x modulo t/2. (Note that if r is a square root of x, then so is -r. This is true even modulo numbers, but beware, modulo some numbers, things can have even more than 2 square roots; notably, this includes powers of 2.) Because our actual square root is less than 2^32, at that point we can actually just check if r or t/2-r are real square roots. In my actual code, I use the following modified loop:
int64 r, t, z;
r = start[(x >> 3) & 1023];
do {
z = x - r * r;
if( z == 0 )
return true;
if( z < 0 )
return false;
t = z & (-z);
r += (z & t) >> 1;
if( r > (t >> 1) )
r = t - r;
} while( t <= (1LL << 33) );
The speedup here is obtained in three ways: precomputed start value (equivalent to ~10 iterations of the loop), earlier exit of the loop, and skipping some t values. For the last part, I look at z = r - x * x
, and set t to be the largest power of 2 dividing z with a bit trick. This allows me to skip t values that wouldn't have affected the value of r anyway. The precomputed start value in my case picks out the "smallest positive" square root modulo 8192.
Even if this code doesn't work faster for you, I hope you enjoy some of the ideas it contains. Complete, tested code follows, including the precomputed tables.
typedef signed long long int int64;
int start[1024] =
{1,3,1769,5,1937,1741,7,1451,479,157,9,91,945,659,1817,11,
1983,707,1321,1211,1071,13,1479,405,415,1501,1609,741,15,339,1703,203,
129,1411,873,1669,17,1715,1145,1835,351,1251,887,1573,975,19,1127,395,
1855,1981,425,453,1105,653,327,21,287,93,713,1691,1935,301,551,587,
257,1277,23,763,1903,1075,1799,1877,223,1437,1783,859,1201,621,25,779,
1727,573,471,1979,815,1293,825,363,159,1315,183,27,241,941,601,971,
385,131,919,901,273,435,647,1493,95,29,1417,805,719,1261,1177,1163,
1599,835,1367,315,1361,1933,1977,747,31,1373,1079,1637,1679,1581,1753,1355,
513,1539,1815,1531,1647,205,505,1109,33,1379,521,1627,1457,1901,1767,1547,
1471,1853,1833,1349,559,1523,967,1131,97,35,1975,795,497,1875,1191,1739,
641,1149,1385,133,529,845,1657,725,161,1309,375,37,463,1555,615,1931,
1343,445,937,1083,1617,883,185,1515,225,1443,1225,869,1423,1235,39,1973,
769,259,489,1797,1391,1485,1287,341,289,99,1271,1701,1713,915,537,1781,
1215,963,41,581,303,243,1337,1899,353,1245,329,1563,753,595,1113,1589,
897,1667,407,635,785,1971,135,43,417,1507,1929,731,207,275,1689,1397,
1087,1725,855,1851,1873,397,1607,1813,481,163,567,101,1167,45,1831,1205,
1025,1021,1303,1029,1135,1331,1017,427,545,1181,1033,933,1969,365,1255,1013,
959,317,1751,187,47,1037,455,1429,609,1571,1463,1765,1009,685,679,821,
1153,387,1897,1403,1041,691,1927,811,673,227,137,1499,49,1005,103,629,
831,1091,1449,1477,1967,1677,697,1045,737,1117,1737,667,911,1325,473,437,
1281,1795,1001,261,879,51,775,1195,801,1635,759,165,1871,1645,1049,245,
703,1597,553,955,209,1779,1849,661,865,291,841,997,1265,1965,1625,53,
1409,893,105,1925,1297,589,377,1579,929,1053,1655,1829,305,1811,1895,139,
575,189,343,709,1711,1139,1095,277,993,1699,55,1435,655,1491,1319,331,
1537,515,791,507,623,1229,1529,1963,1057,355,1545,603,1615,1171,743,523,
447,1219,1239,1723,465,499,57,107,1121,989,951,229,1521,851,167,715,
1665,1923,1687,1157,1553,1869,1415,1749,1185,1763,649,1061,561,531,409,907,
319,1469,1961,59,1455,141,1209,491,1249,419,1847,1893,399,211,985,1099,
1793,765,1513,1275,367,1587,263,1365,1313,925,247,1371,1359,109,1561,1291,
191,61,1065,1605,721,781,1735,875,1377,1827,1353,539,1777,429,1959,1483,
1921,643,617,389,1809,947,889,981,1441,483,1143,293,817,749,1383,1675,
63,1347,169,827,1199,1421,583,1259,1505,861,457,1125,143,1069,807,1867,
2047,2045,279,2043,111,307,2041,597,1569,1891,2039,1957,1103,1389,231,2037,
65,1341,727,837,977,2035,569,1643,1633,547,439,1307,2033,1709,345,1845,
1919,637,1175,379,2031,333,903,213,1697,797,1161,475,1073,2029,921,1653,
193,67,1623,1595,943,1395,1721,2027,1761,1955,1335,357,113,1747,1497,1461,
1791,771,2025,1285,145,973,249,171,1825,611,265,1189,847,1427,2023,1269,
321,1475,1577,69,1233,755,1223,1685,1889,733,1865,2021,1807,1107,1447,1077,
1663,1917,1129,1147,1775,1613,1401,555,1953,2019,631,1243,1329,787,871,885,
449,1213,681,1733,687,115,71,1301,2017,675,969,411,369,467,295,693,
1535,509,233,517,401,1843,1543,939,2015,669,1527,421,591,147,281,501,
577,195,215,699,1489,525,1081,917,1951,2013,73,1253,1551,173,857,309,
1407,899,663,1915,1519,1203,391,1323,1887,739,1673,2011,1585,493,1433,117,
705,1603,1111,965,431,1165,1863,533,1823,605,823,1179,625,813,2009,75,
1279,1789,1559,251,657,563,761,1707,1759,1949,777,347,335,1133,1511,267,
833,1085,2007,1467,1745,1805,711,149,1695,803,1719,485,1295,1453,935,459,
1151,381,1641,1413,1263,77,1913,2005,1631,541,119,1317,1841,1773,359,651,
961,323,1193,197,175,1651,441,235,1567,1885,1481,1947,881,2003,217,843,
1023,1027,745,1019,913,717,1031,1621,1503,867,1015,1115,79,1683,793,1035,
1089,1731,297,1861,2001,1011,1593,619,1439,477,585,283,1039,1363,1369,1227,
895,1661,151,645,1007,1357,121,1237,1375,1821,1911,549,1999,1043,1945,1419,
1217,957,599,571,81,371,1351,1003,1311,931,311,1381,1137,723,1575,1611,
767,253,1047,1787,1169,1997,1273,853,1247,413,1289,1883,177,403,999,1803,
1345,451,1495,1093,1839,269,199,1387,1183,1757,1207,1051,783,83,423,1995,
639,1155,1943,123,751,1459,1671,469,1119,995,393,219,1743,237,153,1909,
1473,1859,1705,1339,337,909,953,1771,1055,349,1993,613,1393,557,729,1717,
511,1533,1257,1541,1425,819,519,85,991,1693,503,1445,433,877,1305,1525,
1601,829,809,325,1583,1549,1991,1941,927,1059,1097,1819,527,1197,1881,1333,
383,125,361,891,495,179,633,299,863,285,1399,987,1487,1517,1639,1141,
1729,579,87,1989,593,1907,839,1557,799,1629,201,155,1649,1837,1063,949,
255,1283,535,773,1681,461,1785,683,735,1123,1801,677,689,1939,487,757,
1857,1987,983,443,1327,1267,313,1173,671,221,695,1509,271,1619,89,565,
127,1405,1431,1659,239,1101,1159,1067,607,1565,905,1755,1231,1299,665,373,
1985,701,1879,1221,849,627,1465,789,543,1187,1591,923,1905,979,1241,181};
bool bad255[512] =
{0,0,1,1,0,1,1,1,1,0,1,1,1,1,1,0,0,1,1,0,1,0,1,1,1,0,1,1,1,1,0,1,
1,1,0,1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,1,0,1,1,1,0,1,1,1,1,0,1,1,1,
0,1,0,1,1,0,0,1,1,1,1,1,0,1,1,1,1,0,1,1,0,0,1,1,1,1,1,1,1,1,0,1,
1,1,1,1,0,1,1,1,1,1,0,1,1,1,1,0,1,1,1,0,1,1,1,1,0,0,1,1,1,1,1,1,
1,1,1,1,1,1,1,0,0,1,1,1,1,1,1,1,0,0,1,1,1,1,1,0,1,1,0,1,1,1,1,1,
1,1,1,1,1,1,0,1,1,0,1,0,1,1,0,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,1,1,
1,1,1,0,0,1,1,1,1,1,1,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,1,1,1,
1,0,1,1,1,0,1,1,1,1,0,1,1,1,1,1,0,1,1,1,1,1,0,1,1,1,1,1,1,1,1,
0,0,1,1,0,1,1,1,1,0,1,1,1,1,1,0,0,1,1,0,1,0,1,1,1,0,1,1,1,1,0,1,
1,1,0,1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,1,0,1,1,1,0,1,1,1,1,0,1,1,1,
0,1,0,1,1,0,0,1,1,1,1,1,0,1,1,1,1,0,1,1,0,0,1,1,1,1,1,1,1,1,0,1,
1,1,1,1,0,1,1,1,1,1,0,1,1,1,1,0,1,1,1,0,1,1,1,1,0,0,1,1,1,1,1,1,
1,1,1,1,1,1,1,0,0,1,1,1,1,1,1,1,0,0,1,1,1,1,1,0,1,1,0,1,1,1,1,1,
1,1,1,1,1,1,0,1,1,0,1,0,1,1,0,1,1,1,1,1,1,1,1,1,1,1,0,1,1,0,1,1,
1,1,1,0,0,1,1,1,1,1,1,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,1,1,1,
1,0,1,1,1,0,1,1,1,1,0,1,1,1,1,1,0,1,1,1,1,1,0,1,1,1,1,1,1,1,1,
0,0};
inline bool square( int64 x ) {
// Quickfail
if( x < 0 || (x&2) || ((x & 7) == 5) || ((x & 11) == 8) )
return false;
if( x == 0 )
return true;
// Check mod 255 = 3 * 5 * 17, for fun
int64 y = x;
y = (y & 4294967295LL) + (y >> 32);
y = (y & 65535) + (y >> 16);
y = (y & 255) + ((y >> 8) & 255) + (y >> 16);
if( bad255[y] )
return false;
// Divide out powers of 4 using binary search
if((x & 4294967295LL) == 0)
x >>= 32;
if((x & 65535) == 0)
x >>= 16;
if((x & 255) == 0)
x >>= 8;
if((x & 15) == 0)
x >>= 4;
if((x & 3) == 0)
x >>= 2;
if((x & 7) != 1)
return false;
// Compute sqrt using something like Hensel's lemma
int64 r, t, z;
r = start[(x >> 3) & 1023];
do {
z = x - r * r;
if( z == 0 )
return true;
if( z < 0 )
return false;
t = z & (-z);
r += (z & t) >> 1;
if( r > (t >> 1) )
r = t - r;
} while( t <= (1LL << 33) );
return false;
}
Best Answer
The JAXB APIs are considered to be Java EE APIs and therefore are no longer contained on the default classpath in Java SE 9. In Java 11, they are completely removed from the JDK.
Java 9 introduces the concepts of modules, and by default, the
java.se
aggregate module is available on the classpath (or rather, module-path). As the name implies, thejava.se
aggregate module does not include the Java EE APIs that have been traditionally bundled with Java 6/7/8.Fortunately, these Java EE APIs that were provided in JDK 6/7/8 are still in the JDK, but they just aren't on the classpath by default. The extra Java EE APIs are provided in the following modules:
Quick and dirty solution: (JDK 9/10 only)
To make the JAXB APIs available at runtime, specify the following command-line option:
--add-modules java.xml.bind
But I still need this to work with Java 8!!!
If you try specifying
--add-modules
with an older JDK, it will blow up because it's an unrecognized option. I suggest one of two options:JDK_JAVA_OPTIONS
environment variable. This environment variable is automatically read by thejava
launcher for Java 9+.-XX:+IgnoreUnrecognizedVMOptions
to make the JVM silently ignore unrecognized options, instead of blowing up. But beware! Any other command-line arguments you use will no longer be validated for you by the JVM. This option works with Oracle/OpenJDK as well as IBM JDK (as of JDK 8sr4).Alternate quick solution: (JDK 9/10 only)
Note that you can make all of the above Java EE modules available at run time by specifying the
--add-modules java.se.ee
option. Thejava.se.ee
module is an aggregate module that includesjava.se.ee
as well as the above Java EE API modules. Note, this doesn't work on Java 11 becausejava.se.ee
was removed in Java 11.Proper long-term solution: (JDK 9 and beyond)
The Java EE API modules listed above are all marked
@Deprecated(forRemoval=true)
because they are scheduled for removal in Java 11. So the--add-module
approach will no longer work in Java 11 out-of-the-box.What you will need to do in Java 11 and forward is include your own copy of the Java EE APIs on the classpath or module path. For example, you can add the JAX-B APIs as a Maven dependency like this:
See the JAXB Reference Implementation page for more details on JAXB.
For full details on Java modularity, see JEP 261: Module System
For Gradle or Android Studio developer: (JDK 9 and beyond)
Add the following dependencies to your
build.gradle
file: