Jackson Core Source Code

Jackson is "the Java JSON library" or "the best JSON parser for Java". Or simply as "JSON for Java".

Jackson Core Source Code files are provided in the source packge (jackson-core-2.14.0-sources.jar). You can download it at Jackson Maven Website.

You can also browse Jackson Core Source Code below:

✍: FYIcenter.com

com/fasterxml/jackson/core/io/doubleparser/FastDoubleMath.java

/**
 * References:
 * <dl>
 *     <dt>This class has been derived from "FastDoubleParser".</dt>
 *     <dd>Copyright (c) Werner Randelshofer. Apache 2.0 License.
 *         <a href="https://github.com/wrandelshofer/FastDoubleParser">github.com</a>.</dd>
 * </dl>
 */

package com.fasterxml.jackson.core.io.doubleparser;

/**
 * This class provides the mathematical functions needed by {@link FastDoubleParser}.
 * <p>
 * References:
 * <dl>
 *     <dt>Daniel Lemire. fast_double_parser. 4x faster than strtod.
 *     Apache License 2.0 or Boost Software License.</dt>
 *     <dd><a href="https://github.com/lemire/fast_double_parser">github.com</a></dd>
 *
 *     <dt>Daniel Lemire. fast_float number parsing library: 4x faster than strtod.
 *     Apache License 2.0.</dt>
 *     <dd><a href="https://github.com/fastfloat/fast_float">github.com</a></dd>
 *
 *     <dt>Daniel Lemire, Number Parsing at a Gigabyte per Second,
 *     Software: Practice and Experience 51 (8), 2021.
 *     arXiv.2101.11408v3 [cs.DS] 24 Feb 2021</dt>
 *     <dd><a href="https://arxiv.org/pdf/2101.11408.pdf">arxiv.org</a></dd>
 *
 *     <dt>Clinger WD (1990). How to read floating point numbers accurately. ACM SIGPLAN Notices.</dt>
 *     <dd>- no link -</dd>
 * </dl>
 * </p>
 */
class FastDoubleMath {
    /**
     * Bias used in the exponent of a double.
     */
    public static final int DOUBLE_EXPONENT_BIAS = 1023;
    /**
     * The number of bits in the significand, including the implicit bit.
     */
    public static final int DOUBLE_SIGNIFICAND_WIDTH = 53;
    /**
     * Smallest power of 10 value of the exponent.
     * <p>
     * The smallest non-zero double is 2^−1074.
     * <p>
     * We take as input numbers of the form w x 10^q where w < 2^64.
     * <p>
     * We have that {@literal w * 10^-343 < 2^(63-343) * 5^-343 < 2^-1076}.
     * <p>
     * However, we have that
     * {@literal (2^64-1) * 10^-342 = (2^64 - 1) * 2^-342 * 5^-342 > 2^−1074}.
     * Thus, it is possible for a number of the form w * 10^-342 where
     * w is a 64-bit value to be a non-zero double.
     * <p>
     * ********
     * <p>
     * If we are solely interested in the *normal* numbers then the
     * smallest value is 2^-1022. We can generate a value larger
     * than 2^-1022 with expressions of the form w * 10^-326.
     * Thus, we need to pick SMALLEST_POWER_OF_TEN >= -326.
     */
    final static int DOUBLE_MIN_EXPONENT_POWER_OF_TEN = -325;
    /**
     * Largest power of 10 value of the exponent.
     * <p>
     * Any number of form w * 10^309 where {@literal w >= 1} is going to be
     * infinite in a double, so we never need to worry about powers
     * of 10 greater than 308.
     */
    final static int DOUBLE_MAX_EXPONENT_POWER_OF_TEN = 308;
    /**
     * When mapping numbers from decimal to binary, we go from w * 10^q to
     * m * 2^p, but we have 10^q = 5^q * 2^q, so effectively we are trying to match
     * w * 2^q * 5^q to m * 2^p.
     * <p>
     * Thus, the powers of two are not a concern since they can be represented
     * exactly using the binary notation, only the powers of five affect the
     * binary significand.
     * <p>
     * The mantissas of powers of ten from -308 to 308, extended out to sixty-four
     * bits. The array contains the powers of ten approximated
     * as a 64-bit mantissa. It goes from 10^{@value #DOUBLE_MIN_EXPONENT_POWER_OF_TEN} to
     * 10^{@value #DOUBLE_MAX_EXPONENT_POWER_OF_TEN} (inclusively). The mantissa is truncated, and
     * never rounded up. Uses about 5 KB.
     * <p>
     * <pre>
     * long getMantissaHigh(int q) {
     *  MANTISSA_64[q - SMALLEST_POWER_OF_TEN];
     * }
     * </pre>
     */
    static final long[] MANTISSA_64 = {
            0xa5ced43b7e3e9188L, 0xcf42894a5dce35eaL,
            0x818995ce7aa0e1b2L, 0xa1ebfb4219491a1fL,
            0xca66fa129f9b60a6L, 0xfd00b897478238d0L,
            0x9e20735e8cb16382L, 0xc5a890362fddbc62L,
            0xf712b443bbd52b7bL, 0x9a6bb0aa55653b2dL,
            0xc1069cd4eabe89f8L, 0xf148440a256e2c76L,
            0x96cd2a865764dbcaL, 0xbc807527ed3e12bcL,
            0xeba09271e88d976bL, 0x93445b8731587ea3L,
            0xb8157268fdae9e4cL, 0xe61acf033d1a45dfL,
            0x8fd0c16206306babL, 0xb3c4f1ba87bc8696L,
            0xe0b62e2929aba83cL, 0x8c71dcd9ba0b4925L,
            0xaf8e5410288e1b6fL, 0xdb71e91432b1a24aL,
            0x892731ac9faf056eL, 0xab70fe17c79ac6caL,
            0xd64d3d9db981787dL, 0x85f0468293f0eb4eL,
            0xa76c582338ed2621L, 0xd1476e2c07286faaL,
            0x82cca4db847945caL, 0xa37fce126597973cL,
            0xcc5fc196fefd7d0cL, 0xff77b1fcbebcdc4fL,
            0x9faacf3df73609b1L, 0xc795830d75038c1dL,
            0xf97ae3d0d2446f25L, 0x9becce62836ac577L,
            0xc2e801fb244576d5L, 0xf3a20279ed56d48aL,
            0x9845418c345644d6L, 0xbe5691ef416bd60cL,
            0xedec366b11c6cb8fL, 0x94b3a202eb1c3f39L,
            0xb9e08a83a5e34f07L, 0xe858ad248f5c22c9L,
            0x91376c36d99995beL, 0xb58547448ffffb2dL,
            0xe2e69915b3fff9f9L, 0x8dd01fad907ffc3bL,
            0xb1442798f49ffb4aL, 0xdd95317f31c7fa1dL,
            0x8a7d3eef7f1cfc52L, 0xad1c8eab5ee43b66L,
            0xd863b256369d4a40L, 0x873e4f75e2224e68L,
            0xa90de3535aaae202L, 0xd3515c2831559a83L,
            0x8412d9991ed58091L, 0xa5178fff668ae0b6L,
            0xce5d73ff402d98e3L, 0x80fa687f881c7f8eL,
            0xa139029f6a239f72L, 0xc987434744ac874eL,
            0xfbe9141915d7a922L, 0x9d71ac8fada6c9b5L,
            0xc4ce17b399107c22L, 0xf6019da07f549b2bL,
            0x99c102844f94e0fbL, 0xc0314325637a1939L,
            0xf03d93eebc589f88L, 0x96267c7535b763b5L,
            0xbbb01b9283253ca2L, 0xea9c227723ee8bcbL,
            0x92a1958a7675175fL, 0xb749faed14125d36L,
            0xe51c79a85916f484L, 0x8f31cc0937ae58d2L,
            0xb2fe3f0b8599ef07L, 0xdfbdcece67006ac9L,
            0x8bd6a141006042bdL, 0xaecc49914078536dL,
            0xda7f5bf590966848L, 0x888f99797a5e012dL,
            0xaab37fd7d8f58178L, 0xd5605fcdcf32e1d6L,
            0x855c3be0a17fcd26L, 0xa6b34ad8c9dfc06fL,
            0xd0601d8efc57b08bL, 0x823c12795db6ce57L,
            0xa2cb1717b52481edL, 0xcb7ddcdda26da268L,
            0xfe5d54150b090b02L, 0x9efa548d26e5a6e1L,
            0xc6b8e9b0709f109aL, 0xf867241c8cc6d4c0L,
            0x9b407691d7fc44f8L, 0xc21094364dfb5636L,
            0xf294b943e17a2bc4L, 0x979cf3ca6cec5b5aL,
            0xbd8430bd08277231L, 0xece53cec4a314ebdL,
            0x940f4613ae5ed136L, 0xb913179899f68584L,
            0xe757dd7ec07426e5L, 0x9096ea6f3848984fL,
            0xb4bca50b065abe63L, 0xe1ebce4dc7f16dfbL,
            0x8d3360f09cf6e4bdL, 0xb080392cc4349decL,
            0xdca04777f541c567L, 0x89e42caaf9491b60L,
            0xac5d37d5b79b6239L, 0xd77485cb25823ac7L,
            0x86a8d39ef77164bcL, 0xa8530886b54dbdebL,
            0xd267caa862a12d66L, 0x8380dea93da4bc60L,
            0xa46116538d0deb78L, 0xcd795be870516656L,
            0x806bd9714632dff6L, 0xa086cfcd97bf97f3L,
            0xc8a883c0fdaf7df0L, 0xfad2a4b13d1b5d6cL,
            0x9cc3a6eec6311a63L, 0xc3f490aa77bd60fcL,
            0xf4f1b4d515acb93bL, 0x991711052d8bf3c5L,
            0xbf5cd54678eef0b6L, 0xef340a98172aace4L,
            0x9580869f0e7aac0eL, 0xbae0a846d2195712L,
            0xe998d258869facd7L, 0x91ff83775423cc06L,
            0xb67f6455292cbf08L, 0xe41f3d6a7377eecaL,
            0x8e938662882af53eL, 0xb23867fb2a35b28dL,
            0xdec681f9f4c31f31L, 0x8b3c113c38f9f37eL,
            0xae0b158b4738705eL, 0xd98ddaee19068c76L,
            0x87f8a8d4cfa417c9L, 0xa9f6d30a038d1dbcL,
            0xd47487cc8470652bL, 0x84c8d4dfd2c63f3bL,
            0xa5fb0a17c777cf09L, 0xcf79cc9db955c2ccL,
            0x81ac1fe293d599bfL, 0xa21727db38cb002fL,
            0xca9cf1d206fdc03bL, 0xfd442e4688bd304aL,
            0x9e4a9cec15763e2eL, 0xc5dd44271ad3cdbaL,
            0xf7549530e188c128L, 0x9a94dd3e8cf578b9L,
            0xc13a148e3032d6e7L, 0xf18899b1bc3f8ca1L,
            0x96f5600f15a7b7e5L, 0xbcb2b812db11a5deL,
            0xebdf661791d60f56L, 0x936b9fcebb25c995L,
            0xb84687c269ef3bfbL, 0xe65829b3046b0afaL,
            0x8ff71a0fe2c2e6dcL, 0xb3f4e093db73a093L,
            0xe0f218b8d25088b8L, 0x8c974f7383725573L,
            0xafbd2350644eeacfL, 0xdbac6c247d62a583L,
            0x894bc396ce5da772L, 0xab9eb47c81f5114fL,
            0xd686619ba27255a2L, 0x8613fd0145877585L,
            0xa798fc4196e952e7L, 0xd17f3b51fca3a7a0L,
            0x82ef85133de648c4L, 0xa3ab66580d5fdaf5L,
            0xcc963fee10b7d1b3L, 0xffbbcfe994e5c61fL,
            0x9fd561f1fd0f9bd3L, 0xc7caba6e7c5382c8L,
            0xf9bd690a1b68637bL, 0x9c1661a651213e2dL,
            0xc31bfa0fe5698db8L, 0xf3e2f893dec3f126L,
            0x986ddb5c6b3a76b7L, 0xbe89523386091465L,
            0xee2ba6c0678b597fL, 0x94db483840b717efL,
            0xba121a4650e4ddebL, 0xe896a0d7e51e1566L,
            0x915e2486ef32cd60L, 0xb5b5ada8aaff80b8L,
            0xe3231912d5bf60e6L, 0x8df5efabc5979c8fL,
            0xb1736b96b6fd83b3L, 0xddd0467c64bce4a0L,
            0x8aa22c0dbef60ee4L, 0xad4ab7112eb3929dL,
            0xd89d64d57a607744L, 0x87625f056c7c4a8bL,
            0xa93af6c6c79b5d2dL, 0xd389b47879823479L,
            0x843610cb4bf160cbL, 0xa54394fe1eedb8feL,
            0xce947a3da6a9273eL, 0x811ccc668829b887L,
            0xa163ff802a3426a8L, 0xc9bcff6034c13052L,
            0xfc2c3f3841f17c67L, 0x9d9ba7832936edc0L,
            0xc5029163f384a931L, 0xf64335bcf065d37dL,
            0x99ea0196163fa42eL, 0xc06481fb9bcf8d39L,
            0xf07da27a82c37088L, 0x964e858c91ba2655L,
            0xbbe226efb628afeaL, 0xeadab0aba3b2dbe5L,
            0x92c8ae6b464fc96fL, 0xb77ada0617e3bbcbL,
            0xe55990879ddcaabdL, 0x8f57fa54c2a9eab6L,
            0xb32df8e9f3546564L, 0xdff9772470297ebdL,
            0x8bfbea76c619ef36L, 0xaefae51477a06b03L,
            0xdab99e59958885c4L, 0x88b402f7fd75539bL,
            0xaae103b5fcd2a881L, 0xd59944a37c0752a2L,
            0x857fcae62d8493a5L, 0xa6dfbd9fb8e5b88eL,
            0xd097ad07a71f26b2L, 0x825ecc24c873782fL,
            0xa2f67f2dfa90563bL, 0xcbb41ef979346bcaL,
            0xfea126b7d78186bcL, 0x9f24b832e6b0f436L,
            0xc6ede63fa05d3143L, 0xf8a95fcf88747d94L,
            0x9b69dbe1b548ce7cL, 0xc24452da229b021bL,
            0xf2d56790ab41c2a2L, 0x97c560ba6b0919a5L,
            0xbdb6b8e905cb600fL, 0xed246723473e3813L,
            0x9436c0760c86e30bL, 0xb94470938fa89bceL,
            0xe7958cb87392c2c2L, 0x90bd77f3483bb9b9L,
            0xb4ecd5f01a4aa828L, 0xe2280b6c20dd5232L,
            0x8d590723948a535fL, 0xb0af48ec79ace837L,
            0xdcdb1b2798182244L, 0x8a08f0f8bf0f156bL,
            0xac8b2d36eed2dac5L, 0xd7adf884aa879177L,
            0x86ccbb52ea94baeaL, 0xa87fea27a539e9a5L,
            0xd29fe4b18e88640eL, 0x83a3eeeef9153e89L,
            0xa48ceaaab75a8e2bL, 0xcdb02555653131b6L,
            0x808e17555f3ebf11L, 0xa0b19d2ab70e6ed6L,
            0xc8de047564d20a8bL, 0xfb158592be068d2eL,
            0x9ced737bb6c4183dL, 0xc428d05aa4751e4cL,
            0xf53304714d9265dfL, 0x993fe2c6d07b7fabL,
            0xbf8fdb78849a5f96L, 0xef73d256a5c0f77cL,
            0x95a8637627989aadL, 0xbb127c53b17ec159L,
            0xe9d71b689dde71afL, 0x9226712162ab070dL,
            0xb6b00d69bb55c8d1L, 0xe45c10c42a2b3b05L,
            0x8eb98a7a9a5b04e3L, 0xb267ed1940f1c61cL,
            0xdf01e85f912e37a3L, 0x8b61313bbabce2c6L,
            0xae397d8aa96c1b77L, 0xd9c7dced53c72255L,
            0x881cea14545c7575L, 0xaa242499697392d2L,
            0xd4ad2dbfc3d07787L, 0x84ec3c97da624ab4L,
            0xa6274bbdd0fadd61L, 0xcfb11ead453994baL,
            0x81ceb32c4b43fcf4L, 0xa2425ff75e14fc31L,
            0xcad2f7f5359a3b3eL, 0xfd87b5f28300ca0dL,
            0x9e74d1b791e07e48L, 0xc612062576589ddaL,
            0xf79687aed3eec551L, 0x9abe14cd44753b52L,
            0xc16d9a0095928a27L, 0xf1c90080baf72cb1L,
            0x971da05074da7beeL, 0xbce5086492111aeaL,
            0xec1e4a7db69561a5L, 0x9392ee8e921d5d07L,
            0xb877aa3236a4b449L, 0xe69594bec44de15bL,
            0x901d7cf73ab0acd9L, 0xb424dc35095cd80fL,
            0xe12e13424bb40e13L, 0x8cbccc096f5088cbL,
            0xafebff0bcb24aafeL, 0xdbe6fecebdedd5beL,
            0x89705f4136b4a597L, 0xabcc77118461cefcL,
            0xd6bf94d5e57a42bcL, 0x8637bd05af6c69b5L,
            0xa7c5ac471b478423L, 0xd1b71758e219652bL,
            0x83126e978d4fdf3bL, 0xa3d70a3d70a3d70aL,
            0xccccccccccccccccL, 0x8000000000000000L,
            0xa000000000000000L, 0xc800000000000000L,
            0xfa00000000000000L, 0x9c40000000000000L,
            0xc350000000000000L, 0xf424000000000000L,
            0x9896800000000000L, 0xbebc200000000000L,
            0xee6b280000000000L, 0x9502f90000000000L,
            0xba43b74000000000L, 0xe8d4a51000000000L,
            0x9184e72a00000000L, 0xb5e620f480000000L,
            0xe35fa931a0000000L, 0x8e1bc9bf04000000L,
            0xb1a2bc2ec5000000L, 0xde0b6b3a76400000L,
            0x8ac7230489e80000L, 0xad78ebc5ac620000L,
            0xd8d726b7177a8000L, 0x878678326eac9000L,
            0xa968163f0a57b400L, 0xd3c21bcecceda100L,
            0x84595161401484a0L, 0xa56fa5b99019a5c8L,
            0xcecb8f27f4200f3aL, 0x813f3978f8940984L,
            0xa18f07d736b90be5L, 0xc9f2c9cd04674edeL,
            0xfc6f7c4045812296L, 0x9dc5ada82b70b59dL,
            0xc5371912364ce305L, 0xf684df56c3e01bc6L,
            0x9a130b963a6c115cL, 0xc097ce7bc90715b3L,
            0xf0bdc21abb48db20L, 0x96769950b50d88f4L,
            0xbc143fa4e250eb31L, 0xeb194f8e1ae525fdL,
            0x92efd1b8d0cf37beL, 0xb7abc627050305adL,
            0xe596b7b0c643c719L, 0x8f7e32ce7bea5c6fL,
            0xb35dbf821ae4f38bL, 0xe0352f62a19e306eL,
            0x8c213d9da502de45L, 0xaf298d050e4395d6L,
            0xdaf3f04651d47b4cL, 0x88d8762bf324cd0fL,
            0xab0e93b6efee0053L, 0xd5d238a4abe98068L,
            0x85a36366eb71f041L, 0xa70c3c40a64e6c51L,
            0xd0cf4b50cfe20765L, 0x82818f1281ed449fL,
            0xa321f2d7226895c7L, 0xcbea6f8ceb02bb39L,
            0xfee50b7025c36a08L, 0x9f4f2726179a2245L,
            0xc722f0ef9d80aad6L, 0xf8ebad2b84e0d58bL,
            0x9b934c3b330c8577L, 0xc2781f49ffcfa6d5L,
            0xf316271c7fc3908aL, 0x97edd871cfda3a56L,
            0xbde94e8e43d0c8ecL, 0xed63a231d4c4fb27L,
            0x945e455f24fb1cf8L, 0xb975d6b6ee39e436L,
            0xe7d34c64a9c85d44L, 0x90e40fbeea1d3a4aL,
            0xb51d13aea4a488ddL, 0xe264589a4dcdab14L,
            0x8d7eb76070a08aecL, 0xb0de65388cc8ada8L,
            0xdd15fe86affad912L, 0x8a2dbf142dfcc7abL,
            0xacb92ed9397bf996L, 0xd7e77a8f87daf7fbL,
            0x86f0ac99b4e8dafdL, 0xa8acd7c0222311bcL,
            0xd2d80db02aabd62bL, 0x83c7088e1aab65dbL,
            0xa4b8cab1a1563f52L, 0xcde6fd5e09abcf26L,
            0x80b05e5ac60b6178L, 0xa0dc75f1778e39d6L,
            0xc913936dd571c84cL, 0xfb5878494ace3a5fL,
            0x9d174b2dcec0e47bL, 0xc45d1df942711d9aL,
            0xf5746577930d6500L, 0x9968bf6abbe85f20L,
            0xbfc2ef456ae276e8L, 0xefb3ab16c59b14a2L,
            0x95d04aee3b80ece5L, 0xbb445da9ca61281fL,
            0xea1575143cf97226L, 0x924d692ca61be758L,
            0xb6e0c377cfa2e12eL, 0xe498f455c38b997aL,
            0x8edf98b59a373fecL, 0xb2977ee300c50fe7L,
            0xdf3d5e9bc0f653e1L, 0x8b865b215899f46cL,
            0xae67f1e9aec07187L, 0xda01ee641a708de9L,
            0x884134fe908658b2L, 0xaa51823e34a7eedeL,
            0xd4e5e2cdc1d1ea96L, 0x850fadc09923329eL,
            0xa6539930bf6bff45L, 0xcfe87f7cef46ff16L,
            0x81f14fae158c5f6eL, 0xa26da3999aef7749L,
            0xcb090c8001ab551cL, 0xfdcb4fa002162a63L,
            0x9e9f11c4014dda7eL, 0xc646d63501a1511dL,
            0xf7d88bc24209a565L, 0x9ae757596946075fL,
            0xc1a12d2fc3978937L, 0xf209787bb47d6b84L,
            0x9745eb4d50ce6332L, 0xbd176620a501fbffL,
            0xec5d3fa8ce427affL, 0x93ba47c980e98cdfL,
            0xb8a8d9bbe123f017L, 0xe6d3102ad96cec1dL,
            0x9043ea1ac7e41392L, 0xb454e4a179dd1877L,
            0xe16a1dc9d8545e94L, 0x8ce2529e2734bb1dL,
            0xb01ae745b101e9e4L, 0xdc21a1171d42645dL,
            0x899504ae72497ebaL, 0xabfa45da0edbde69L,
            0xd6f8d7509292d603L, 0x865b86925b9bc5c2L,
            0xa7f26836f282b732L, 0xd1ef0244af2364ffL,
            0x8335616aed761f1fL, 0xa402b9c5a8d3a6e7L,
            0xcd036837130890a1L, 0x802221226be55a64L,
            0xa02aa96b06deb0fdL, 0xc83553c5c8965d3dL,
            0xfa42a8b73abbf48cL, 0x9c69a97284b578d7L,
            0xc38413cf25e2d70dL, 0xf46518c2ef5b8cd1L,
            0x98bf2f79d5993802L, 0xbeeefb584aff8603L,
            0xeeaaba2e5dbf6784L, 0x952ab45cfa97a0b2L,
            0xba756174393d88dfL, 0xe912b9d1478ceb17L,
            0x91abb422ccb812eeL, 0xb616a12b7fe617aaL,
            0xe39c49765fdf9d94L, 0x8e41ade9fbebc27dL,
            0xb1d219647ae6b31cL, 0xde469fbd99a05fe3L,
            0x8aec23d680043beeL, 0xada72ccc20054ae9L,
            0xd910f7ff28069da4L, 0x87aa9aff79042286L,
            0xa99541bf57452b28L, 0xd3fa922f2d1675f2L,
            0x847c9b5d7c2e09b7L, 0xa59bc234db398c25L,
            0xcf02b2c21207ef2eL, 0x8161afb94b44f57dL,
            0xa1ba1ba79e1632dcL, 0xca28a291859bbf93L,
            0xfcb2cb35e702af78L, 0x9defbf01b061adabL,
            0xc56baec21c7a1916L, 0xf6c69a72a3989f5bL,
            0x9a3c2087a63f6399L, 0xc0cb28a98fcf3c7fL,
            0xf0fdf2d3f3c30b9fL, 0x969eb7c47859e743L,
            0xbc4665b596706114L, 0xeb57ff22fc0c7959L,
            0x9316ff75dd87cbd8L, 0xb7dcbf5354e9beceL,
            0xe5d3ef282a242e81L, 0x8fa475791a569d10L,
            0xb38d92d760ec4455L, 0xe070f78d3927556aL,
            0x8c469ab843b89562L, 0xaf58416654a6babbL,
            0xdb2e51bfe9d0696aL, 0x88fcf317f22241e2L,
            0xab3c2fddeeaad25aL, 0xd60b3bd56a5586f1L,
            0x85c7056562757456L, 0xa738c6bebb12d16cL,
            0xd106f86e69d785c7L, 0x82a45b450226b39cL,
            0xa34d721642b06084L, 0xcc20ce9bd35c78a5L,
            0xff290242c83396ceL, 0x9f79a169bd203e41L,
            0xc75809c42c684dd1L, 0xf92e0c3537826145L,
            0x9bbcc7a142b17ccbL, 0xc2abf989935ddbfeL,
            0xf356f7ebf83552feL, 0x98165af37b2153deL,
            0xbe1bf1b059e9a8d6L, 0xeda2ee1c7064130cL,
            0x9485d4d1c63e8be7L, 0xb9a74a0637ce2ee1L,
            0xe8111c87c5c1ba99L, 0x910ab1d4db9914a0L,
            0xb54d5e4a127f59c8L, 0xe2a0b5dc971f303aL,
            0x8da471a9de737e24L, 0xb10d8e1456105dadL,
            0xdd50f1996b947518L, 0x8a5296ffe33cc92fL,
            0xace73cbfdc0bfb7bL, 0xd8210befd30efa5aL,
            0x8714a775e3e95c78L, 0xa8d9d1535ce3b396L,
            0xd31045a8341ca07cL, 0x83ea2b892091e44dL,
            0xa4e4b66b68b65d60L, 0xce1de40642e3f4b9L,
            0x80d2ae83e9ce78f3L, 0xa1075a24e4421730L,
            0xc94930ae1d529cfcL, 0xfb9b7cd9a4a7443cL,
            0x9d412e0806e88aa5L, 0xc491798a08a2ad4eL,
            0xf5b5d7ec8acb58a2L, 0x9991a6f3d6bf1765L,
            0xbff610b0cc6edd3fL, 0xeff394dcff8a948eL,
            0x95f83d0a1fb69cd9L, 0xbb764c4ca7a4440fL,
            0xea53df5fd18d5513L, 0x92746b9be2f8552cL,
            0xb7118682dbb66a77L, 0xe4d5e82392a40515L,
            0x8f05b1163ba6832dL, 0xb2c71d5bca9023f8L,
            0xdf78e4b2bd342cf6L, 0x8bab8eefb6409c1aL,
            0xae9672aba3d0c320L, 0xda3c0f568cc4f3e8L,
            0x8865899617fb1871L, 0xaa7eebfb9df9de8dL,
            0xd51ea6fa85785631L, 0x8533285c936b35deL,
            0xa67ff273b8460356L, 0xd01fef10a657842cL,
            0x8213f56a67f6b29bL, 0xa298f2c501f45f42L,
            0xcb3f2f7642717713L, 0xfe0efb53d30dd4d7L,
            0x9ec95d1463e8a506L, 0xc67bb4597ce2ce48L,
            0xf81aa16fdc1b81daL, 0x9b10a4e5e9913128L,
            0xc1d4ce1f63f57d72L, 0xf24a01a73cf2dccfL,
            0x976e41088617ca01L, 0xbd49d14aa79dbc82L,
            0xec9c459d51852ba2L, 0x93e1ab8252f33b45L,
            0xb8da1662e7b00a17L, 0xe7109bfba19c0c9dL,
            0x906a617d450187e2L, 0xb484f9dc9641e9daL,
            0xe1a63853bbd26451L, 0x8d07e33455637eb2L,
            0xb049dc016abc5e5fL, 0xdc5c5301c56b75f7L,
            0x89b9b3e11b6329baL, 0xac2820d9623bf429L,
            0xd732290fbacaf133L, 0x867f59a9d4bed6c0L,
            0xa81f301449ee8c70L, 0xd226fc195c6a2f8cL,
            0x83585d8fd9c25db7L, 0xa42e74f3d032f525L,
            0xcd3a1230c43fb26fL, 0x80444b5e7aa7cf85L,
            0xa0555e361951c366L, 0xc86ab5c39fa63440L,
            0xfa856334878fc150L, 0x9c935e00d4b9d8d2L,
            0xc3b8358109e84f07L, 0xf4a642e14c6262c8L,
            0x98e7e9cccfbd7dbdL, 0xbf21e44003acdd2cL,
            0xeeea5d5004981478L, 0x95527a5202df0ccbL,
            0xbaa718e68396cffdL, 0xe950df20247c83fdL,
            0x91d28b7416cdd27eL, 0xb6472e511c81471dL,
            0xe3d8f9e563a198e5L, 0x8e679c2f5e44ff8fL};
    /**
     * A complement to mantissa_64 complete to a
     * 128-bit mantissa.
     * <p>
     * Uses about 5KB but is rarely accessed.
     * <pre>
     * UInt128 getMantissa128(int q) {
     *     return new UInt128(
     *        MANTISSA_64[q - SMALLEST_POWER_OF_TEN],
     *        MANTISSA_128[q - SMALLEST_POWER_OF_TEN];
     *     );
     * }
     * </pre>
     */
    final static long[] MANTISSA_128 = {
            0x419ea3bd35385e2dL, 0x52064cac828675b9L,
            0x7343efebd1940993L, 0x1014ebe6c5f90bf8L,
            0xd41a26e077774ef6L, 0x8920b098955522b4L,
            0x55b46e5f5d5535b0L, 0xeb2189f734aa831dL,
            0xa5e9ec7501d523e4L, 0x47b233c92125366eL,
            0x999ec0bb696e840aL, 0xc00670ea43ca250dL,
            0x380406926a5e5728L, 0xc605083704f5ecf2L,
            0xf7864a44c633682eL, 0x7ab3ee6afbe0211dL,
            0x5960ea05bad82964L, 0x6fb92487298e33bdL,
            0xa5d3b6d479f8e056L, 0x8f48a4899877186cL,
            0x331acdabfe94de87L, 0x9ff0c08b7f1d0b14L,
            0x7ecf0ae5ee44dd9L, 0xc9e82cd9f69d6150L,
            0xbe311c083a225cd2L, 0x6dbd630a48aaf406L,
            0x92cbbccdad5b108L, 0x25bbf56008c58ea5L,
            0xaf2af2b80af6f24eL, 0x1af5af660db4aee1L,
            0x50d98d9fc890ed4dL, 0xe50ff107bab528a0L,
            0x1e53ed49a96272c8L, 0x25e8e89c13bb0f7aL,
            0x77b191618c54e9acL, 0xd59df5b9ef6a2417L,
            0x4b0573286b44ad1dL, 0x4ee367f9430aec32L,
            0x229c41f793cda73fL, 0x6b43527578c1110fL,
            0x830a13896b78aaa9L, 0x23cc986bc656d553L,
            0x2cbfbe86b7ec8aa8L, 0x7bf7d71432f3d6a9L,
            0xdaf5ccd93fb0cc53L, 0xd1b3400f8f9cff68L,
            0x23100809b9c21fa1L, 0xabd40a0c2832a78aL,
            0x16c90c8f323f516cL, 0xae3da7d97f6792e3L,
            0x99cd11cfdf41779cL, 0x40405643d711d583L,
            0x482835ea666b2572L, 0xda3243650005eecfL,
            0x90bed43e40076a82L, 0x5a7744a6e804a291L,
            0x711515d0a205cb36L, 0xd5a5b44ca873e03L,
            0xe858790afe9486c2L, 0x626e974dbe39a872L,
            0xfb0a3d212dc8128fL, 0x7ce66634bc9d0b99L,
            0x1c1fffc1ebc44e80L, 0xa327ffb266b56220L,
            0x4bf1ff9f0062baa8L, 0x6f773fc3603db4a9L,
            0xcb550fb4384d21d3L, 0x7e2a53a146606a48L,
            0x2eda7444cbfc426dL, 0xfa911155fefb5308L,
            0x793555ab7eba27caL, 0x4bc1558b2f3458deL,
            0x9eb1aaedfb016f16L, 0x465e15a979c1cadcL,
            0xbfacd89ec191ec9L, 0xcef980ec671f667bL,
            0x82b7e12780e7401aL, 0xd1b2ecb8b0908810L,
            0x861fa7e6dcb4aa15L, 0x67a791e093e1d49aL,
            0xe0c8bb2c5c6d24e0L, 0x58fae9f773886e18L,
            0xaf39a475506a899eL, 0x6d8406c952429603L,
            0xc8e5087ba6d33b83L, 0xfb1e4a9a90880a64L,
            0x5cf2eea09a55067fL, 0xf42faa48c0ea481eL,
            0xf13b94daf124da26L, 0x76c53d08d6b70858L,
            0x54768c4b0c64ca6eL, 0xa9942f5dcf7dfd09L,
            0xd3f93b35435d7c4cL, 0xc47bc5014a1a6dafL,
            0x359ab6419ca1091bL, 0xc30163d203c94b62L,
            0x79e0de63425dcf1dL, 0x985915fc12f542e4L,
            0x3e6f5b7b17b2939dL, 0xa705992ceecf9c42L,
            0x50c6ff782a838353L, 0xa4f8bf5635246428L,
            0x871b7795e136be99L, 0x28e2557b59846e3fL,
            0x331aeada2fe589cfL, 0x3ff0d2c85def7621L,
            0xfed077a756b53a9L, 0xd3e8495912c62894L,
            0x64712dd7abbbd95cL, 0xbd8d794d96aacfb3L,
            0xecf0d7a0fc5583a0L, 0xf41686c49db57244L,
            0x311c2875c522ced5L, 0x7d633293366b828bL,
            0xae5dff9c02033197L, 0xd9f57f830283fdfcL,
            0xd072df63c324fd7bL, 0x4247cb9e59f71e6dL,
            0x52d9be85f074e608L, 0x67902e276c921f8bL,
            0xba1cd8a3db53b6L, 0x80e8a40eccd228a4L,
            0x6122cd128006b2cdL, 0x796b805720085f81L,
            0xcbe3303674053bb0L, 0xbedbfc4411068a9cL,
            0xee92fb5515482d44L, 0x751bdd152d4d1c4aL,
            0xd262d45a78a0635dL, 0x86fb897116c87c34L,
            0xd45d35e6ae3d4da0L, 0x8974836059cca109L,
            0x2bd1a438703fc94bL, 0x7b6306a34627ddcfL,
            0x1a3bc84c17b1d542L, 0x20caba5f1d9e4a93L,
            0x547eb47b7282ee9cL, 0xe99e619a4f23aa43L,
            0x6405fa00e2ec94d4L, 0xde83bc408dd3dd04L,
            0x9624ab50b148d445L, 0x3badd624dd9b0957L,
            0xe54ca5d70a80e5d6L, 0x5e9fcf4ccd211f4cL,
            0x7647c3200069671fL, 0x29ecd9f40041e073L,
            0xf468107100525890L, 0x7182148d4066eeb4L,
            0xc6f14cd848405530L, 0xb8ada00e5a506a7cL,
            0xa6d90811f0e4851cL, 0x908f4a166d1da663L,
            0x9a598e4e043287feL, 0x40eff1e1853f29fdL,
            0xd12bee59e68ef47cL, 0x82bb74f8301958ceL,
            0xe36a52363c1faf01L, 0xdc44e6c3cb279ac1L,
            0x29ab103a5ef8c0b9L, 0x7415d448f6b6f0e7L,
            0x111b495b3464ad21L, 0xcab10dd900beec34L,
            0x3d5d514f40eea742L, 0xcb4a5a3112a5112L,
            0x47f0e785eaba72abL, 0x59ed216765690f56L,
            0x306869c13ec3532cL, 0x1e414218c73a13fbL,
            0xe5d1929ef90898faL, 0xdf45f746b74abf39L,
            0x6b8bba8c328eb783L, 0x66ea92f3f326564L,
            0xc80a537b0efefebdL, 0xbd06742ce95f5f36L,
            0x2c48113823b73704L, 0xf75a15862ca504c5L,
            0x9a984d73dbe722fbL, 0xc13e60d0d2e0ebbaL,
            0x318df905079926a8L, 0xfdf17746497f7052L,
            0xfeb6ea8bedefa633L, 0xfe64a52ee96b8fc0L,
            0x3dfdce7aa3c673b0L, 0x6bea10ca65c084eL,
            0x486e494fcff30a62L, 0x5a89dba3c3efccfaL,
            0xf89629465a75e01cL, 0xf6bbb397f1135823L,
            0x746aa07ded582e2cL, 0xa8c2a44eb4571cdcL,
            0x92f34d62616ce413L, 0x77b020baf9c81d17L,
            0xace1474dc1d122eL, 0xd819992132456baL,
            0x10e1fff697ed6c69L, 0xca8d3ffa1ef463c1L,
            0xbd308ff8a6b17cb2L, 0xac7cb3f6d05ddbdeL,
            0x6bcdf07a423aa96bL, 0x86c16c98d2c953c6L,
            0xe871c7bf077ba8b7L, 0x11471cd764ad4972L,
            0xd598e40d3dd89bcfL, 0x4aff1d108d4ec2c3L,
            0xcedf722a585139baL, 0xc2974eb4ee658828L,
            0x733d226229feea32L, 0x806357d5a3f525fL,
            0xca07c2dcb0cf26f7L, 0xfc89b393dd02f0b5L,
            0xbbac2078d443ace2L, 0xd54b944b84aa4c0dL,
            0xa9e795e65d4df11L, 0x4d4617b5ff4a16d5L,
            0x504bced1bf8e4e45L, 0xe45ec2862f71e1d6L,
            0x5d767327bb4e5a4cL, 0x3a6a07f8d510f86fL,
            0x890489f70a55368bL, 0x2b45ac74ccea842eL,
            0x3b0b8bc90012929dL, 0x9ce6ebb40173744L,
            0xcc420a6a101d0515L, 0x9fa946824a12232dL,
            0x47939822dc96abf9L, 0x59787e2b93bc56f7L,
            0x57eb4edb3c55b65aL, 0xede622920b6b23f1L,
            0xe95fab368e45ecedL, 0x11dbcb0218ebb414L,
            0xd652bdc29f26a119L, 0x4be76d3346f0495fL,
            0x6f70a4400c562ddbL, 0xcb4ccd500f6bb952L,
            0x7e2000a41346a7a7L, 0x8ed400668c0c28c8L,
            0x728900802f0f32faL, 0x4f2b40a03ad2ffb9L,
            0xe2f610c84987bfa8L, 0xdd9ca7d2df4d7c9L,
            0x91503d1c79720dbbL, 0x75a44c6397ce912aL,
            0xc986afbe3ee11abaL, 0xfbe85badce996168L,
            0xfae27299423fb9c3L, 0xdccd879fc967d41aL,
            0x5400e987bbc1c920L, 0x290123e9aab23b68L,
            0xf9a0b6720aaf6521L, 0xf808e40e8d5b3e69L,
            0xb60b1d1230b20e04L, 0xb1c6f22b5e6f48c2L,
            0x1e38aeb6360b1af3L, 0x25c6da63c38de1b0L,
            0x579c487e5a38ad0eL, 0x2d835a9df0c6d851L,
            0xf8e431456cf88e65L, 0x1b8e9ecb641b58ffL,
            0xe272467e3d222f3fL, 0x5b0ed81dcc6abb0fL,
            0x98e947129fc2b4e9L, 0x3f2398d747b36224L,
            0x8eec7f0d19a03aadL, 0x1953cf68300424acL,
            0x5fa8c3423c052dd7L, 0x3792f412cb06794dL,
            0xe2bbd88bbee40bd0L, 0x5b6aceaeae9d0ec4L,
            0xf245825a5a445275L, 0xeed6e2f0f0d56712L,
            0x55464dd69685606bL, 0xaa97e14c3c26b886L,
            0xd53dd99f4b3066a8L, 0xe546a8038efe4029L,
            0xde98520472bdd033L, 0x963e66858f6d4440L,
            0xdde7001379a44aa8L, 0x5560c018580d5d52L,
            0xaab8f01e6e10b4a6L, 0xcab3961304ca70e8L,
            0x3d607b97c5fd0d22L, 0x8cb89a7db77c506aL,
            0x77f3608e92adb242L, 0x55f038b237591ed3L,
            0x6b6c46dec52f6688L, 0x2323ac4b3b3da015L,
            0xabec975e0a0d081aL, 0x96e7bd358c904a21L,
            0x7e50d64177da2e54L, 0xdde50bd1d5d0b9e9L,
            0x955e4ec64b44e864L, 0xbd5af13bef0b113eL,
            0xecb1ad8aeacdd58eL, 0x67de18eda5814af2L,
            0x80eacf948770ced7L, 0xa1258379a94d028dL,
            0x96ee45813a04330L, 0x8bca9d6e188853fcL,
            0x775ea264cf55347dL, 0x95364afe032a819dL,
            0x3a83ddbd83f52204L, 0xc4926a9672793542L,
            0x75b7053c0f178293L, 0x5324c68b12dd6338L,
            0xd3f6fc16ebca5e03L, 0x88f4bb1ca6bcf584L,
            0x2b31e9e3d06c32e5L, 0x3aff322e62439fcfL,
            0x9befeb9fad487c2L, 0x4c2ebe687989a9b3L,
            0xf9d37014bf60a10L, 0x538484c19ef38c94L,
            0x2865a5f206b06fb9L, 0xf93f87b7442e45d3L,
            0xf78f69a51539d748L, 0xb573440e5a884d1bL,
            0x31680a88f8953030L, 0xfdc20d2b36ba7c3dL,
            0x3d32907604691b4cL, 0xa63f9a49c2c1b10fL,
            0xfcf80dc33721d53L, 0xd3c36113404ea4a8L,
            0x645a1cac083126e9L, 0x3d70a3d70a3d70a3L,
            0xccccccccccccccccL, 0x0L,
            0x0L, 0x0L,
            0x0L, 0x0L,
            0x0L, 0x0L,
            0x0L, 0x0L,
            0x0L, 0x0L,
            0x0L, 0x0L,
            0x0L, 0x0L,
            0x0L, 0x0L,
            0x0L, 0x0L,
            0x0L, 0x0L,
            0x0L, 0x0L,
            0x0L, 0x0L,
            0x0L, 0x0L,
            0x0L, 0x4000000000000000L,
            0x5000000000000000L, 0xa400000000000000L,
            0x4d00000000000000L, 0xf020000000000000L,
            0x6c28000000000000L, 0xc732000000000000L,
            0x3c7f400000000000L, 0x4b9f100000000000L,
            0x1e86d40000000000L, 0x1314448000000000L,
            0x17d955a000000000L, 0x5dcfab0800000000L,
            0x5aa1cae500000000L, 0xf14a3d9e40000000L,
            0x6d9ccd05d0000000L, 0xe4820023a2000000L,
            0xdda2802c8a800000L, 0xd50b2037ad200000L,
            0x4526f422cc340000L, 0x9670b12b7f410000L,
            0x3c0cdd765f114000L, 0xa5880a69fb6ac800L,
            0x8eea0d047a457a00L, 0x72a4904598d6d880L,
            0x47a6da2b7f864750L, 0x999090b65f67d924L,
            0xfff4b4e3f741cf6dL, 0xbff8f10e7a8921a4L,
            0xaff72d52192b6a0dL, 0x9bf4f8a69f764490L,
            0x2f236d04753d5b4L, 0x1d762422c946590L,
            0x424d3ad2b7b97ef5L, 0xd2e0898765a7deb2L,
            0x63cc55f49f88eb2fL, 0x3cbf6b71c76b25fbL,
            0x8bef464e3945ef7aL, 0x97758bf0e3cbb5acL,
            0x3d52eeed1cbea317L, 0x4ca7aaa863ee4bddL,
            0x8fe8caa93e74ef6aL, 0xb3e2fd538e122b44L,
            0x60dbbca87196b616L, 0xbc8955e946fe31cdL,
            0x6babab6398bdbe41L, 0xc696963c7eed2dd1L,
            0xfc1e1de5cf543ca2L, 0x3b25a55f43294bcbL,
            0x49ef0eb713f39ebeL, 0x6e3569326c784337L,
            0x49c2c37f07965404L, 0xdc33745ec97be906L,
            0x69a028bb3ded71a3L, 0xc40832ea0d68ce0cL,
            0xf50a3fa490c30190L, 0x792667c6da79e0faL,
            0x577001b891185938L, 0xed4c0226b55e6f86L,
            0x544f8158315b05b4L, 0x696361ae3db1c721L,
            0x3bc3a19cd1e38e9L, 0x4ab48a04065c723L,
            0x62eb0d64283f9c76L, 0x3ba5d0bd324f8394L,
            0xca8f44ec7ee36479L, 0x7e998b13cf4e1ecbL,
            0x9e3fedd8c321a67eL, 0xc5cfe94ef3ea101eL,
            0xbba1f1d158724a12L, 0x2a8a6e45ae8edc97L,
            0xf52d09d71a3293bdL, 0x593c2626705f9c56L,
            0x6f8b2fb00c77836cL, 0xb6dfb9c0f956447L,
            0x4724bd4189bd5eacL, 0x58edec91ec2cb657L,
            0x2f2967b66737e3edL, 0xbd79e0d20082ee74L,
            0xecd8590680a3aa11L, 0xe80e6f4820cc9495L,
            0x3109058d147fdcddL, 0xbd4b46f0599fd415L,
            0x6c9e18ac7007c91aL, 0x3e2cf6bc604ddb0L,
            0x84db8346b786151cL, 0xe612641865679a63L,
            0x4fcb7e8f3f60c07eL, 0xe3be5e330f38f09dL,
            0x5cadf5bfd3072cc5L, 0x73d9732fc7c8f7f6L,
            0x2867e7fddcdd9afaL, 0xb281e1fd541501b8L,
            0x1f225a7ca91a4226L, 0x3375788de9b06958L,
            0x52d6b1641c83aeL, 0xc0678c5dbd23a49aL,
            0xf840b7ba963646e0L, 0xb650e5a93bc3d898L,
            0xa3e51f138ab4cebeL, 0xc66f336c36b10137L,
            0xb80b0047445d4184L, 0xa60dc059157491e5L,
            0x87c89837ad68db2fL, 0x29babe4598c311fbL,
            0xf4296dd6fef3d67aL, 0x1899e4a65f58660cL,
            0x5ec05dcff72e7f8fL, 0x76707543f4fa1f73L,
            0x6a06494a791c53a8L, 0x487db9d17636892L,
            0x45a9d2845d3c42b6L, 0xb8a2392ba45a9b2L,
            0x8e6cac7768d7141eL, 0x3207d795430cd926L,
            0x7f44e6bd49e807b8L, 0x5f16206c9c6209a6L,
            0x36dba887c37a8c0fL, 0xc2494954da2c9789L,
            0xf2db9baa10b7bd6cL, 0x6f92829494e5acc7L,
            0xcb772339ba1f17f9L, 0xff2a760414536efbL,
            0xfef5138519684abaL, 0x7eb258665fc25d69L,
            0xef2f773ffbd97a61L, 0xaafb550ffacfd8faL,
            0x95ba2a53f983cf38L, 0xdd945a747bf26183L,
            0x94f971119aeef9e4L, 0x7a37cd5601aab85dL,
            0xac62e055c10ab33aL, 0x577b986b314d6009L,
            0xed5a7e85fda0b80bL, 0x14588f13be847307L,
            0x596eb2d8ae258fc8L, 0x6fca5f8ed9aef3bbL,
            0x25de7bb9480d5854L, 0xaf561aa79a10ae6aL,
            0x1b2ba1518094da04L, 0x90fb44d2f05d0842L,
            0x353a1607ac744a53L, 0x42889b8997915ce8L,
            0x69956135febada11L, 0x43fab9837e699095L,
            0x94f967e45e03f4bbL, 0x1d1be0eebac278f5L,
            0x6462d92a69731732L, 0x7d7b8f7503cfdcfeL,
            0x5cda735244c3d43eL, 0x3a0888136afa64a7L,
            0x88aaa1845b8fdd0L, 0x8aad549e57273d45L,
            0x36ac54e2f678864bL, 0x84576a1bb416a7ddL,
            0x656d44a2a11c51d5L, 0x9f644ae5a4b1b325L,
            0x873d5d9f0dde1feeL, 0xa90cb506d155a7eaL,
            0x9a7f12442d588f2L, 0xc11ed6d538aeb2fL,
            0x8f1668c8a86da5faL, 0xf96e017d694487bcL,
            0x37c981dcc395a9acL, 0x85bbe253f47b1417L,
            0x93956d7478ccec8eL, 0x387ac8d1970027b2L,
            0x6997b05fcc0319eL, 0x441fece3bdf81f03L,
            0xd527e81cad7626c3L, 0x8a71e223d8d3b074L,
            0xf6872d5667844e49L, 0xb428f8ac016561dbL,
            0xe13336d701beba52L, 0xecc0024661173473L,
            0x27f002d7f95d0190L, 0x31ec038df7b441f4L,
            0x7e67047175a15271L, 0xf0062c6e984d386L,
            0x52c07b78a3e60868L, 0xa7709a56ccdf8a82L,
            0x88a66076400bb691L, 0x6acff893d00ea435L,
            0x583f6b8c4124d43L, 0xc3727a337a8b704aL,
            0x744f18c0592e4c5cL, 0x1162def06f79df73L,
            0x8addcb5645ac2ba8L, 0x6d953e2bd7173692L,
            0xc8fa8db6ccdd0437L, 0x1d9c9892400a22a2L,
            0x2503beb6d00cab4bL, 0x2e44ae64840fd61dL,
            0x5ceaecfed289e5d2L, 0x7425a83e872c5f47L,
            0xd12f124e28f77719L, 0x82bd6b70d99aaa6fL,
            0x636cc64d1001550bL, 0x3c47f7e05401aa4eL,
            0x65acfaec34810a71L, 0x7f1839a741a14d0dL,
            0x1ede48111209a050L, 0x934aed0aab460432L,
            0xf81da84d5617853fL, 0x36251260ab9d668eL,
            0xc1d72b7c6b426019L, 0xb24cf65b8612f81fL,
            0xdee033f26797b627L, 0x169840ef017da3b1L,
            0x8e1f289560ee864eL, 0xf1a6f2bab92a27e2L,
            0xae10af696774b1dbL, 0xacca6da1e0a8ef29L,
            0x17fd090a58d32af3L, 0xddfc4b4cef07f5b0L,
            0x4abdaf101564f98eL, 0x9d6d1ad41abe37f1L,
            0x84c86189216dc5edL, 0x32fd3cf5b4e49bb4L,
            0x3fbc8c33221dc2a1L, 0xfabaf3feaa5334aL,
            0x29cb4d87f2a7400eL, 0x743e20e9ef511012L,
            0x914da9246b255416L, 0x1ad089b6c2f7548eL,
            0xa184ac2473b529b1L, 0xc9e5d72d90a2741eL,
            0x7e2fa67c7a658892L, 0xddbb901b98feeab7L,
            0x552a74227f3ea565L, 0xd53a88958f87275fL,
            0x8a892abaf368f137L, 0x2d2b7569b0432d85L,
            0x9c3b29620e29fc73L, 0x8349f3ba91b47b8fL,
            0x241c70a936219a73L, 0xed238cd383aa0110L,
            0xf4363804324a40aaL, 0xb143c6053edcd0d5L,
            0xdd94b7868e94050aL, 0xca7cf2b4191c8326L,
            0xfd1c2f611f63a3f0L, 0xbc633b39673c8cecL,
            0xd5be0503e085d813L, 0x4b2d8644d8a74e18L,
            0xddf8e7d60ed1219eL, 0xcabb90e5c942b503L,
            0x3d6a751f3b936243L, 0xcc512670a783ad4L,
            0x27fb2b80668b24c5L, 0xb1f9f660802dedf6L,
            0x5e7873f8a0396973L, 0xdb0b487b6423e1e8L,
            0x91ce1a9a3d2cda62L, 0x7641a140cc7810fbL,
            0xa9e904c87fcb0a9dL, 0x546345fa9fbdcd44L,
            0xa97c177947ad4095L, 0x49ed8eabcccc485dL,
            0x5c68f256bfff5a74L, 0x73832eec6fff3111L,
            0xc831fd53c5ff7eabL, 0xba3e7ca8b77f5e55L,
            0x28ce1bd2e55f35ebL, 0x7980d163cf5b81b3L,
            0xd7e105bcc332621fL, 0x8dd9472bf3fefaa7L,
            0xb14f98f6f0feb951L, 0x6ed1bf9a569f33d3L,
            0xa862f80ec4700c8L, 0xcd27bb612758c0faL,
            0x8038d51cb897789cL, 0xe0470a63e6bd56c3L,
            0x1858ccfce06cac74L, 0xf37801e0c43ebc8L,
            0xd30560258f54e6baL, 0x47c6b82ef32a2069L,
            0x4cdc331d57fa5441L, 0xe0133fe4adf8e952L,
            0x58180fddd97723a6L, 0x570f09eaa7ea7648L};
    private final static int DOUBLE_MIN_EXPONENT_POWER_OF_TWO = Double.MIN_EXPONENT;
    private final static int DOUBLE_MAX_EXPONENT_POWER_OF_TWO = Double.MAX_EXPONENT;
    /**
     * Precomputed powers of ten from 10^0 to 10^22. These
     * can be represented exactly using the double type.
     */
    private static final double[] DOUBLE_POWERS_OF_TEN = {
            1e0, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9, 1e10, 1e11,
            1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19, 1e20, 1e21, 1e22};

    /**
     * Don't let anyone instantiate this class.
     */
    private FastDoubleMath() {

    }

    /**
     * Computes {@code uint128 product = (uint64)x * (uint64)y}.
     * <p>
     * References:
     * <dl>
     *     <dt>Getting the high part of 64 bit integer multiplication</dt>
     *     <dd><a href="https://stackoverflow.com/questions/28868367/getting-the-high-part-of-64-bit-integer-multiplication">
     *         stackoverflow</a></dd>
     * </dl>
     *
     * @param x uint64 factor x
     * @param y uint64 factor y
     * @return uint128 product of x and y
     */
    static UInt128 fullMultiplication(long x, long y) {//before Java 18
        long x0 = x & 0xffffffffL, x1 = x >>> 32;
        long y0 = y & 0xffffffffL, y1 = y >>> 32;
        long p11 = x1 * y1, p01 = x0 * y1;
        long p10 = x1 * y0, p00 = x0 * y0;

        // 64-bit product + two 32-bit values
        long middle = p10 + (p00 >>> 32) + (p01 & 0xffffffffL);
        return new UInt128(
                // 64-bit product + two 32-bit values
                p11 + (middle >>> 32) + (p01 >>> 32),
                // Add LOW PART and lower half of MIDDLE PART
                (middle << 32) | (p00 & 0xffffffffL));
    }

    /**
     * Tries to compute {@code significand * 10^power} exactly using
     * a fast algorithm; and if {@code isNegative} is true, negate the result.
     * <p>
     * This function will only work in some cases, when it does not work it
     * returns NaN. This should work *most of the time* (like 99% of the time).
     * We assume that power is in the
     * [{@value #DOUBLE_MIN_EXPONENT_POWER_OF_TEN}, {@value #DOUBLE_MAX_EXPONENT_POWER_OF_TEN}]
     * interval: the caller is responsible for this check.
     *
     * @param isNegative  whether the number is negative
     * @param significand uint64 the significand
     * @param power       the exponent number (the power)
     * @return the computed double on success, {@link Double#NaN} on failure
     */
    static double tryDecFloatToDouble(boolean isNegative, long significand, int power) {
        // we start with a fast path
        // It was described in Clinger WD (1990).
        if (-22 <= power && power <= 22 && Long.compareUnsigned(significand, (1L << DOUBLE_SIGNIFICAND_WIDTH) - 1) <= 0) {
            // convert the integer into a double. This is lossless since
            // 0 <= i <= 2^53 - 1.
            double d = (double) significand;
            //
            // The general idea is as follows.
            // If 0 <= s < 2^53 and if 10^0 <= p <= 10^22 then
            // 1) Both s and p can be represented exactly as 64-bit floating-point values
            // 2) Because s and p can be represented exactly as floating-point values,
            // then s * p and s / p will produce correctly rounded values.
            //
            if (power < 0) {
                d = d / DOUBLE_POWERS_OF_TEN[-power];
            } else {
                d = d * DOUBLE_POWERS_OF_TEN[power];
            }
            return (isNegative) ? -d : d;
        }


        // The fast path has now failed, so we are falling back on the slower path.

        // We are going to need to do some 64-bit arithmetic to get a more precise product.
        // We use a table lookup approach.
        // It is safe because
        // power >= DOUBLE_MIN_EXPONENT_POWER_OF_TEN
        // and power <= DOUBLE_MAX_EXPONENT_POWER_OF_TEN
        // We recover the mantissa of the power, it has a leading 1. It is always
        // rounded down.
        long factorMantissa = MANTISSA_64[power - DOUBLE_MIN_EXPONENT_POWER_OF_TEN];


        // The exponent is 1023 + 64 + power + floor(log(5**power)/log(2)).
        //
        // 1023 is the exponent bias.
        // The 64 comes from the fact that we use a 64-bit word.
        //
        // Computing floor(log(5**power)/log(2)) could be
        // slow. Instead, we use a fast function.
        //
        // For power in (-400,350), we have that
        // (((152170 + 65536) * power ) >> 16);
        // is equal to
        //  floor(log(5**power)/log(2)) + power when power >= 0,
        // and it is equal to
        //  ceil(log(5**-power)/log(2)) + power when power < 0
        //
        //
        // The 65536 is (1<<16) and corresponds to
        // (65536 * power) >> 16 ---> power
        //
        // ((152170 * power ) >> 16) is equal to
        // floor(log(5**power)/log(2))
        //
        // Note that this is not magic: 152170/(1<<16) is
        // approximately equal to log(5)/log(2).
        // The 1<<16 value is a power of two; we could use a
        // larger power of 2 if we wanted to.
        //
        long exponent = (((152170L + 65536L) * power) >> 16) + DOUBLE_EXPONENT_BIAS + 64;
        // We want the most significant bit of digits to be 1. Shift if needed.
        int lz = Long.numberOfLeadingZeros(significand);
        significand <<= lz;
        // We want the most significant 64 bits of the product. We know
        // this will be non-zero because the most significant bit of digits is
        // 1.
        UInt128 product = fullMultiplication(significand, factorMantissa);
        long lower = product.low;
        long upper = product.high;
        // We know that upper has at most one leading zero because
        // both i and factor_mantissa have a leading one. This means
        // that the result is at least as large as ((1<<63)*(1<<63))/(1<<64).

        // As long as the first 9 bits of "upper" are not "1", then we
        // know that we have an exact computed value for the leading
        // 55 bits because any imprecision would play out as a +1, in
        // the worst case.
        // Having 55 bits is necessary because
        // we need 53 bits for the mantissa, but we have to have one rounding bit and
        // we can waste a bit if the most significant bit of the product is zero.
        // We expect this next branch to be rarely taken (say 1% of the time).
        // When (upper & 0x1FF) == 0x1FF, it can be common for
        // lower + i < lower to be true (proba. much higher than 1%).
        if ((upper & 0x1ffL) == 0x1ffL && Long.compareUnsigned(lower + significand, lower) < 0) {
            long factorMantissaLow =
                    MANTISSA_128[power - DOUBLE_MIN_EXPONENT_POWER_OF_TEN];
            // next, we compute the 64-bit x 128-bit multiplication, getting a 192-bit
            // result (three 64-bit values)
            product = fullMultiplication(significand, factorMantissaLow);
            long productLow = product.low;
            long productMiddle2 = product.high;
            long productMiddle1 = lower;
            long productHigh = upper;
            long productMiddle = productMiddle1 + productMiddle2;
            if (Long.compareUnsigned(productMiddle, productMiddle1) < 0) {
                productHigh++; // overflow carry
            }


            // we want to check whether mantissa *i + i would affect our result
            // This does happen, e.g. with 7.3177701707893310e+15
            if (((productMiddle + 1 == 0) && ((productHigh & 0x1ffL) == 0x1ffL) &&
                    (productLow + Long.compareUnsigned(significand, productLow) < 0))) { // let us be prudent and bail out.
                return Double.NaN;
            }
            upper = productHigh;
            //lower = product_middle;
        }

        // The final mantissa should be 53 bits with a leading 1.
        // We shift it so that it occupies 54 bits with a leading 1.
        long upperbit = upper >>> 63;
        long mantissa = upper >>> (upperbit + 9);
        lz += (int) (1 ^ upperbit);
        // Here we have mantissa < (1<<54).

        // We have to round to even. The "to even" part
        // is only a problem when we are right in between two floating-point values
        // which we guard against.
        // If we have lots of trailing zeros, we may fall right between two
        // floating-point values.
        if (((upper & 0x1ff) == 0x1ff)
                || ((upper & 0x1ff) == 0) && (mantissa & 3) == 1) {
            // if mantissa & 1 == 1 we might need to round up.
            //
            // Scenarios:
            // 1. We are not in the middle. Then we should round up.
            //
            // 2. We are right in the middle. Whether we round up depends
            // on the last significant bit: if it is "one" then we round
            // up (round to even) otherwise, we do not.
            //
            // So if the last significant bit is 1, we can safely round up.
            // Hence, we only need to bail out if (mantissa & 3) == 1.
            // Otherwise, we may need more accuracy or analysis to determine whether
            // we are exactly between two floating-point numbers.
            // It can be triggered with 1e23.
            // Note: because the factor_mantissa and factor_mantissa_low are
            // almost always rounded down (except for small positive powers),
            // almost always should round up.
            return Double.NaN;
        }

        mantissa += 1;
        mantissa >>>= 1;

        // Here we have mantissa < (1<<53), unless there was an overflow
        if (mantissa >= (1L << DOUBLE_SIGNIFICAND_WIDTH)) {
            // This will happen when parsing values such as 7.2057594037927933e+16
            mantissa = (1L << (DOUBLE_SIGNIFICAND_WIDTH - 1));
            lz--; // undo previous addition
        }

        mantissa &= ~(1L << (DOUBLE_SIGNIFICAND_WIDTH - 1));

        long realExponent = exponent - lz;
        // we have to check that realExponent is in range, otherwise we bail out
        if ((realExponent < 1) || (realExponent > DOUBLE_MAX_EXPONENT_POWER_OF_TWO + DOUBLE_EXPONENT_BIAS)) {
            return Double.NaN;
        }

        long bits = mantissa | realExponent << (DOUBLE_SIGNIFICAND_WIDTH - 1)
                | (isNegative ? 1L << 63 : 0L);
        return Double.longBitsToDouble(bits);
    }

    /**
     * Tries to compute {@code significand * 10^exponent} exactly using a fast
     * algorithm; and if {@code isNegative} is true, negate the result;
     * the significand can be truncated.
     *
     * @param isNegative                     true if the sign is negative
     * @param significand                    the significand
     * @param exponent                       the exponent number (the power)
     * @param isSignificandTruncated         true if significand has been truncated
     * @param exponentOfTruncatedSignificand the exponent number of the truncated significand
     * @return the double value,
     * or {@link Double#NaN} if the fast path failed.
     */
    static double tryDecFloatToDoubleTruncated(boolean isNegative, long significand, int exponent,
                                               boolean isSignificandTruncated,
                                               final int exponentOfTruncatedSignificand) {
        if (significand == 0) {
            return isNegative ? -0.0 : 0.0;
        }

        final double result;
        if (isSignificandTruncated) {
            // We have too many digits. We may have to round up.
            // To know whether rounding up is needed, we may have to examine up to 768 digits.

            // There are cases, in which rounding has no effect.
            if (DOUBLE_MIN_EXPONENT_POWER_OF_TEN <= exponentOfTruncatedSignificand
                    && exponentOfTruncatedSignificand <= DOUBLE_MAX_EXPONENT_POWER_OF_TEN) {
                double withoutRounding = tryDecFloatToDouble(isNegative, significand, exponentOfTruncatedSignificand);
                double roundedUp = tryDecFloatToDouble(isNegative, significand + 1, exponentOfTruncatedSignificand);
                if (!Double.isNaN(withoutRounding) && roundedUp == withoutRounding) {
                    return withoutRounding;
                }
            }

            // We have to take a slow path.
            //return Double.parseDouble(str.toString());
            result = Double.NaN;

        } else if (DOUBLE_MIN_EXPONENT_POWER_OF_TEN <= exponent && exponent <= DOUBLE_MAX_EXPONENT_POWER_OF_TEN) {
            result = tryDecFloatToDouble(isNegative, significand, exponent);
        } else {
            result = Double.NaN;
        }
        return result;
    }

    /**
     * Tries to compute {@code significand * 2^power} exactly using a fast
     * algorithm; and if {@code isNegative} is true, negate the result.
     *
     * @param isNegative  true if the sign is negative
     * @param significand the significand
     * @param power       the power of the exponent
     * @return the double value,
     * or {@link Double#NaN} if the fast path failed.
     */
    static double tryHexFloatToDouble(boolean isNegative, long significand, int power) {

        // we start with a fast path
        // We try to mimic the fast described by Clinger WD for decimal
        // float number literals. How to read floating point numbers accurately.
        // ACM SIGPLAN Notices. 1990
        if (Long.compareUnsigned(significand, 0x1fffffffffffffL) <= 0) {
            // convert the integer into a double. This is lossless since
            // 0 <= i <= 2^53 - 1.
            double d = (double) significand;
            //
            // The general idea is as follows.
            // If 0 <= s < 2^53  then
            // 1) Both s and p can be represented exactly as 64-bit floating-point
            // values (binary64).
            // 2) Because s and p can be represented exactly as floating-point values,
            // then s * p will produce correctly rounded values.
            //
            d = d * Math.scalb(1d, power);
            if (isNegative) {
                d = -d;
            }
            return d;
        }

        // The fast path has failed
        return Double.NaN;
    }

    /**
     * Tries to compute {@code significand * 2^exponent} exactly using a fast
     * algorithm; and if {@code isNegative} is true, negate the result;
     * the significand can be truncated.
     *
     * @param isNegative                     true if the sign is negative
     * @param significand                    the significand
     * @param exponent                       the exponent number (the power)
     * @param isSignificandTruncated         true if significand has been truncated
     * @param exponentOfTruncatedSignificand the exponent number of the truncated significand
     * @return the double value,
     * or {@link Double#NaN} if the fast path failed.
     */
    static double tryHexFloatToDoubleTruncated(boolean isNegative, long significand, long exponent, boolean isSignificandTruncated,
                                               long exponentOfTruncatedSignificand) {
        if (significand == 0) {
            return isNegative ? -0.0 : 0.0;
        }

        final double outDouble;
        if (isSignificandTruncated) {

            // We have too many digits. We may have to round up.
            // To know whether rounding up is needed, we may have to examine up to 768 digits.

            // There are cases, in which rounding has no effect.
            if (DOUBLE_MIN_EXPONENT_POWER_OF_TWO <= exponentOfTruncatedSignificand && exponentOfTruncatedSignificand <= DOUBLE_MAX_EXPONENT_POWER_OF_TWO) {
                double withoutRounding = tryHexFloatToDouble(isNegative, significand, (int) exponentOfTruncatedSignificand);
                double roundedUp = tryHexFloatToDouble(isNegative, significand + 1, (int) exponentOfTruncatedSignificand);
                if (!Double.isNaN(withoutRounding) && roundedUp == withoutRounding) {
                    return withoutRounding;
                }
            }

            // We have to take a slow path.
            outDouble = Double.NaN;

        } else if (DOUBLE_MIN_EXPONENT_POWER_OF_TWO <= exponent && exponent <= DOUBLE_MAX_EXPONENT_POWER_OF_TWO) {
            outDouble = tryHexFloatToDouble(isNegative, significand, (int) exponent);
        } else {
            outDouble = Double.NaN;
        }
        return outDouble;
    }

    static class UInt128 {
        final long high, low;

        private UInt128(long high, long low) {
            this.high = high;
            this.low = low;
        }
    }

    public static long clamp(long value, long min, long max) {
        //noinspection ManualMinMaxCalculation
        return value < min ? min : (value > max ? max : value);
    }
}

com/fasterxml/jackson/core/io/doubleparser/FastDoubleMath.java

 

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File size: 497693 bytes
Release date: 2022-11-05
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