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[51.6.255.108]) by smtp.gmail.com with ESMTPSA id ffacd0b85a97d-36bbcf0cc9csm10961723f8f.17.2024.08.05.13.29.16 (version=TLS1_3 cipher=TLS_AES_256_GCM_SHA384 bits=256/256); Mon, 05 Aug 2024 13:29:16 -0700 (PDT) From: Cassio Neri To: libstdc++@gcc.gnu.org, gcc-patches@gcc.gnu.org Cc: Cassio Neri Subject: [RFC] libstdc++: Replace Ryu with Teju Jagua for float. Date: Mon, 5 Aug 2024 21:29:04 +0100 Message-ID: <20240805202904.14254-1-cassio.neri@gmail.com> X-Mailer: git-send-email 2.43.0 MIME-Version: 1.0 X-Spam-Status: No, score=-12.0 required=5.0 tests=BAYES_00, DKIM_SIGNED, DKIM_VALID, DKIM_VALID_AU, DKIM_VALID_EF, FREEMAIL_FROM, GIT_PATCH_0, KAM_SHORT, RCVD_IN_DNSWL_NONE, SPF_HELO_NONE, SPF_PASS, TXREP autolearn=ham autolearn_force=no version=3.4.6 X-Spam-Checker-Version: SpamAssassin 3.4.6 (2021-04-09) on server2.sourceware.org X-BeenThere: gcc-patches@gcc.gnu.org X-Mailman-Version: 2.1.30 Precedence: list List-Id: Gcc-patches mailing list List-Unsubscribe: , List-Archive: List-Post: List-Help: List-Subscribe: , Errors-To: gcc-patches-bounces~incoming=patchwork.ozlabs.org@gcc.gnu.org Implement the template function teju_jagua which finds the shortest representation of a floating-point number. The floating-point type is a template parameter and the implementation is generic enough to handle all floating-point types of interest, namely, IEEE 754, std::bfloat16_t, x86 80-bit and IBM128. The function uses a few helpers and traits classes to handle the different floating-point types. This patch only provides those required for float and shows the pattern to be followed by subsequent patches that will handle the remaining floating-point types. Add new static data to the existent struct floating_type_traits which are required by teju_jagua. Some of these duplicate information that is already there and used by Ryu. A subsequent clean-up patch will remove duplications once the shortest representation for all types is found by teju_jagua. libstdc++-v3/ChangeLog: * src/c++17/floating_to_chars.cc: implement teju_jagua and helpers (for float). * src/c++17/floating_to_chars_tables.h: tables with constants required at runtime. --- libstdc++-v3/src/c++17/floating_to_chars.cc | 321 +++++++- .../src/c++17/floating_to_chars_tables.h | 716 ++++++++++++++++++ 2 files changed, 1027 insertions(+), 10 deletions(-) create mode 100644 libstdc++-v3/src/c++17/floating_to_chars_tables.h -- 2.43.0 diff --git a/libstdc++-v3/src/c++17/floating_to_chars.cc b/libstdc++-v3/src/c++17/floating_to_chars.cc index 2c9da977cd9..b6adbf9cf5f 100644 --- a/libstdc++-v3/src/c++17/floating_to_chars.cc +++ b/libstdc++-v3/src/c++17/floating_to_chars.cc @@ -96,6 +96,8 @@ using F128_type = void; #include +#include "floating_to_chars_tables.h" + namespace { #if defined __SIZEOF_INT128__ @@ -132,6 +134,26 @@ namespace { return generic128::generic_to_chars(v, result); } } // namespace ryu + // Binary or decimal field representation of a positive floating-point number. + // The base is implicitly given by the context and the represented number is: + // mantissa * base ^ exponent. + template + struct fields { + Mantissa mantissa; + int32_t exponent; + }; + + // Binary or decimal field representation of a floating-point number. + // The base is implicitly given by the context and the represented number is: + // (sign ? -1 : 1) * mantissa * base ^ exponent. + template + struct signed_fields; + + // TODO (CN): Remove. + template <> + struct signed_fields : ryu::floating_decimal_32 + { }; + // A traits class that contains pertinent information about the binary // format of each of the floating-point types we support. template @@ -149,6 +171,12 @@ namespace static constexpr uint64_t pow10_adjustment_tab[] = { 0b0000000000011101011100110101100101101110000000000000000000000000 }; + + using fields_t = signed_fields; + + static constexpr uint32_t mantissa_size = 23; + static constexpr uint32_t exponent_size = 8; + static constexpr int32_t exponent_minimum = -149; }; template<> @@ -430,6 +458,276 @@ namespace bool sign; }; + // Return 2^k. + template + constexpr T + pow2(const uint32_t k) + { + return T(1) << k; + } + + // Return the k least significant bits of n (i.e., e % 2^k.) + template + T + lsb(const T n, const uint32_t k) + { + return n % pow2(k); + } + + // Return the binary field representation of an IEEE Float value. + template > + typename Traits::fields_t + to_binary_ieee(const Float value) + { + using mantissa_t = typename Traits::mantissa_t; + + mantissa_t bits; + memcpy(&bits, &value, sizeof(value)); + + const mantissa_t mantissa_bits = lsb(bits, Traits::mantissa_size); + bits >>= Traits::mantissa_size; + + const int32_t exponent_bits = lsb(bits, Traits::exponent_size); + bits >>= Traits::exponent_size; + + const bool negative = bits != 0; + + int32_t exponent = int32_t(exponent_bits) + Traits::exponent_minimum; + mantissa_t mantissa = mantissa_bits; + + if (exponent_bits != 0) + { + // Number is normal. + exponent -= 1; + mantissa += pow2(Traits::mantissa_size); + } + + return { mantissa, exponent, negative } ; + } + + // Return the binary field representation of a float value. + typename floating_type_traits::fields_t + to_binary(const float value) + { + return to_binary_ieee(value); + } + + // Check whether m is multiple of 2^k. + template + bool + is_multiple_of_pow2(const Mantissa m, const uint32_t k) + { + return lsb(m, k) == 0; + } + + // Check whether the binary field representation of a number is that of a + // small integer. + template + bool + is_small_integer(const Mantissa m, const int32_t e) + { + return -static_cast(Traits::mantissa_size) <= e && e <= 0 && + is_multiple_of_pow2(m, -e); + } + + // Return the quotient n / 10. + uint32_t + div10(const uint32_t n) + { + __glibcxx_assert(n < 1'073'741'829); + constexpr uint64_t M = 429'496'730; // M ~= 2^32 / 10. + return static_cast(M * n >> 32); + } + + // Return the largest exponent f such that 10^f <= 2^e, i.e., f is the integer + // part of e * log_10(2). + int32_t + log10_pow2(const int32_t e) + { + __glibcxx_assert(-112'815 <= e && e <= 112'815); + constexpr uint64_t M = 1'292'913'987; // M ~= 2^32 * log_10(2) + return static_cast(M * e >> 32); + } + + // Return the residual r = e - e_0, where e_0 is the smallest exponent such + // that log10_pow2(e_0) = log10_pow2(e). + uint32_t + log10_pow2_residual(const int32_t e) + { + __glibcxx_assert(-112'815 <= e && e <= 112'815); + constexpr uint64_t M = 1'292'913'987; + return static_cast(M * e) / M; + } + + // Shorten the decimal field representation of a given number by removing + // mantissa's trailing zeros and increasing the exponent accordingly. + template + fields + remove_trailing_zeros(Mantissa m, int32_t e) + { + while (true) + { + const Mantissa q = div10(m); + if (m != 10 * q) + return { m, e }; + ++e; + m = q; + } + } + + // Return N * m / 2^64 (multiply and shift), where N is a 64-bits number split + // in two 32-bits parts. + uint32_t + mshift(const uint32_t m, const parts N) + { + const uint64_t s0 = static_cast(N.lower) * m; + const uint64_t s1 = static_cast(N.upper) * m; + return (s1 + (s0 >> 32)) >> 32; + } + + // Return N * 2^k / 2^64 (multiply and shift for power of 2), where N is a + // 64-bits number split in two 32-bits parts. + uint32_t + mshift_pow2(const uint32_t k, const parts N) + { + const int32_t s = int32_t(k) - 32; + if (s <= 0) + return N.upper >> -s; + return (N.upper << s) | (N.lower >> (32 - s)); + } + + // Check whether m is multiple of 5^f. + // pre: 0 <= f && f < static_cast(size(minverse::data)) + template + bool + is_multiple_of_pow5(const Mantissa m, const int32_t f) + { + const auto [multiplier, bound] = minverse::data[f]; + return m * multiplier <= bound; + } + + // Check whether m, for m in { m_a, m_b, c_ 2 }, yields a tie. + template + bool + is_tie(const Mantissa m, const int32_t f) + { + return 0 <= f + && f < static_cast(std::size(minverse::data)) + && is_multiple_of_pow5(m, f); + } + + // Check whether the mantissa of an uncentred value (whose decimal exponent is + // f) yields a tie. + template + bool + is_tie_uncentred(const int32_t f) + { + return f > 0 && Traits::mantissa_size % 4 == 2; + } + + // Return the shortest decimal field representation of a positive + // floating-point number whose binary field representation is given. + // Implementation based on: https://github.com/cassioneri/teju_jagua. + template + fields + teju_jagua(const fields binary) + { + const auto [m , e] = binary; + + if (is_small_integer(m, e)) + return remove_trailing_zeros(m >> -e, 0); + + const int32_t f = log10_pow2(e); + const uint32_t r = log10_pow2_residual(e); + const auto N = to_chars_table::get_multiplier(Mantissa{}, f); + + constexpr uint32_t mantissa_size = Traits::mantissa_size; + constexpr Mantissa m_0 = pow2(mantissa_size); + if (m != m_0 || e == Traits::exponent_minimum) + { + const Mantissa m_a = (2 * m - 1) << r; + const Mantissa a = mshift(m_a, N); + const Mantissa m_b = (2 * m + 1) << r; + const Mantissa b = mshift(m_b, N); + const Mantissa q = div10(b); + const Mantissa s = 10 * q; + + if (s >= a) + { + if (s == b) + { + if (m % 2 == 0 || !is_tie(m_b, f)) + return remove_trailing_zeros(q, f + 1); + } + else if (s > a || (m % 2 == 0 && is_tie(m_a, f))) + return remove_trailing_zeros(q, f + 1); + } + + if ((a + b) % 2 == 1) + return { (a + b) / 2 + 1, f }; + + const Mantissa m_c = (2 * 2 * m) << r; + const Mantissa c_2 = mshift(m_c, N); + const Mantissa c = c_2 / 2; + + if (c_2 % 2 == 0 || (c % 2 == 0 && is_tie(c_2, -f))) + return { c, f }; + + return { c + 1, f }; + } + + const Mantissa m_b = 2 * m_0 + 1; + const Mantissa b = mshift(m_b << r, N); + + const Mantissa m_a = 4 * m_0 - 1; + const Mantissa a = mshift(m_a << r, N) / 2; + + if (b > a) + { + const Mantissa q = div10(b); + const Mantissa s = 10 * q; + + if (s > a || (s == a && is_tie_uncentred(f))) + return remove_trailing_zeros(q, f + 1); + + // m_c = 2 * 2 * m_0 = 2 * 2 * 2^{mantissa_size} + // c_2 = mshift(m_c << r, N); + const uint32_t log2_m_c = mantissa_size + 2; + const Mantissa c_2 = mshift_pow2(log2_m_c + r, N); + const Mantissa c = c_2 / 2; + + if (c == a && !is_tie_uncentred(f)) + return { c + 1, f }; + + if (c_2 % 2 == 0 || (c % 2 == 0 && is_tie(c_2, -f))) + return { c, f }; + + return { c + 1, f }; + } + + else if (is_tie_uncentred(f)) + return remove_trailing_zeros(a, f); + + const Mantissa m_c = 10 * 2 * 2 * m_0; + const Mantissa c_2 = mshift(m_c << r, N); + const Mantissa c = c_2 / 2; + + if (c_2 % 2 == 0 || (c % 2 == 0 && is_tie(c_2, -f))) + return { c, f - 1 }; + + return { c + 1, f - 1 }; + } + + // Return the shortest decimal field representation of a Float value. + template > + typename Traits::fields_t + teju_jagua(const Float value) + { + auto const [m, e, sign] = to_binary(value); + auto const abs = teju_jagua({ m, e }); + return { abs.mantissa, abs.exponent, sign }; + } + // Decompose the floating-point value into its IEEE components. template ieee_t @@ -454,12 +752,12 @@ namespace ieee_t ieee_repr; ieee_repr.mantissa - = static_cast(value_bits & ((uint_t{1} << mantissa_bits) - 1u)); + = static_cast(lsb(value_bits, mantissa_bits)); value_bits >>= mantissa_bits; ieee_repr.biased_exponent - = static_cast(value_bits & ((uint_t{1} << exponent_bits) - 1u)); + = static_cast(lsb(value_bits, exponent_bits)); value_bits >>= exponent_bits; - ieee_repr.sign = (value_bits & 1) != 0; + ieee_repr.sign = lsb(value_bits, 1) != 0; return ieee_repr; } @@ -542,12 +840,12 @@ namespace ieee_t ieee_repr; ieee_repr.mantissa - = static_cast(value_bits & ((uint32_t{1} << mantissa_bits) - 1u)); + = static_cast(lsb(value_bits, mantissa_bits)); value_bits >>= mantissa_bits; ieee_repr.biased_exponent - = static_cast(value_bits & ((uint32_t{1} << exponent_bits) - 1u)); + = static_cast(lsb(value_bits, exponent_bits)); value_bits >>= exponent_bits; - ieee_repr.sign = (value_bits & 1) != 0; + ieee_repr.sign = lsb(value_bits, 1) != 0; // We have mantissa and biased_exponent from the float (originally // float16_t converted to float). // Transform that to float16_t mantissa and biased_exponent. @@ -589,12 +887,12 @@ namespace ieee_t ieee_repr; ieee_repr.mantissa - = static_cast(value_bits & ((uint32_t{1} << mantissa_bits) - 1u)); + = static_cast(lsb(value_bits, mantissa_bits)); value_bits >>= mantissa_bits; ieee_repr.biased_exponent - = static_cast(value_bits & ((uint32_t{1} << exponent_bits) - 1u)); + = static_cast(lsb(value_bits, exponent_bits)); value_bits >>= exponent_bits; - ieee_repr.sign = (value_bits & 1) != 0; + ieee_repr.sign = lsb(value_bits, 1) != 0; // We have mantissa and biased_exponent from the float (originally // bfloat16_t converted to float). // Transform that to bfloat16_t mantissa and biased_exponent. @@ -609,7 +907,10 @@ namespace floating_to_shortest_scientific(const T value) { if constexpr (std::is_same_v) - return ryu::floating_to_fd32(value); + { + (void) &ryu::floating_to_fd32; // TODO (CN): remove. + return teju_jagua(value); + } else if constexpr (std::is_same_v) return ryu::floating_to_fd64(value); else if constexpr (std::is_same_v diff --git a/libstdc++-v3/src/c++17/floating_to_chars_tables.h b/libstdc++-v3/src/c++17/floating_to_chars_tables.h new file mode 100644 index 00000000000..5cb648713b7 --- /dev/null +++ b/libstdc++-v3/src/c++17/floating_to_chars_tables.h @@ -0,0 +1,716 @@ +// Tables used by std::to_chars implementation for floating-point types -*- C++ -*- + +// Copyright (C) 2020-2024 Free Software Foundation, Inc. +// +// This file is part of the GNU ISO C++ Library. This library is free +// software; you can redistribute it and/or modify it under the +// terms of the GNU General Public License as published by the +// Free Software Foundation; either version 3, or (at your option) +// any later version. + +// This library 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 for more details. + +// Under Section 7 of GPL version 3, you are granted additional +// permissions described in the GCC Runtime Library Exception, version +// 3.1, as published by the Free Software Foundation. + +// You should have received a copy of the GNU General Public License and +// a copy of the GCC Runtime Library Exception along with this program; +// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see +// . + +#ifndef _GLIBCXX_FLOATING_TO_CHARS_TABLES_H +#define _GLIBCXX_FLOATING_TO_CHARS_TABLES_H + +#include + +namespace { + +// Upper and lower parts of mantissa multipliers. +template +struct parts { + const Mantissa upper; + const Mantissa lower; +}; + +// Taken from: https://github.com/cassioneri/teju_jagua. +struct to_chars_table { + + // Get the multiplier for mantissas of type uint32_t at index f. + // Suits IEEE Binary32. + static parts + get_multiplier(uint32_t /*tag_dispatch*/, const int32_t f) { + const uint64_t N = _get_multiplier(f).upper + 1; + return { static_cast(N >> 32), static_cast(N) }; + } + +private: + + static constexpr int32_t _index_minimum = -324; + static constexpr int32_t _index_maximum = 292; + + static const parts& + _get_multiplier(const int32_t f) + { + __glibcxx_assert(_index_minimum <= f && f <= _index_maximum); + return _entry[f - _index_minimum]; + } + + // Multipliers for mantissas of type uint64_t. + // Suits IEEE Binary64. + static constexpr parts _entry[] = + { + { 0x9e19db92b4e31ba9, 0x6c07a2c26a8346d2 }, // -324 + { 0xfcf62c1dee382c42, 0x46729e03dd9ed7b6 }, // -323 + { 0xca5e89b18b602368, 0x385bb19cb14bdfc5 }, // -322 + { 0xa1e53af46f801c53, 0x60495ae3c1097fd1 }, // -321 + { 0x81842f29f2cce375, 0xe6a1158300d46641 }, // -320 + { 0xcf39e50feae16bef, 0xd768226b34870a01 }, // -319 + { 0xa5c7ea73224deff3, 0x12b9b522906c0801 }, // -318 + { 0x849feec281d7f328, 0xdbc7c41ba6bcd334 }, // -317 + { 0xd433179d9c8cb841, 0x5fa60692a46151ec }, // -316 + { 0xa9c2794ae3a3c69a, 0xb2eb3875504ddb23 }, // -315 + { 0x87cec76f1c830548, 0x8f2293910d0b15b6 }, // -314 + { 0xd94ad8b1c7380874, 0x18375281ae7822bd }, // -313 + { 0xadd57a27d29339f6, 0x79c5db9af1f9b564 }, // -312 + { 0x8b112e86420f6191, 0xfb04afaf27faf783 }, // -311 + { 0xde81e40a034bcf4f, 0xf8077f7ea65e58d2 }, // -310 + { 0xb201833b35d63f73, 0x2cd2cc6551e513db }, // -309 + { 0x8e679c2f5e44ff8f, 0x570f09eaa7ea7649 }, // -308 + { 0xe3d8f9e563a198e5, 0x58180fddd97723a7 }, // -307 + { 0xb6472e511c81471d, 0xe0133fe4adf8e953 }, // -306 + { 0x91d28b7416cdd27e, 0x4cdc331d57fa5442 }, // -305 + { 0xe950df20247c83fd, 0x47c6b82ef32a206a }, // -304 + { 0xbaa718e68396cffd, 0xd30560258f54e6bb }, // -303 + { 0x95527a5202df0ccb, 0x0f37801e0c43ebc9 }, // -302 + { 0xeeea5d5004981478, 0x1858ccfce06cac75 }, // -301 + { 0xbf21e44003acdd2c, 0xe0470a63e6bd56c4 }, // -300 + { 0x98e7e9cccfbd7dbd, 0x8038d51cb897789d }, // -299 + { 0xf4a642e14c6262c8, 0xcd27bb612758c0fb }, // -298 + { 0xc3b8358109e84f07, 0x0a862f80ec4700c9 }, // -297 + { 0x9c935e00d4b9d8d2, 0x6ed1bf9a569f33d4 }, // -296 + { 0xfa856334878fc150, 0xb14f98f6f0feb952 }, // -295 + { 0xc86ab5c39fa63440, 0x8dd9472bf3fefaa8 }, // -294 + { 0xa0555e361951c366, 0xd7e105bcc3326220 }, // -293 + { 0x80444b5e7aa7cf85, 0x7980d163cf5b81b4 }, // -292 + { 0xcd3a1230c43fb26f, 0x28ce1bd2e55f35ec }, // -291 + { 0xa42e74f3d032f525, 0xba3e7ca8b77f5e56 }, // -290 + { 0x83585d8fd9c25db7, 0xc831fd53c5ff7eac }, // -289 + { 0xd226fc195c6a2f8c, 0x73832eec6fff3112 }, // -288 + { 0xa81f301449ee8c70, 0x5c68f256bfff5a75 }, // -287 + { 0x867f59a9d4bed6c0, 0x49ed8eabcccc485e }, // -286 + { 0xd732290fbacaf133, 0xa97c177947ad4096 }, // -285 + { 0xac2820d9623bf429, 0x546345fa9fbdcd45 }, // -284 + { 0x89b9b3e11b6329ba, 0xa9e904c87fcb0a9e }, // -283 + { 0xdc5c5301c56b75f7, 0x7641a140cc7810fc }, // -282 + { 0xb049dc016abc5e5f, 0x91ce1a9a3d2cda63 }, // -281 + { 0x8d07e33455637eb2, 0xdb0b487b6423e1e9 }, // -280 + { 0xe1a63853bbd26451, 0x5e7873f8a0396974 }, // -279 + { 0xb484f9dc9641e9da, 0xb1f9f660802dedf7 }, // -278 + { 0x906a617d450187e2, 0x27fb2b80668b24c6 }, // -277 + { 0xe7109bfba19c0c9d, 0x0cc512670a783ad5 }, // -276 + { 0xb8da1662e7b00a17, 0x3d6a751f3b936244 }, // -275 + { 0x93e1ab8252f33b45, 0xcabb90e5c942b504 }, // -274 + { 0xec9c459d51852ba2, 0xddf8e7d60ed1219f }, // -273 + { 0xbd49d14aa79dbc82, 0x4b2d8644d8a74e19 }, // -272 + { 0x976e41088617ca01, 0xd5be0503e085d814 }, // -271 + { 0xf24a01a73cf2dccf, 0xbc633b39673c8ced }, // -270 + { 0xc1d4ce1f63f57d72, 0xfd1c2f611f63a3f1 }, // -269 + { 0x9b10a4e5e9913128, 0xca7cf2b4191c8327 }, // -268 + { 0xf81aa16fdc1b81da, 0xdd94b7868e94050b }, // -267 + { 0xc67bb4597ce2ce48, 0xb143c6053edcd0d6 }, // -266 + { 0x9ec95d1463e8a506, 0xf4363804324a40ab }, // -265 + { 0xfe0efb53d30dd4d7, 0xed238cd383aa0111 }, // -264 + { 0xcb3f2f7642717713, 0x241c70a936219a74 }, // -263 + { 0xa298f2c501f45f42, 0x8349f3ba91b47b90 }, // -262 + { 0x8213f56a67f6b29b, 0x9c3b29620e29fc74 }, // -261 + { 0xd01fef10a657842c, 0x2d2b7569b0432d86 }, // -260 + { 0xa67ff273b8460356, 0x8a892abaf368f138 }, // -259 + { 0x8533285c936b35de, 0xd53a88958f872760 }, // -258 + { 0xd51ea6fa85785631, 0x552a74227f3ea566 }, // -257 + { 0xaa7eebfb9df9de8d, 0xddbb901b98feeab8 }, // -256 + { 0x8865899617fb1871, 0x7e2fa67c7a658893 }, // -255 + { 0xda3c0f568cc4f3e8, 0xc9e5d72d90a2741f }, // -254 + { 0xae9672aba3d0c320, 0xa184ac2473b529b2 }, // -253 + { 0x8bab8eefb6409c1a, 0x1ad089b6c2f7548f }, // -252 + { 0xdf78e4b2bd342cf6, 0x914da9246b255417 }, // -251 + { 0xb2c71d5bca9023f8, 0x743e20e9ef511013 }, // -250 + { 0x8f05b1163ba6832d, 0x29cb4d87f2a7400f }, // -249 + { 0xe4d5e82392a40515, 0x0fabaf3feaa5334b }, // -248 + { 0xb7118682dbb66a77, 0x3fbc8c33221dc2a2 }, // -247 + { 0x92746b9be2f8552c, 0x32fd3cf5b4e49bb5 }, // -246 + { 0xea53df5fd18d5513, 0x84c86189216dc5ee }, // -245 + { 0xbb764c4ca7a4440f, 0x9d6d1ad41abe37f2 }, // -244 + { 0x95f83d0a1fb69cd9, 0x4abdaf101564f98f }, // -243 + { 0xeff394dcff8a948e, 0xddfc4b4cef07f5b1 }, // -242 + { 0xbff610b0cc6edd3f, 0x17fd090a58d32af4 }, // -241 + { 0x9991a6f3d6bf1765, 0xacca6da1e0a8ef2a }, // -240 + { 0xf5b5d7ec8acb58a2, 0xae10af696774b1dc }, // -239 + { 0xc491798a08a2ad4e, 0xf1a6f2bab92a27e3 }, // -238 + { 0x9d412e0806e88aa5, 0x8e1f289560ee864f }, // -237 + { 0xfb9b7cd9a4a7443c, 0x169840ef017da3b2 }, // -236 + { 0xc94930ae1d529cfc, 0xdee033f26797b628 }, // -235 + { 0xa1075a24e4421730, 0xb24cf65b8612f820 }, // -234 + { 0x80d2ae83e9ce78f3, 0xc1d72b7c6b42601a }, // -233 + { 0xce1de40642e3f4b9, 0x36251260ab9d668f }, // -232 + { 0xa4e4b66b68b65d60, 0xf81da84d56178540 }, // -231 + { 0x83ea2b892091e44d, 0x934aed0aab460433 }, // -230 + { 0xd31045a8341ca07c, 0x1ede48111209a051 }, // -229 + { 0xa8d9d1535ce3b396, 0x7f1839a741a14d0e }, // -228 + { 0x8714a775e3e95c78, 0x65acfaec34810a72 }, // -227 + { 0xd8210befd30efa5a, 0x3c47f7e05401aa4f }, // -226 + { 0xace73cbfdc0bfb7b, 0x636cc64d1001550c }, // -225 + { 0x8a5296ffe33cc92f, 0x82bd6b70d99aaa70 }, // -224 + { 0xdd50f1996b947518, 0xd12f124e28f7771a }, // -223 + { 0xb10d8e1456105dad, 0x7425a83e872c5f48 }, // -222 + { 0x8da471a9de737e24, 0x5ceaecfed289e5d3 }, // -221 + { 0xe2a0b5dc971f303a, 0x2e44ae64840fd61e }, // -220 + { 0xb54d5e4a127f59c8, 0x2503beb6d00cab4c }, // -219 + { 0x910ab1d4db9914a0, 0x1d9c9892400a22a3 }, // -218 + { 0xe8111c87c5c1ba99, 0xc8fa8db6ccdd0438 }, // -217 + { 0xb9a74a0637ce2ee1, 0x6d953e2bd7173693 }, // -216 + { 0x9485d4d1c63e8be7, 0x8addcb5645ac2ba9 }, // -215 + { 0xeda2ee1c7064130c, 0x1162def06f79df74 }, // -214 + { 0xbe1bf1b059e9a8d6, 0x744f18c0592e4c5d }, // -213 + { 0x98165af37b2153de, 0xc3727a337a8b704b }, // -212 + { 0xf356f7ebf83552fe, 0x0583f6b8c4124d44 }, // -211 + { 0xc2abf989935ddbfe, 0x6acff893d00ea436 }, // -210 + { 0x9bbcc7a142b17ccb, 0x88a66076400bb692 }, // -209 + { 0xf92e0c3537826145, 0xa7709a56ccdf8a83 }, // -208 + { 0xc75809c42c684dd1, 0x52c07b78a3e60869 }, // -207 + { 0x9f79a169bd203e41, 0x0f0062c6e984d387 }, // -206 + { 0xff290242c83396ce, 0x7e67047175a15272 }, // -205 + { 0xcc20ce9bd35c78a5, 0x31ec038df7b441f5 }, // -204 + { 0xa34d721642b06084, 0x27f002d7f95d0191 }, // -203 + { 0x82a45b450226b39c, 0xecc0024661173474 }, // -202 + { 0xd106f86e69d785c7, 0xe13336d701beba53 }, // -201 + { 0xa738c6bebb12d16c, 0xb428f8ac016561dc }, // -200 + { 0x85c7056562757456, 0xf6872d5667844e4a }, // -199 + { 0xd60b3bd56a5586f1, 0x8a71e223d8d3b075 }, // -198 + { 0xab3c2fddeeaad25a, 0xd527e81cad7626c4 }, // -197 + { 0x88fcf317f22241e2, 0x441fece3bdf81f04 }, // -196 + { 0xdb2e51bfe9d0696a, 0x06997b05fcc0319f }, // -195 + { 0xaf58416654a6babb, 0x387ac8d1970027b3 }, // -194 + { 0x8c469ab843b89562, 0x93956d7478ccec8f }, // -193 + { 0xe070f78d3927556a, 0x85bbe253f47b1418 }, // -192 + { 0xb38d92d760ec4455, 0x37c981dcc395a9ad }, // -191 + { 0x8fa475791a569d10, 0xf96e017d694487bd }, // -190 + { 0xe5d3ef282a242e81, 0x8f1668c8a86da5fb }, // -189 + { 0xb7dcbf5354e9bece, 0x0c11ed6d538aeb30 }, // -188 + { 0x9316ff75dd87cbd8, 0x09a7f12442d588f3 }, // -187 + { 0xeb57ff22fc0c7959, 0xa90cb506d155a7eb }, // -186 + { 0xbc4665b596706114, 0x873d5d9f0dde1fef }, // -185 + { 0x969eb7c47859e743, 0x9f644ae5a4b1b326 }, // -184 + { 0xf0fdf2d3f3c30b9f, 0x656d44a2a11c51d6 }, // -183 + { 0xc0cb28a98fcf3c7f, 0x84576a1bb416a7de }, // -182 + { 0x9a3c2087a63f6399, 0x36ac54e2f678864c }, // -181 + { 0xf6c69a72a3989f5b, 0x8aad549e57273d46 }, // -180 + { 0xc56baec21c7a1916, 0x088aaa1845b8fdd1 }, // -179 + { 0x9defbf01b061adab, 0x3a0888136afa64a8 }, // -178 + { 0xfcb2cb35e702af78, 0x5cda735244c3d43f }, // -177 + { 0xca28a291859bbf93, 0x7d7b8f7503cfdcff }, // -176 + { 0xa1ba1ba79e1632dc, 0x6462d92a69731733 }, // -175 + { 0x8161afb94b44f57d, 0x1d1be0eebac278f6 }, // -174 + { 0xcf02b2c21207ef2e, 0x94f967e45e03f4bc }, // -173 + { 0xa59bc234db398c25, 0x43fab9837e699096 }, // -172 + { 0x847c9b5d7c2e09b7, 0x69956135febada12 }, // -171 + { 0xd3fa922f2d1675f2, 0x42889b8997915ce9 }, // -170 + { 0xa99541bf57452b28, 0x353a1607ac744a54 }, // -169 + { 0x87aa9aff79042286, 0x90fb44d2f05d0843 }, // -168 + { 0xd910f7ff28069da4, 0x1b2ba1518094da05 }, // -167 + { 0xada72ccc20054ae9, 0xaf561aa79a10ae6b }, // -166 + { 0x8aec23d680043bee, 0x25de7bb9480d5855 }, // -165 + { 0xde469fbd99a05fe3, 0x6fca5f8ed9aef3bc }, // -164 + { 0xb1d219647ae6b31c, 0x596eb2d8ae258fc9 }, // -163 + { 0x8e41ade9fbebc27d, 0x14588f13be847308 }, // -162 + { 0xe39c49765fdf9d94, 0xed5a7e85fda0b80c }, // -161 + { 0xb616a12b7fe617aa, 0x577b986b314d600a }, // -160 + { 0x91abb422ccb812ee, 0xac62e055c10ab33b }, // -159 + { 0xe912b9d1478ceb17, 0x7a37cd5601aab85e }, // -158 + { 0xba756174393d88df, 0x94f971119aeef9e5 }, // -157 + { 0x952ab45cfa97a0b2, 0xdd945a747bf26184 }, // -156 + { 0xeeaaba2e5dbf6784, 0x95ba2a53f983cf39 }, // -155 + { 0xbeeefb584aff8603, 0xaafb550ffacfd8fb }, // -154 + { 0x98bf2f79d5993802, 0xef2f773ffbd97a62 }, // -153 + { 0xf46518c2ef5b8cd1, 0x7eb258665fc25d6a }, // -152 + { 0xc38413cf25e2d70d, 0xfef5138519684abb }, // -151 + { 0x9c69a97284b578d7, 0xff2a760414536efc }, // -150 + { 0xfa42a8b73abbf48c, 0xcb772339ba1f17fa }, // -149 + { 0xc83553c5c8965d3d, 0x6f92829494e5acc8 }, // -148 + { 0xa02aa96b06deb0fd, 0xf2db9baa10b7bd6d }, // -147 + { 0x802221226be55a64, 0xc2494954da2c978a }, // -146 + { 0xcd036837130890a1, 0x36dba887c37a8c10 }, // -145 + { 0xa402b9c5a8d3a6e7, 0x5f16206c9c6209a7 }, // -144 + { 0x8335616aed761f1f, 0x7f44e6bd49e807b9 }, // -143 + { 0xd1ef0244af2364ff, 0x3207d795430cd927 }, // -142 + { 0xa7f26836f282b732, 0x8e6cac7768d7141f }, // -141 + { 0x865b86925b9bc5c2, 0x0b8a2392ba45a9b3 }, // -140 + { 0xd6f8d7509292d603, 0x45a9d2845d3c42b7 }, // -139 + { 0xabfa45da0edbde69, 0x0487db9d17636893 }, // -138 + { 0x899504ae72497eba, 0x6a06494a791c53a9 }, // -137 + { 0xdc21a1171d42645d, 0x76707543f4fa1f74 }, // -136 + { 0xb01ae745b101e9e4, 0x5ec05dcff72e7f90 }, // -135 + { 0x8ce2529e2734bb1d, 0x1899e4a65f58660d }, // -134 + { 0xe16a1dc9d8545e94, 0xf4296dd6fef3d67b }, // -133 + { 0xb454e4a179dd1877, 0x29babe4598c311fc }, // -132 + { 0x9043ea1ac7e41392, 0x87c89837ad68db30 }, // -131 + { 0xe6d3102ad96cec1d, 0xa60dc059157491e6 }, // -130 + { 0xb8a8d9bbe123f017, 0xb80b0047445d4185 }, // -129 + { 0x93ba47c980e98cdf, 0xc66f336c36b10138 }, // -128 + { 0xec5d3fa8ce427aff, 0xa3e51f138ab4cebf }, // -127 + { 0xbd176620a501fbff, 0xb650e5a93bc3d899 }, // -126 + { 0x9745eb4d50ce6332, 0xf840b7ba963646e1 }, // -125 + { 0xf209787bb47d6b84, 0xc0678c5dbd23a49b }, // -124 + { 0xc1a12d2fc3978937, 0x0052d6b1641c83af }, // -123 + { 0x9ae757596946075f, 0x3375788de9b06959 }, // -122 + { 0xf7d88bc24209a565, 0x1f225a7ca91a4227 }, // -121 + { 0xc646d63501a1511d, 0xb281e1fd541501b9 }, // -120 + { 0x9e9f11c4014dda7e, 0x2867e7fddcdd9afb }, // -119 + { 0xfdcb4fa002162a63, 0x73d9732fc7c8f7f7 }, // -118 + { 0xcb090c8001ab551c, 0x5cadf5bfd3072cc6 }, // -117 + { 0xa26da3999aef7749, 0xe3be5e330f38f09e }, // -116 + { 0x81f14fae158c5f6e, 0x4fcb7e8f3f60c07f }, // -115 + { 0xcfe87f7cef46ff16, 0xe612641865679a64 }, // -114 + { 0xa6539930bf6bff45, 0x84db8346b786151d }, // -113 + { 0x850fadc09923329e, 0x03e2cf6bc604ddb1 }, // -112 + { 0xd4e5e2cdc1d1ea96, 0x6c9e18ac7007c91b }, // -111 + { 0xaa51823e34a7eede, 0xbd4b46f0599fd416 }, // -110 + { 0x884134fe908658b2, 0x3109058d147fdcde }, // -109 + { 0xda01ee641a708de9, 0xe80e6f4820cc9496 }, // -108 + { 0xae67f1e9aec07187, 0xecd8590680a3aa12 }, // -107 + { 0x8b865b215899f46c, 0xbd79e0d20082ee75 }, // -106 + { 0xdf3d5e9bc0f653e1, 0x2f2967b66737e3ee }, // -105 + { 0xb2977ee300c50fe7, 0x58edec91ec2cb658 }, // -104 + { 0x8edf98b59a373fec, 0x4724bd4189bd5ead }, // -103 + { 0xe498f455c38b997a, 0x0b6dfb9c0f956448 }, // -102 + { 0xb6e0c377cfa2e12e, 0x6f8b2fb00c77836d }, // -101 + { 0x924d692ca61be758, 0x593c2626705f9c57 }, // -100 + { 0xea1575143cf97226, 0xf52d09d71a3293be }, // -99 + { 0xbb445da9ca61281f, 0x2a8a6e45ae8edc98 }, // -98 + { 0x95d04aee3b80ece5, 0xbba1f1d158724a13 }, // -97 + { 0xefb3ab16c59b14a2, 0xc5cfe94ef3ea101f }, // -96 + { 0xbfc2ef456ae276e8, 0x9e3fedd8c321a67f }, // -95 + { 0x9968bf6abbe85f20, 0x7e998b13cf4e1ecc }, // -94 + { 0xf5746577930d6500, 0xca8f44ec7ee3647a }, // -93 + { 0xc45d1df942711d9a, 0x3ba5d0bd324f8395 }, // -92 + { 0x9d174b2dcec0e47b, 0x62eb0d64283f9c77 }, // -91 + { 0xfb5878494ace3a5f, 0x04ab48a04065c724 }, // -90 + { 0xc913936dd571c84c, 0x03bc3a19cd1e38ea }, // -89 + { 0xa0dc75f1778e39d6, 0x696361ae3db1c722 }, // -88 + { 0x80b05e5ac60b6178, 0x544f8158315b05b5 }, // -87 + { 0xcde6fd5e09abcf26, 0xed4c0226b55e6f87 }, // -86 + { 0xa4b8cab1a1563f52, 0x577001b891185939 }, // -85 + { 0x83c7088e1aab65db, 0x792667c6da79e0fb }, // -84 + { 0xd2d80db02aabd62b, 0xf50a3fa490c30191 }, // -83 + { 0xa8acd7c0222311bc, 0xc40832ea0d68ce0d }, // -82 + { 0x86f0ac99b4e8dafd, 0x69a028bb3ded71a4 }, // -81 + { 0xd7e77a8f87daf7fb, 0xdc33745ec97be907 }, // -80 + { 0xacb92ed9397bf996, 0x49c2c37f07965405 }, // -79 + { 0x8a2dbf142dfcc7ab, 0x6e3569326c784338 }, // -78 + { 0xdd15fe86affad912, 0x49ef0eb713f39ebf }, // -77 + { 0xb0de65388cc8ada8, 0x3b25a55f43294bcc }, // -76 + { 0x8d7eb76070a08aec, 0xfc1e1de5cf543ca3 }, // -75 + { 0xe264589a4dcdab14, 0xc696963c7eed2dd2 }, // -74 + { 0xb51d13aea4a488dd, 0x6babab6398bdbe42 }, // -73 + { 0x90e40fbeea1d3a4a, 0xbc8955e946fe31ce }, // -72 + { 0xe7d34c64a9c85d44, 0x60dbbca87196b617 }, // -71 + { 0xb975d6b6ee39e436, 0xb3e2fd538e122b45 }, // -70 + { 0x945e455f24fb1cf8, 0x8fe8caa93e74ef6b }, // -69 + { 0xed63a231d4c4fb27, 0x4ca7aaa863ee4bde }, // -68 + { 0xbde94e8e43d0c8ec, 0x3d52eeed1cbea318 }, // -67 + { 0x97edd871cfda3a56, 0x97758bf0e3cbb5ad }, // -66 + { 0xf316271c7fc3908a, 0x8bef464e3945ef7b }, // -65 + { 0xc2781f49ffcfa6d5, 0x3cbf6b71c76b25fc }, // -64 + { 0x9b934c3b330c8577, 0x63cc55f49f88eb30 }, // -63 + { 0xf8ebad2b84e0d58b, 0xd2e0898765a7deb3 }, // -62 + { 0xc722f0ef9d80aad6, 0x424d3ad2b7b97ef6 }, // -61 + { 0x9f4f2726179a2245, 0x01d762422c946591 }, // -60 + { 0xfee50b7025c36a08, 0x02f236d04753d5b5 }, // -59 + { 0xcbea6f8ceb02bb39, 0x9bf4f8a69f764491 }, // -58 + { 0xa321f2d7226895c7, 0xaff72d52192b6a0e }, // -57 + { 0x82818f1281ed449f, 0xbff8f10e7a8921a5 }, // -56 + { 0xd0cf4b50cfe20765, 0xfff4b4e3f741cf6e }, // -55 + { 0xa70c3c40a64e6c51, 0x999090b65f67d925 }, // -54 + { 0x85a36366eb71f041, 0x47a6da2b7f864751 }, // -53 + { 0xd5d238a4abe98068, 0x72a4904598d6d881 }, // -52 + { 0xab0e93b6efee0053, 0x8eea0d047a457a01 }, // -51 + { 0x88d8762bf324cd0f, 0xa5880a69fb6ac801 }, // -50 + { 0xdaf3f04651d47b4c, 0x3c0cdd765f114001 }, // -49 + { 0xaf298d050e4395d6, 0x9670b12b7f410001 }, // -48 + { 0x8c213d9da502de45, 0x4526f422cc340001 }, // -47 + { 0xe0352f62a19e306e, 0xd50b2037ad200001 }, // -46 + { 0xb35dbf821ae4f38b, 0xdda2802c8a800001 }, // -45 + { 0x8f7e32ce7bea5c6f, 0xe4820023a2000001 }, // -44 + { 0xe596b7b0c643c719, 0x6d9ccd05d0000001 }, // -43 + { 0xb7abc627050305ad, 0xf14a3d9e40000001 }, // -42 + { 0x92efd1b8d0cf37be, 0x5aa1cae500000001 }, // -41 + { 0xeb194f8e1ae525fd, 0x5dcfab0800000001 }, // -40 + { 0xbc143fa4e250eb31, 0x17d955a000000001 }, // -39 + { 0x96769950b50d88f4, 0x1314448000000001 }, // -38 + { 0xf0bdc21abb48db20, 0x1e86d40000000001 }, // -37 + { 0xc097ce7bc90715b3, 0x4b9f100000000001 }, // -36 + { 0x9a130b963a6c115c, 0x3c7f400000000001 }, // -35 + { 0xf684df56c3e01bc6, 0xc732000000000001 }, // -34 + { 0xc5371912364ce305, 0x6c28000000000001 }, // -33 + { 0x9dc5ada82b70b59d, 0xf020000000000001 }, // -32 + { 0xfc6f7c4045812296, 0x4d00000000000001 }, // -31 + { 0xc9f2c9cd04674ede, 0xa400000000000001 }, // -30 + { 0xa18f07d736b90be5, 0x5000000000000001 }, // -29 + { 0x813f3978f8940984, 0x4000000000000001 }, // -28 + { 0xcecb8f27f4200f3a, 0x0000000000000001 }, // -27 + { 0xa56fa5b99019a5c8, 0x0000000000000001 }, // -26 + { 0x84595161401484a0, 0x0000000000000001 }, // -25 + { 0xd3c21bcecceda100, 0x0000000000000001 }, // -24 + { 0xa968163f0a57b400, 0x0000000000000001 }, // -23 + { 0x878678326eac9000, 0x0000000000000001 }, // -22 + { 0xd8d726b7177a8000, 0x0000000000000001 }, // -21 + { 0xad78ebc5ac620000, 0x0000000000000001 }, // -20 + { 0x8ac7230489e80000, 0x0000000000000001 }, // -19 + { 0xde0b6b3a76400000, 0x0000000000000001 }, // -18 + { 0xb1a2bc2ec5000000, 0x0000000000000001 }, // -17 + { 0x8e1bc9bf04000000, 0x0000000000000001 }, // -16 + { 0xe35fa931a0000000, 0x0000000000000001 }, // -15 + { 0xb5e620f480000000, 0x0000000000000001 }, // -14 + { 0x9184e72a00000000, 0x0000000000000001 }, // -13 + { 0xe8d4a51000000000, 0x0000000000000001 }, // -12 + { 0xba43b74000000000, 0x0000000000000001 }, // -11 + { 0x9502f90000000000, 0x0000000000000001 }, // -10 + { 0xee6b280000000000, 0x0000000000000001 }, // -9 + { 0xbebc200000000000, 0x0000000000000001 }, // -8 + { 0x9896800000000000, 0x0000000000000001 }, // -7 + { 0xf424000000000000, 0x0000000000000001 }, // -6 + { 0xc350000000000000, 0x0000000000000001 }, // -5 + { 0x9c40000000000000, 0x0000000000000001 }, // -4 + { 0xfa00000000000000, 0x0000000000000001 }, // -3 + { 0xc800000000000000, 0x0000000000000001 }, // -2 + { 0xa000000000000000, 0x0000000000000001 }, // -1 + { 0x8000000000000000, 0x0000000000000001 }, // 0 + { 0xcccccccccccccccc, 0xcccccccccccccccd }, // 1 + { 0xa3d70a3d70a3d70a, 0x3d70a3d70a3d70a4 }, // 2 + { 0x83126e978d4fdf3b, 0x645a1cac083126ea }, // 3 + { 0xd1b71758e219652b, 0xd3c36113404ea4a9 }, // 4 + { 0xa7c5ac471b478423, 0x0fcf80dc33721d54 }, // 5 + { 0x8637bd05af6c69b5, 0xa63f9a49c2c1b110 }, // 6 + { 0xd6bf94d5e57a42bc, 0x3d32907604691b4d }, // 7 + { 0xabcc77118461cefc, 0xfdc20d2b36ba7c3e }, // 8 + { 0x89705f4136b4a597, 0x31680a88f8953031 }, // 9 + { 0xdbe6fecebdedd5be, 0xb573440e5a884d1c }, // 10 + { 0xafebff0bcb24aafe, 0xf78f69a51539d749 }, // 11 + { 0x8cbccc096f5088cb, 0xf93f87b7442e45d4 }, // 12 + { 0xe12e13424bb40e13, 0x2865a5f206b06fba }, // 13 + { 0xb424dc35095cd80f, 0x538484c19ef38c95 }, // 14 + { 0x901d7cf73ab0acd9, 0x0f9d37014bf60a11 }, // 15 + { 0xe69594bec44de15b, 0x4c2ebe687989a9b4 }, // 16 + { 0xb877aa3236a4b449, 0x09befeb9fad487c3 }, // 17 + { 0x9392ee8e921d5d07, 0x3aff322e62439fd0 }, // 18 + { 0xec1e4a7db69561a5, 0x2b31e9e3d06c32e6 }, // 19 + { 0xbce5086492111aea, 0x88f4bb1ca6bcf585 }, // 20 + { 0x971da05074da7bee, 0xd3f6fc16ebca5e04 }, // 21 + { 0xf1c90080baf72cb1, 0x5324c68b12dd6339 }, // 22 + { 0xc16d9a0095928a27, 0x75b7053c0f178294 }, // 23 + { 0x9abe14cd44753b52, 0xc4926a9672793543 }, // 24 + { 0xf79687aed3eec551, 0x3a83ddbd83f52205 }, // 25 + { 0xc612062576589dda, 0x95364afe032a819e }, // 26 + { 0x9e74d1b791e07e48, 0x775ea264cf55347e }, // 27 + { 0xfd87b5f28300ca0d, 0x8bca9d6e188853fd }, // 28 + { 0xcad2f7f5359a3b3e, 0x096ee45813a04331 }, // 29 + { 0xa2425ff75e14fc31, 0xa1258379a94d028e }, // 30 + { 0x81ceb32c4b43fcf4, 0x80eacf948770ced8 }, // 31 + { 0xcfb11ead453994ba, 0x67de18eda5814af3 }, // 32 + { 0xa6274bbdd0fadd61, 0xecb1ad8aeacdd58f }, // 33 + { 0x84ec3c97da624ab4, 0xbd5af13bef0b113f }, // 34 + { 0xd4ad2dbfc3d07787, 0x955e4ec64b44e865 }, // 35 + { 0xaa242499697392d2, 0xdde50bd1d5d0b9ea }, // 36 + { 0x881cea14545c7575, 0x7e50d64177da2e55 }, // 37 + { 0xd9c7dced53c72255, 0x96e7bd358c904a22 }, // 38 + { 0xae397d8aa96c1b77, 0xabec975e0a0d081b }, // 39 + { 0x8b61313bbabce2c6, 0x2323ac4b3b3da016 }, // 40 + { 0xdf01e85f912e37a3, 0x6b6c46dec52f6689 }, // 41 + { 0xb267ed1940f1c61c, 0x55f038b237591ed4 }, // 42 + { 0x8eb98a7a9a5b04e3, 0x77f3608e92adb243 }, // 43 + { 0xe45c10c42a2b3b05, 0x8cb89a7db77c506b }, // 44 + { 0xb6b00d69bb55c8d1, 0x3d607b97c5fd0d23 }, // 45 + { 0x9226712162ab070d, 0xcab3961304ca70e9 }, // 46 + { 0xe9d71b689dde71af, 0xaab8f01e6e10b4a7 }, // 47 + { 0xbb127c53b17ec159, 0x5560c018580d5d53 }, // 48 + { 0x95a8637627989aad, 0xdde7001379a44aa9 }, // 49 + { 0xef73d256a5c0f77c, 0x963e66858f6d4441 }, // 50 + { 0xbf8fdb78849a5f96, 0xde98520472bdd034 }, // 51 + { 0x993fe2c6d07b7fab, 0xe546a8038efe402a }, // 52 + { 0xf53304714d9265df, 0xd53dd99f4b3066a9 }, // 53 + { 0xc428d05aa4751e4c, 0xaa97e14c3c26b887 }, // 54 + { 0x9ced737bb6c4183d, 0x55464dd69685606c }, // 55 + { 0xfb158592be068d2e, 0xeed6e2f0f0d56713 }, // 56 + { 0xc8de047564d20a8b, 0xf245825a5a445276 }, // 57 + { 0xa0b19d2ab70e6ed6, 0x5b6aceaeae9d0ec5 }, // 58 + { 0x808e17555f3ebf11, 0xe2bbd88bbee40bd1 }, // 59 + { 0xcdb02555653131b6, 0x3792f412cb06794e }, // 60 + { 0xa48ceaaab75a8e2b, 0x5fa8c3423c052dd8 }, // 61 + { 0x83a3eeeef9153e89, 0x1953cf68300424ad }, // 62 + { 0xd29fe4b18e88640e, 0x8eec7f0d19a03aae }, // 63 + { 0xa87fea27a539e9a5, 0x3f2398d747b36225 }, // 64 + { 0x86ccbb52ea94baea, 0x98e947129fc2b4ea }, // 65 + { 0xd7adf884aa879177, 0x5b0ed81dcc6abb10 }, // 66 + { 0xac8b2d36eed2dac5, 0xe272467e3d222f40 }, // 67 + { 0x8a08f0f8bf0f156b, 0x1b8e9ecb641b5900 }, // 68 + { 0xdcdb1b2798182244, 0xf8e431456cf88e66 }, // 69 + { 0xb0af48ec79ace837, 0x2d835a9df0c6d852 }, // 70 + { 0x8d590723948a535f, 0x579c487e5a38ad0f }, // 71 + { 0xe2280b6c20dd5232, 0x25c6da63c38de1b1 }, // 72 + { 0xb4ecd5f01a4aa828, 0x1e38aeb6360b1af4 }, // 73 + { 0x90bd77f3483bb9b9, 0xb1c6f22b5e6f48c3 }, // 74 + { 0xe7958cb87392c2c2, 0xb60b1d1230b20e05 }, // 75 + { 0xb94470938fa89bce, 0xf808e40e8d5b3e6a }, // 76 + { 0x9436c0760c86e30b, 0xf9a0b6720aaf6522 }, // 77 + { 0xed246723473e3813, 0x290123e9aab23b69 }, // 78 + { 0xbdb6b8e905cb600f, 0x5400e987bbc1c921 }, // 79 + { 0x97c560ba6b0919a5, 0xdccd879fc967d41b }, // 80 + { 0xf2d56790ab41c2a2, 0xfae27299423fb9c4 }, // 81 + { 0xc24452da229b021b, 0xfbe85badce996169 }, // 82 + { 0x9b69dbe1b548ce7c, 0xc986afbe3ee11abb }, // 83 + { 0xf8a95fcf88747d94, 0x75a44c6397ce912b }, // 84 + { 0xc6ede63fa05d3143, 0x91503d1c79720dbc }, // 85 + { 0x9f24b832e6b0f436, 0x0dd9ca7d2df4d7ca }, // 86 + { 0xfea126b7d78186bc, 0xe2f610c84987bfa9 }, // 87 + { 0xcbb41ef979346bca, 0x4f2b40a03ad2ffba }, // 88 + { 0xa2f67f2dfa90563b, 0x728900802f0f32fb }, // 89 + { 0x825ecc24c873782f, 0x8ed400668c0c28c9 }, // 90 + { 0xd097ad07a71f26b2, 0x7e2000a41346a7a8 }, // 91 + { 0xa6dfbd9fb8e5b88e, 0xcb4ccd500f6bb953 }, // 92 + { 0x857fcae62d8493a5, 0x6f70a4400c562ddc }, // 93 + { 0xd59944a37c0752a2, 0x4be76d3346f04960 }, // 94 + { 0xaae103b5fcd2a881, 0xd652bdc29f26a11a }, // 95 + { 0x88b402f7fd75539b, 0x11dbcb0218ebb415 }, // 96 + { 0xdab99e59958885c4, 0xe95fab368e45ecee }, // 97 + { 0xaefae51477a06b03, 0xede622920b6b23f2 }, // 98 + { 0x8bfbea76c619ef36, 0x57eb4edb3c55b65b }, // 99 + { 0xdff9772470297ebd, 0x59787e2b93bc56f8 }, // 100 + { 0xb32df8e9f3546564, 0x47939822dc96abfa }, // 101 + { 0x8f57fa54c2a9eab6, 0x9fa946824a12232e }, // 102 + { 0xe55990879ddcaabd, 0xcc420a6a101d0516 }, // 103 + { 0xb77ada0617e3bbcb, 0x09ce6ebb40173745 }, // 104 + { 0x92c8ae6b464fc96f, 0x3b0b8bc90012929e }, // 105 + { 0xeadab0aba3b2dbe5, 0x2b45ac74ccea842f }, // 106 + { 0xbbe226efb628afea, 0x890489f70a55368c }, // 107 + { 0x964e858c91ba2655, 0x3a6a07f8d510f870 }, // 108 + { 0xf07da27a82c37088, 0x5d767327bb4e5a4d }, // 109 + { 0xc06481fb9bcf8d39, 0xe45ec2862f71e1d7 }, // 110 + { 0x99ea0196163fa42e, 0x504bced1bf8e4e46 }, // 111 + { 0xf64335bcf065d37d, 0x4d4617b5ff4a16d6 }, // 112 + { 0xc5029163f384a931, 0x0a9e795e65d4df12 }, // 113 + { 0x9d9ba7832936edc0, 0xd54b944b84aa4c0e }, // 114 + { 0xfc2c3f3841f17c67, 0xbbac2078d443ace3 }, // 115 + { 0xc9bcff6034c13052, 0xfc89b393dd02f0b6 }, // 116 + { 0xa163ff802a3426a8, 0xca07c2dcb0cf26f8 }, // 117 + { 0x811ccc668829b887, 0x0806357d5a3f5260 }, // 118 + { 0xce947a3da6a9273e, 0x733d226229feea33 }, // 119 + { 0xa54394fe1eedb8fe, 0xc2974eb4ee658829 }, // 120 + { 0x843610cb4bf160cb, 0xcedf722a585139bb }, // 121 + { 0xd389b47879823479, 0x4aff1d108d4ec2c4 }, // 122 + { 0xa93af6c6c79b5d2d, 0xd598e40d3dd89bd0 }, // 123 + { 0x87625f056c7c4a8b, 0x11471cd764ad4973 }, // 124 + { 0xd89d64d57a607744, 0xe871c7bf077ba8b8 }, // 125 + { 0xad4ab7112eb3929d, 0x86c16c98d2c953c7 }, // 126 + { 0x8aa22c0dbef60ee4, 0x6bcdf07a423aa96c }, // 127 + { 0xddd0467c64bce4a0, 0xac7cb3f6d05ddbdf }, // 128 + { 0xb1736b96b6fd83b3, 0xbd308ff8a6b17cb3 }, // 129 + { 0x8df5efabc5979c8f, 0xca8d3ffa1ef463c2 }, // 130 + { 0xe3231912d5bf60e6, 0x10e1fff697ed6c6a }, // 131 + { 0xb5b5ada8aaff80b8, 0x0d819992132456bb }, // 132 + { 0x915e2486ef32cd60, 0x0ace1474dc1d122f }, // 133 + { 0xe896a0d7e51e1566, 0x77b020baf9c81d18 }, // 134 + { 0xba121a4650e4ddeb, 0x92f34d62616ce414 }, // 135 + { 0x94db483840b717ef, 0xa8c2a44eb4571cdd }, // 136 + { 0xee2ba6c0678b597f, 0x746aa07ded582e2d }, // 137 + { 0xbe89523386091465, 0xf6bbb397f1135824 }, // 138 + { 0x986ddb5c6b3a76b7, 0xf89629465a75e01d }, // 139 + { 0xf3e2f893dec3f126, 0x5a89dba3c3efccfb }, // 140 + { 0xc31bfa0fe5698db8, 0x486e494fcff30a63 }, // 141 + { 0x9c1661a651213e2d, 0x06bea10ca65c084f }, // 142 + { 0xf9bd690a1b68637b, 0x3dfdce7aa3c673b1 }, // 143 + { 0xc7caba6e7c5382c8, 0xfe64a52ee96b8fc1 }, // 144 + { 0x9fd561f1fd0f9bd3, 0xfeb6ea8bedefa634 }, // 145 + { 0xffbbcfe994e5c61f, 0xfdf17746497f7053 }, // 146 + { 0xcc963fee10b7d1b3, 0x318df905079926a9 }, // 147 + { 0xa3ab66580d5fdaf5, 0xc13e60d0d2e0ebbb }, // 148 + { 0x82ef85133de648c4, 0x9a984d73dbe722fc }, // 149 + { 0xd17f3b51fca3a7a0, 0xf75a15862ca504c6 }, // 150 + { 0xa798fc4196e952e7, 0x2c48113823b73705 }, // 151 + { 0x8613fd0145877585, 0xbd06742ce95f5f37 }, // 152 + { 0xd686619ba27255a2, 0xc80a537b0efefebe }, // 153 + { 0xab9eb47c81f5114f, 0x066ea92f3f326565 }, // 154 + { 0x894bc396ce5da772, 0x6b8bba8c328eb784 }, // 155 + { 0xdbac6c247d62a583, 0xdf45f746b74abf3a }, // 156 + { 0xafbd2350644eeacf, 0xe5d1929ef90898fb }, // 157 + { 0x8c974f7383725573, 0x1e414218c73a13fc }, // 158 + { 0xe0f218b8d25088b8, 0x306869c13ec3532d }, // 159 + { 0xb3f4e093db73a093, 0x59ed216765690f57 }, // 160 + { 0x8ff71a0fe2c2e6dc, 0x47f0e785eaba72ac }, // 161 + { 0xe65829b3046b0afa, 0x0cb4a5a3112a5113 }, // 162 + { 0xb84687c269ef3bfb, 0x3d5d514f40eea743 }, // 163 + { 0x936b9fcebb25c995, 0xcab10dd900beec35 }, // 164 + { 0xebdf661791d60f56, 0x111b495b3464ad22 }, // 165 + { 0xbcb2b812db11a5de, 0x7415d448f6b6f0e8 }, // 166 + { 0x96f5600f15a7b7e5, 0x29ab103a5ef8c0ba }, // 167 + { 0xf18899b1bc3f8ca1, 0xdc44e6c3cb279ac2 }, // 168 + { 0xc13a148e3032d6e7, 0xe36a52363c1faf02 }, // 169 + { 0x9a94dd3e8cf578b9, 0x82bb74f8301958cf }, // 170 + { 0xf7549530e188c128, 0xd12bee59e68ef47d }, // 171 + { 0xc5dd44271ad3cdba, 0x40eff1e1853f29fe }, // 172 + { 0x9e4a9cec15763e2e, 0x9a598e4e043287ff }, // 173 + { 0xfd442e4688bd304a, 0x908f4a166d1da664 }, // 174 + { 0xca9cf1d206fdc03b, 0xa6d90811f0e4851d }, // 175 + { 0xa21727db38cb002f, 0xb8ada00e5a506a7d }, // 176 + { 0x81ac1fe293d599bf, 0xc6f14cd848405531 }, // 177 + { 0xcf79cc9db955c2cc, 0x7182148d4066eeb5 }, // 178 + { 0xa5fb0a17c777cf09, 0xf468107100525891 }, // 179 + { 0x84c8d4dfd2c63f3b, 0x29ecd9f40041e074 }, // 180 + { 0xd47487cc8470652b, 0x7647c32000696720 }, // 181 + { 0xa9f6d30a038d1dbc, 0x5e9fcf4ccd211f4d }, // 182 + { 0x87f8a8d4cfa417c9, 0xe54ca5d70a80e5d7 }, // 183 + { 0xd98ddaee19068c76, 0x3badd624dd9b0958 }, // 184 + { 0xae0b158b4738705e, 0x9624ab50b148d446 }, // 185 + { 0x8b3c113c38f9f37e, 0xde83bc408dd3dd05 }, // 186 + { 0xdec681f9f4c31f31, 0x6405fa00e2ec94d5 }, // 187 + { 0xb23867fb2a35b28d, 0xe99e619a4f23aa44 }, // 188 + { 0x8e938662882af53e, 0x547eb47b7282ee9d }, // 189 + { 0xe41f3d6a7377eeca, 0x20caba5f1d9e4a94 }, // 190 + { 0xb67f6455292cbf08, 0x1a3bc84c17b1d543 }, // 191 + { 0x91ff83775423cc06, 0x7b6306a34627ddd0 }, // 192 + { 0xe998d258869facd7, 0x2bd1a438703fc94c }, // 193 + { 0xbae0a846d2195712, 0x8974836059cca10a }, // 194 + { 0x9580869f0e7aac0e, 0xd45d35e6ae3d4da1 }, // 195 + { 0xef340a98172aace4, 0x86fb897116c87c35 }, // 196 + { 0xbf5cd54678eef0b6, 0xd262d45a78a0635e }, // 197 + { 0x991711052d8bf3c5, 0x751bdd152d4d1c4b }, // 198 + { 0xf4f1b4d515acb93b, 0xee92fb5515482d45 }, // 199 + { 0xc3f490aa77bd60fc, 0xbedbfc4411068a9d }, // 200 + { 0x9cc3a6eec6311a63, 0xcbe3303674053bb1 }, // 201 + { 0xfad2a4b13d1b5d6c, 0x796b805720085f82 }, // 202 + { 0xc8a883c0fdaf7df0, 0x6122cd128006b2ce }, // 203 + { 0xa086cfcd97bf97f3, 0x80e8a40eccd228a5 }, // 204 + { 0x806bd9714632dff6, 0x00ba1cd8a3db53b7 }, // 205 + { 0xcd795be870516656, 0x67902e276c921f8c }, // 206 + { 0xa46116538d0deb78, 0x52d9be85f074e609 }, // 207 + { 0x8380dea93da4bc60, 0x4247cb9e59f71e6e }, // 208 + { 0xd267caa862a12d66, 0xd072df63c324fd7c }, // 209 + { 0xa8530886b54dbdeb, 0xd9f57f830283fdfd }, // 210 + { 0x86a8d39ef77164bc, 0xae5dff9c02033198 }, // 211 + { 0xd77485cb25823ac7, 0x7d633293366b828c }, // 212 + { 0xac5d37d5b79b6239, 0x311c2875c522ced6 }, // 213 + { 0x89e42caaf9491b60, 0xf41686c49db57245 }, // 214 + { 0xdca04777f541c567, 0xecf0d7a0fc5583a1 }, // 215 + { 0xb080392cc4349dec, 0xbd8d794d96aacfb4 }, // 216 + { 0x8d3360f09cf6e4bd, 0x64712dd7abbbd95d }, // 217 + { 0xe1ebce4dc7f16dfb, 0xd3e8495912c62895 }, // 218 + { 0xb4bca50b065abe63, 0x0fed077a756b53aa }, // 219 + { 0x9096ea6f3848984f, 0x3ff0d2c85def7622 }, // 220 + { 0xe757dd7ec07426e5, 0x331aeada2fe589d0 }, // 221 + { 0xb913179899f68584, 0x28e2557b59846e40 }, // 222 + { 0x940f4613ae5ed136, 0x871b7795e136be9a }, // 223 + { 0xece53cec4a314ebd, 0xa4f8bf5635246429 }, // 224 + { 0xbd8430bd08277231, 0x50c6ff782a838354 }, // 225 + { 0x979cf3ca6cec5b5a, 0xa705992ceecf9c43 }, // 226 + { 0xf294b943e17a2bc4, 0x3e6f5b7b17b2939e }, // 227 + { 0xc21094364dfb5636, 0x985915fc12f542e5 }, // 228 + { 0x9b407691d7fc44f8, 0x79e0de63425dcf1e }, // 229 + { 0xf867241c8cc6d4c0, 0xc30163d203c94b63 }, // 230 + { 0xc6b8e9b0709f109a, 0x359ab6419ca1091c }, // 231 + { 0x9efa548d26e5a6e1, 0xc47bc5014a1a6db0 }, // 232 + { 0xfe5d54150b090b02, 0xd3f93b35435d7c4d }, // 233 + { 0xcb7ddcdda26da268, 0xa9942f5dcf7dfd0a }, // 234 + { 0xa2cb1717b52481ed, 0x54768c4b0c64ca6f }, // 235 + { 0x823c12795db6ce57, 0x76c53d08d6b70859 }, // 236 + { 0xd0601d8efc57b08b, 0xf13b94daf124da27 }, // 237 + { 0xa6b34ad8c9dfc06f, 0xf42faa48c0ea481f }, // 238 + { 0x855c3be0a17fcd26, 0x5cf2eea09a550680 }, // 239 + { 0xd5605fcdcf32e1d6, 0xfb1e4a9a90880a65 }, // 240 + { 0xaab37fd7d8f58178, 0xc8e5087ba6d33b84 }, // 241 + { 0x888f99797a5e012d, 0x6d8406c952429604 }, // 242 + { 0xda7f5bf590966848, 0xaf39a475506a899f }, // 243 + { 0xaecc49914078536d, 0x58fae9f773886e19 }, // 244 + { 0x8bd6a141006042bd, 0xe0c8bb2c5c6d24e1 }, // 245 + { 0xdfbdcece67006ac9, 0x67a791e093e1d49b }, // 246 + { 0xb2fe3f0b8599ef07, 0x861fa7e6dcb4aa16 }, // 247 + { 0x8f31cc0937ae58d2, 0xd1b2ecb8b0908811 }, // 248 + { 0xe51c79a85916f484, 0x82b7e12780e7401b }, // 249 + { 0xb749faed14125d36, 0xcef980ec671f667c }, // 250 + { 0x92a1958a7675175f, 0x0bfacd89ec191eca }, // 251 + { 0xea9c227723ee8bcb, 0x465e15a979c1cadd }, // 252 + { 0xbbb01b9283253ca2, 0x9eb1aaedfb016f17 }, // 253 + { 0x96267c7535b763b5, 0x4bc1558b2f3458df }, // 254 + { 0xf03d93eebc589f88, 0x793555ab7eba27cb }, // 255 + { 0xc0314325637a1939, 0xfa911155fefb5309 }, // 256 + { 0x99c102844f94e0fb, 0x2eda7444cbfc426e }, // 257 + { 0xf6019da07f549b2b, 0x7e2a53a146606a49 }, // 258 + { 0xc4ce17b399107c22, 0xcb550fb4384d21d4 }, // 259 + { 0x9d71ac8fada6c9b5, 0x6f773fc3603db4aa }, // 260 + { 0xfbe9141915d7a922, 0x4bf1ff9f0062baa9 }, // 261 + { 0xc987434744ac874e, 0xa327ffb266b56221 }, // 262 + { 0xa139029f6a239f72, 0x1c1fffc1ebc44e81 }, // 263 + { 0x80fa687f881c7f8e, 0x7ce66634bc9d0b9a }, // 264 + { 0xce5d73ff402d98e3, 0xfb0a3d212dc81290 }, // 265 + { 0xa5178fff668ae0b6, 0x626e974dbe39a873 }, // 266 + { 0x8412d9991ed58091, 0xe858790afe9486c3 }, // 267 + { 0xd3515c2831559a83, 0x0d5a5b44ca873e04 }, // 268 + { 0xa90de3535aaae202, 0x711515d0a205cb37 }, // 269 + { 0x873e4f75e2224e68, 0x5a7744a6e804a292 }, // 270 + { 0xd863b256369d4a40, 0x90bed43e40076a83 }, // 271 + { 0xad1c8eab5ee43b66, 0xda3243650005eed0 }, // 272 + { 0x8a7d3eef7f1cfc52, 0x482835ea666b2573 }, // 273 + { 0xdd95317f31c7fa1d, 0x40405643d711d584 }, // 274 + { 0xb1442798f49ffb4a, 0x99cd11cfdf41779d }, // 275 + { 0x8dd01fad907ffc3b, 0xae3da7d97f6792e4 }, // 276 + { 0xe2e69915b3fff9f9, 0x16c90c8f323f516d }, // 277 + { 0xb58547448ffffb2d, 0xabd40a0c2832a78b }, // 278 + { 0x91376c36d99995be, 0x23100809b9c21fa2 }, // 279 + { 0xe858ad248f5c22c9, 0xd1b3400f8f9cff69 }, // 280 + { 0xb9e08a83a5e34f07, 0xdaf5ccd93fb0cc54 }, // 281 + { 0x94b3a202eb1c3f39, 0x7bf7d71432f3d6aa }, // 282 + { 0xedec366b11c6cb8f, 0x2cbfbe86b7ec8aa9 }, // 283 + { 0xbe5691ef416bd60c, 0x23cc986bc656d554 }, // 284 + { 0x9845418c345644d6, 0x830a13896b78aaaa }, // 285 + { 0xf3a20279ed56d48a, 0x6b43527578c11110 }, // 286 + { 0xc2e801fb244576d5, 0x229c41f793cda740 }, // 287 + { 0x9becce62836ac577, 0x4ee367f9430aec33 }, // 288 + { 0xf97ae3d0d2446f25, 0x4b0573286b44ad1e }, // 289 + { 0xc795830d75038c1d, 0xd59df5b9ef6a2418 }, // 290 + { 0x9faacf3df73609b1, 0x77b191618c54e9ad }, // 291 + { 0xff77b1fcbebcdc4f, 0x25e8e89c13bb0f7b }, // 292 + }; +}; + +template +struct minverse; + +// Taken from: https://github.com/cassioneri/teju_jagua. +template <> +struct minverse { + static struct { + const uint32_t multiplier; + const uint32_t bound; + } constexpr data[] = + { + { 0x00000001, 0xffffffff }, + { 0xcccccccd, 0x33333333 }, + { 0xc28f5c29, 0x0a3d70a3 }, + { 0x26e978d5, 0x020c49ba }, + { 0x3afb7e91, 0x0068db8b }, + { 0x0bcbe61d, 0x0014f8b5 }, + { 0x68c26139, 0x000431bd }, + { 0xae8d46a5, 0x0000d6bf }, + { 0x22e90e21, 0x00002af3 }, + { 0x3a2e9c6d, 0x00000897 }, + { 0x3ed61f49, 0x000001b7 }, + { 0x0c913975, 0x00000057 }, + { 0xcf503eb1, 0x00000011 }, + { 0xf6433fbd, 0x00000003 }, + }; +}; + +} // namespace + +#endif // _GLIBCXX_FLOATING_TO_CHARS_TABLES_H