26002766134

Committed 17 May 2026 09:00PM UTC coverage: 84.106% (+0.01%) from 84.094%

Build # 26002766134

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Pull #861

github

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web-flow

Commit Message

Merge b4625f823 into 088ccd080

Pull Request Pull Request #861: fix(einteger): correct Knuth Algorithm D step D4 borrow propagation

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85.1

/include/sw/universal/number/einteger/einteger_impl.hpp

#pragma once
// einteger_impl.hpp: implementation of an adaptive precision binary integer
//
// Copyright (C) 2017-2023 Stillwater Supercomputing, Inc.
//
// This file is part of the universal numbers project, which is released under an MIT Open Source license.
#include <string>
#include <sstream>
#include <iostream>
#include <iomanip>
#include <regex>
#include <map>
#include <vector>
#include <limits>
#include <type_traits>

#include <universal/number/einteger/exceptions.hpp>
#include <universal/number/einteger/einteger_fwd.hpp>

// supporting types and functions
#include <universal/native/ieee754.hpp>

namespace sw { namespace universal {

// einteger is an adaptive precision integer type
template<typename BlockType = std::uint32_t>
class einteger {
public:
        using bt = BlockType;
        static constexpr unsigned bitsInBlock = sizeof(BlockType) * 8;
        static constexpr bt       ALL_ONES = bt(0xFFFF'FFFF'FFFF'FFFFull); // block type specific all 1's value
        static constexpr uint64_t BASE = uint64_t(ALL_ONES) + 1ull;
        static_assert(bitsInBlock <= 32, "BlockType must be one of [uint8_t, uint16_t, uint32_t]");

        // Partial-constexpr surface (issue #748): einteger carries a
        // std::vector<BlockType> _block member, so any non-empty digit
        // storage escapes constant evaluation under C++20's transient-
        // allocation rules.  The default ctor already initializes _block
        // to an empty vector (no push_back), so it is constexpr-friendly
        // without is_constant_evaluated() dispatch -- iszero() recognizes
        // the empty state via the size() == 0 branch.  Sign-only and
        // state-only selectors / modifiers are constexpr-clean; free
        // comparison operators (type vs type) are pure reads of the digit
        // vector and thus also constexpr.  All digit-mutating paths
        // (setbits, setblock, parse, native-type ctors / operator=,
        // arithmetic operators, conversion-out) remain non-constexpr until
        // C++23 P2738 relaxes the transient-allocation rule.
        constexpr einteger() noexcept : _sign(false), _block{} { }

        constexpr einteger(const einteger&) = default;
        constexpr einteger(einteger&&) = default;

        constexpr einteger& operator=(const einteger&) = default;
        constexpr einteger& operator=(einteger&&) = default;

        // initializers for native types
        einteger(short initial_value)              { *this = initial_value; }
        einteger(int initial_value)                { *this = initial_value; }
        einteger(long initial_value)               { *this = initial_value; }
        einteger(long long initial_value)          { *this = initial_value; }
        einteger(unsigned int initial_value)       { *this = initial_value; }
        einteger(unsigned long initial_value)      { *this = initial_value; }
        einteger(unsigned long long initial_value) { *this = initial_value; }
        einteger(float initial_value)              { *this = initial_value; }
        einteger(double initial_value)             { *this = initial_value; }

        // assignment operators for native types
        einteger& operator=(int rhs)                noexcept { return convert_signed(rhs); }
        einteger& operator=(long rhs)               noexcept { return convert_signed(rhs); }
        einteger& operator=(long long rhs)          noexcept { return convert_signed(rhs); }
        einteger& operator=(unsigned int rhs)       noexcept { return convert_unsigned(rhs); }
        einteger& operator=(unsigned long rhs)      noexcept { return convert_unsigned(rhs); }
        einteger& operator=(unsigned long long rhs) noexcept { return convert_unsigned(rhs); }
        einteger& operator=(float rhs)              noexcept { return convert_ieee754(rhs); }
        einteger& operator=(double rhs)             noexcept { return convert_ieee754(rhs); }

        // conversion operators
        explicit operator int() const noexcept         { return convert_to_native_integer<int>(); }
        explicit operator long() const noexcept        { return convert_to_native_integer<long>(); }
        explicit operator long long() const noexcept   { return convert_to_native_integer<long long>(); }
        explicit operator float() const noexcept       { return convert_to_native_ieee<float>(); }
        explicit operator double() const noexcept      { return convert_to_native_ieee<double>(); }

#if LONG_DOUBLE_SUPPORT
        einteger(long double initial_value) { *this = initial_value; }
        einteger& operator=(long double rhs)        noexcept { return convert_ieee754(rhs); }
        explicit operator long double() const noexcept { return convert_to_native_ieee<long double>(); }
#endif

        // logic shift operators
        einteger& operator<<=(int shift) {
                if (shift == 0) return *this;
                if (shift < 0) return operator>>=(-shift);

                // by default extend the limbs by 1: TODO: can this be improved?
                _block.push_back(0);
                size_t MSU = _block.size() - 1;
                if (shift >= static_cast<int>(bitsInBlock)) {
                        int blockShift = shift / static_cast<int>(bitsInBlock);
                        if (blockShift > 0) _block.resize(_block.size() + blockShift, 0ul);
                        MSU = _block.size() - 1;
                        for (int i = static_cast<int>(MSU); i >= blockShift; --i) {
                                _block[static_cast<size_t>(i)] = _block[static_cast<size_t>(i) - static_cast<size_t>(blockShift)];
                        }
                        for (int i = blockShift - 1; i >= 0; --i) {
                                _block[static_cast<size_t>(i)] = BlockType(0);
                        }
                        // adjust the shift
                        shift -= static_cast<int>(blockShift * bitsInBlock);
                        if (shift == 0) return *this;
                }
                if (MSU > 0) {
                        // construct the mask for the upper bits in the block that needs to move to the higher word
                        BlockType mask = static_cast<BlockType>(0xFFFF'FFFFul << (bitsInBlock - shift));
                        for (size_t i = MSU; i > 0; --i) {
                                _block[static_cast<size_t>(i)] <<= shift;
                                // mix in the bits from the right
                                BlockType bits = BlockType(mask & _block[i - 1]);
                                _block[static_cast<size_t>(i)] |= (bits >> (bitsInBlock - shift));
                        }
                        _block[0] <<= shift;
                }
                else {
                        _block[0] <<= shift;
                }
                remove_leading_zeros();
                return *this;
        }
        einteger& operator>>=(int shift) {
                if (shift == 0) return *this;
                if (shift < 0) return operator>>=(-shift);
                if (shift > static_cast<int>(nbits())) {
                        setzero();
                        return *this;
                }
                size_t MSU = _block.size() - 1;
                size_t blockShift = 0;
                if (shift >= static_cast<int>(bitsInBlock)) {
                        blockShift = shift / bitsInBlock;
                        if (MSU >= blockShift) {
                                // shift by blocks
                                for (size_t i = 0; i <= MSU - blockShift; ++i) {
                                        _block[i] = _block[i + blockShift];
                                        _block[i + blockShift] = 0; // null the upper block
                                }
                        }
                        // adjust the shift
                        shift -= static_cast<int>(blockShift * bitsInBlock);
                        if (shift == 0) {
                                remove_leading_zeros();
                                return *this;
                        }
                }
                if (MSU > 0) {
                        BlockType mask = static_cast<BlockType>(0xFFFF'FFFFul);
                        mask >>= (bitsInBlock - shift); // this is a mask for the lower bits in the block that need to move to the lower word
                        for (size_t i = 0; i < MSU; ++i) {
                                _block[i] >>= shift;
                                // mix in the bits from the left
                                BlockType bits = BlockType(mask & _block[i + 1]);
                                _block[i] |= (bits << (bitsInBlock - shift));
                        }
                        _block[MSU] >>= shift;
                }
                else {
                        _block[0] >>= shift;
                }
                remove_leading_zeros();
                return *this;
        }
        
        // unary arithmetic operators
        einteger operator-() const {
                einteger negated(*this);
                negated.setsign(!_sign);
                return negated;
        }
        einteger operator++(int) {
                einteger tmp(*this);
                operator++();
                return tmp;
        }
        einteger& operator++() {
                *this += 1;
                return *this;
        }
        einteger operator--(int) {
                einteger tmp(*this);
                operator--();
                return tmp;
        }
        einteger& operator--() {
                *this -= 1;
                return *this;
        }

        // binary arithmetic operators
        einteger& operator+=(const einteger& rhs) {
                if (sign() != rhs.sign()) {
                        if (sign()) {
                                einteger negated(*this);
                                negated.setsign(false);
                                *this = rhs - negated;
                                return *this;
                        }
                        else {
                                einteger negated(rhs);
                                negated.setsign(false);
                                *this -= negated;
                                return *this;
                        }
                }
                auto lhsSize = _block.size();
                auto rhsSize = rhs._block.size();
                if (lhsSize < rhsSize) _block.resize(rhsSize, 0);

                std::uint64_t carry{ 0 };
                typename std::vector<BlockType>::iterator li = _block.begin();
                typename std::vector<BlockType>::const_iterator ri = rhs._block.begin();
                while (li != _block.end()) {
                        if (ri != rhs._block.end()) {
                                carry += static_cast<std::uint64_t>(*li) + static_cast<std::uint64_t>(*ri);
                                ++ri;
                        }
                        else {
                                carry += static_cast<std::uint64_t>(*li);
                        }
                        *li = static_cast<BlockType>(carry);
                        carry >>= bitsInBlock;
                        ++li; 
                }
                if (carry == 0x1ull) {
                        _block.push_back(static_cast<BlockType>(carry));
                }
                return *this;
        }
        einteger& operator+=(long long rhs) {
                return *this += einteger(rhs);
        }
        einteger& operator-=(const einteger& rhs) {
                if (rhs.sign()) {
                        einteger negated(rhs);
                        negated.setsign(false);
                        return *this += negated;
                }
                auto lhsSize = _block.size();
                if (lhsSize == 0) {
                        *this = -rhs;
                        return *this;
                }
                auto rhsSize = rhs._block.size();
                int magnitude = compare_magnitude(*this, rhs); // if -1 result is going to be negative

                auto overlap = (lhsSize < rhsSize ? lhsSize : rhsSize);
                auto extent = (lhsSize < rhsSize ? rhsSize : lhsSize);

                // prep storage
                if (lhsSize < rhsSize) _block.resize(rhsSize, 0);

                typename std::vector<BlockType>::const_iterator aIter, bIter;
                if (magnitude == 1) {
                        aIter = _block.begin();
                        bIter = rhs._block.begin();
                }
                else {
                        aIter = rhs._block.begin();
                        bIter = _block.begin();
                }
                std::uint64_t borrow{ 0 };
                unsigned i{ 0 };
                while (i < overlap) {
                        borrow = static_cast<std::uint64_t>(*aIter) - static_cast<std::uint64_t>(*bIter) - borrow;
                        _block[i] = static_cast<BlockType>(borrow);
                        borrow = (borrow >> bitsInBlock) & 0x1u;
                        ++i; ++aIter; ++bIter;
                }
                while ((i < extent)) {
                        borrow = static_cast<BlockType>(*aIter) - borrow;
                        _block[i] = static_cast<BlockType>(borrow);
                        borrow = (borrow >> bitsInBlock) & 0x1u;
                        ++i; ++aIter;
                }
                remove_leading_zeros();
                setsign(magnitude == -1);
                return *this;
        }
        einteger& operator-=(long long rhs) {
                return *this -= einteger(rhs);
        }
        einteger& operator*=(const einteger& rhs) {
                if (iszero() || rhs.iszero()) {
                        clear();
                        return *this;
                }
                einteger base(*this);
                bool ls = sign();
                unsigned ll = limbs();
                bool rs = rhs.sign();
                unsigned rl = rhs.limbs();

                clear();
                std::uint64_t segment(0);
                for (unsigned i = 0; i < ll; ++i) {
                        for (unsigned j = 0; j < rl; ++j) {
                                segment += static_cast<std::uint64_t>(base.block(i)) * static_cast<std::uint64_t>(rhs.block(j));
                                segment += block(i + j);
                                setblock(i + j, static_cast<bt>(segment));
                                segment >>= bitsInBlock;
                        }
                }
                if (segment != 0) setblock(ll + rl - 1, static_cast<bt>(segment));
                setsign(ls ^ rs);
                return *this;
        }
        einteger& operator*=(long long rhs) {
                return *this *= einteger(rhs);
        }
        einteger& operator/=(const einteger& rhs) {
                einteger q, r;
                q.reduce(*this, rhs, r);
                *this = q;
                return *this;
        }
        einteger& operator/=(long long rhs) {
                return *this /= einteger(rhs);
        }
        einteger& operator%=(const einteger& rhs) {
                einteger q, a(*this), r;
                q.reduce(a, rhs, r);
                *this = r;
                return *this;
        }
        einteger& operator%=(long long rhs) {
                return *this %= einteger(rhs);
        }
        
        // reduce returns the ratio and remainder of a and b in *this and r
        void reduce(const einteger& a, const einteger& b, einteger& r) {
                if (b.iszero()) {
#if EINTEGER_THROW_ARITHMETIC_EXCEPTION
                        throw einteger_divide_by_zero{};
#else
                        std::cerr << "einteger_divide_by_zero\n";
                        return;
#endif // EINTEGER_THROW_ARITHMETIC_EXCEPTION
                }
                clear();
                r.clear();
                if (a.iszero()) return;

                size_t aBlocks = a.limbs();
                size_t bBlocks = b.limbs();
                if (aBlocks == 1 && aBlocks == bBlocks) { // completely reduce this to native div and rem
                        std::uint64_t a0 = a._block[0];
                        std::uint64_t b0 = b._block[0];
                        *this = static_cast<BlockType>(a0 / b0);
                        r = static_cast<BlockType>(a0 % b0);
                }
                        else {
                                // filter out the easy stuff
                                if (a < b) { r = a; clear(); return; }
                                if (a == b) { *this = 1; r.clear(); return; }

                        // determine first non-zero limbs
                        unsigned m{ 0 }, n{ 0 };
                        for (size_t i = aBlocks; i > 0; --i) {
                                if (a._block[i - 1] != 0) {
                                        m = static_cast<unsigned>(i);
                                        break;
                                }
                        }
                        for (size_t i = bBlocks; i > 0; --i) {
                                if (b._block[i - 1] != 0) {
                                        n = static_cast<unsigned>(i);
                                        break;
                                }
                        }

                        // single limb divisor
                        if (n == 1) {
                                _block.resize(m);
                                std::uint64_t remainder{ 0 };
                                auto divisor = b.block(0);
                                for (unsigned j = m; j > 0; --j) {
                                        std::uint64_t dividend = remainder * BASE + static_cast<std::uint64_t>(a.block(j - 1));
                                        std::uint64_t limbQuotient = dividend / divisor;
                                        _block[j - 1] = static_cast<BlockType>(limbQuotient);
                                        remainder = dividend - limbQuotient * divisor;
                                }
                                remove_leading_zeros();
                                r.setblock(0, static_cast<BlockType>(remainder));
                                return;
                        }

                        // Knuth's algorithm calculates a normalization factor d
                        // that perfectly aligns b so that b0 >= floor(BASE/2),
                        // a requirement for the relationship: (qHat - 2) <= q <= qHat

                        int shift = nlz(b.block(n - 1));
                        einteger normalized_a;
                        normalized_a.setblock(m, static_cast<BlockType>((a.block(m - 1) >> (bitsInBlock - shift))));
                        for (unsigned i = m - 1; i > 0; --i) {
                                normalized_a.setblock(i, static_cast<BlockType>((a.block(i) << shift) | (a.block(i - 1) >> (bitsInBlock - shift))));
                        }
                        normalized_a.setblock(0, static_cast<BlockType>(a.block(0) << shift));
                        // normalize b
                        einteger normalized_b;
                        unsigned n_minus_1 = n - 1;
                        for (unsigned i = n_minus_1; i > 0; --i) {
                                normalized_b.setblock(i, static_cast<BlockType>((b.block(i) << shift) | (b.block(i - 1) >> (bitsInBlock - shift))));
                        }
                        normalized_b.setblock(0, static_cast<BlockType>(b.block(0) << shift));

                        //std::cout << "normalized a : " << normalized_a.showLimbs() << " : " << normalized_a.showLimbValues() << '\n';
                        //std::cout << "normalized b :             " << normalized_b.showLimbs() << " : " << normalized_b.showLimbValues() << '\n';

                        // divide by limb
                        std::uint64_t divisor = normalized_b._block[n - 1];
                        std::uint64_t v_nminus2 = normalized_b._block[n - 2]; // n > 1 at this point
                        for (int j = static_cast<int>(m - n); j >= 0; --j) {
                                std::uint64_t dividend = normalized_a.block(j + n) * BASE + normalized_a.block(j + n - 1);
                                std::uint64_t qhat = dividend / divisor;
                                std::uint64_t rhat = dividend - qhat * divisor;

                                while (qhat >= BASE || qhat * v_nminus2 > BASE * rhat + normalized_a.block(j + n - 2)) {
                                        --qhat;
                                        rhat += divisor;
                                        if (rhat < BASE) continue;
                                }
                                // Knuth Algorithm D, step D4 (multi-precision subtraction
                                // with borrow propagation). `diff` MUST be signed so that
                                // the arithmetic right shift `diff >> bitsInBlock` yields
                                // the -1/0 underflow indicator. With uint64_t the shift
                                // instead returns all-ones, and `p_hi - (diff >> shift)`
                                // wraps to a huge value that corrupts the next iteration
                                // and ultimately produces a quotient that is too large
                                // by exactly one limb (issue #842). The low-bits mask
                                // must also track bitsInBlock so the algorithm works for
                                // uint8_t and uint16_t blocks in addition to uint32_t.
                                constexpr std::uint64_t LOW_MASK = BASE - 1u;
                                std::int64_t borrow{ 0 };
                                std::int64_t diff{ 0 };
                                for (unsigned i = 0; i < n; ++i) {
                                        std::uint64_t p     = qhat * normalized_b.block(i);
                                        std::uint64_t p_lo  = p & LOW_MASK;
                                        std::uint64_t p_hi  = p >> bitsInBlock;
                                        diff = static_cast<std::int64_t>(normalized_a.block(i + j))
                                             - static_cast<std::int64_t>(p_lo)
                                             - borrow;
                                        normalized_a.setblock(i + j, static_cast<BlockType>(diff));
                                        // borrow_out = p_hi + (1 if underflow else 0). The
                                        // signed shift `diff >> bitsInBlock` yields -1 on
                                        // underflow and 0 otherwise, so subtracting it adds
                                        // 1 in the underflow case.
                                        borrow = static_cast<std::int64_t>(p_hi) - (diff >> bitsInBlock);
                                }
                                std::int64_t signedBorrow = static_cast<std::int64_t>(normalized_a.block(j + n)) - borrow;
                                normalized_a.setblock(j + n, static_cast<BlockType>(signedBorrow));

                                //std::cout << "   updated a : " << normalized_a.showLimbs() << " : " << normalized_a.showLimbValues() << '\n';

                                setblock(static_cast<unsigned>(j), static_cast<BlockType>(qhat));
                                if (signedBorrow < 0) { // subtracted too much, add back
                                        setblock(static_cast<size_t>(j), static_cast<BlockType>(_block[static_cast<size_t>(j)] - 1));
                                        std::uint64_t carry{ 0 };
                                        for (unsigned i = 0; i < n; ++i) {
                                                carry += static_cast<std::uint64_t>(normalized_a.block(i + j)) + static_cast<std::uint64_t>(normalized_b.block(i));
                                                normalized_a.setblock(i + j, static_cast<BlockType>(carry));
                                                carry >>= bitsInBlock;
                                        }
                                        BlockType rectified = static_cast<BlockType>(normalized_a.block(j + n) + carry);
                                        normalized_a.setblock(j + n, rectified);
                                }
                                //std::cout << "   updated a : " << normalized_a.showLimbs() << " : " << normalized_a.showLimbValues() << '\n';
                        }

                        // remainder needs to be normalized
                        for (unsigned i = 0; i < n - 1; ++i) {
                                std::uint64_t remainder = static_cast<std::uint64_t>(normalized_a.block(i) >> shift);
                                remainder |= (static_cast<std::uint64_t>(normalized_a.block(i + 1)) << (bitsInBlock - shift));
                                r.setblock(i, static_cast<BlockType>(remainder));
                        }
                        r.setblock(n - 1, static_cast<BlockType>(normalized_a.block(n - 1) >> shift));
                }
                remove_leading_zeros();
                _sign = a.sign() ^ b.sign();
        }

        // modifiers (vector::clear is constexpr in C++20; on the empty
        // constant-evaluated state both clear() and setzero() are trivially
        // constexpr-clean.  setsign writes only the bool member.)
        constexpr void clear() noexcept { _sign = false; _block.clear(); }
        constexpr void setzero() noexcept { clear(); }
        constexpr void setsign(bool sign = true) noexcept { _sign = sign; }
        // use un-interpreted raw bits to set the bits of the einteger
        void setbits(unsigned long long value) {
                clear();
                if constexpr (bitsInBlock == 8) {
                        std::uint8_t byte0 = static_cast<std::uint8_t>(value & 0x0000'0000'0000'00FF);
                        std::uint8_t byte1 = static_cast<std::uint8_t>((value & 0x0000'0000'0000'FF00) >> 8);
                        std::uint8_t byte2 = static_cast<std::uint8_t>((value & 0x0000'0000'00FF'0000) >> 16);
                        std::uint8_t byte3 = static_cast<std::uint8_t>((value & 0x0000'0000'FF00'0000) >> 24);
                        std::uint8_t byte4 = static_cast<std::uint8_t>((value & 0x0000'00FF'0000'0000) >> 32);
                        std::uint8_t byte5 = static_cast<std::uint8_t>((value & 0x0000'FF00'0000'0000) >> 40);
                        std::uint8_t byte6 = static_cast<std::uint8_t>((value & 0x00FF'0000'0000'0000) >> 48);
                        std::uint8_t byte7 = static_cast<std::uint8_t>((value & 0xFF00'0000'0000'0000) >> 56);
                        if (byte7 > 0) {
                                _block.push_back(byte0);
                                _block.push_back(byte1);
                                _block.push_back(byte2);
                                _block.push_back(byte3);
                                _block.push_back(byte4);
                                _block.push_back(byte5);
                                _block.push_back(byte6);
                                _block.push_back(byte7);
                        }
                        else if (byte6 > 0) {
                                _block.push_back(byte0);
                                _block.push_back(byte1);
                                _block.push_back(byte2);
                                _block.push_back(byte3);
                                _block.push_back(byte4);
                                _block.push_back(byte5);
                                _block.push_back(byte6);
                        }
                        else if (byte5 > 0) {
                                _block.push_back(byte0);
                                _block.push_back(byte1);
                                _block.push_back(byte2);
                                _block.push_back(byte3);
                                _block.push_back(byte4);
                                _block.push_back(byte5);
                        }
                        else if (byte4 > 0) {
                                _block.push_back(byte0);
                                _block.push_back(byte1);
                                _block.push_back(byte2);
                                _block.push_back(byte3);
                                _block.push_back(byte4);
                        }
                        else if (byte3 > 0) {
                                _block.push_back(byte0);
                                _block.push_back(byte1);
                                _block.push_back(byte2);
                                _block.push_back(byte3);
                        }
                        else if (byte2 > 0) {
                                _block.push_back(byte0);
                                _block.push_back(byte1);
                                _block.push_back(byte2);
                        }
                        else if (byte1 > 0) {
                                _block.push_back(byte0);
                                _block.push_back(byte1);
                        }
                        else if (byte0 > 0) {
                                _block.push_back(byte0);
                        }
                        else {
                                _block.clear();
                        }
                }
                else if constexpr (bitsInBlock == 16) {
                        std::uint16_t word0 = static_cast<std::uint16_t>( value & 0x0000'0000'0000'FFFF);
                        std::uint16_t word1 = static_cast<std::uint16_t>((value & 0x0000'0000'FFFF'0000) >> 16);
                        std::uint16_t word2 = static_cast<std::uint16_t>((value & 0x0000'FFFF'0000'0000) >> 32);
                        std::uint16_t word3 = static_cast<std::uint16_t>((value & 0xFFFF'0000'0000'0000) >> 48);
                        if (word3 > 0) {
                                _block.push_back(word0);
                                _block.push_back(word1);
                                _block.push_back(word2);
                                _block.push_back(word3);
                        }
                        else if (word2 > 0) {
                                _block.push_back(word0);
                                _block.push_back(word1);
                                _block.push_back(word2);
                        }
                        else if (word1 > 0) {
                                _block.push_back(word0);
                                _block.push_back(word1);
                        }
                        else if (word0 > 0) {
                                _block.push_back(word0);
                        }
                        else {
                                _block.clear();
                        }
                }
                else if constexpr (bitsInBlock == 32) {
                        std::uint32_t low  = static_cast<std::uint32_t>(value & 0x0000'0000'FFFF'FFFF);
                        std::uint32_t high = static_cast<std::uint32_t>((value & 0xFFFF'FFFF'0000'0000) >> bitsInBlock);
                        if (high > 0) {
                                _block.push_back(low);
                                _block.push_back(high);
                        }
                        else if (low > 0) {
                                _block.push_back(low);
                        }
                        else {
                                _block.clear();
                        }
                }
        }
        void setblock(unsigned i, BlockType value) noexcept {
                if (i >= _block.size()) _block.resize(i+1ull);
                _block[i] = value;
        }
        void setbyte(unsigned i, std::uint8_t byte) noexcept {
                std::cerr << "setbyte(" << i << ", " << int(byte) << ") TBD\n";
        }
        einteger& assign(const std::string& txt) {
                if (!parse(txt, *this)) {
                        std::cerr << "Unable to parse: " << txt << std::endl;
                }
                return *this;
        }

        // selectors (read-only access to _sign and _block; constexpr-callable
        // on the empty default-constructed state and on any persisted
        // constexpr einteger -- which by definition has empty _block)
        constexpr bool iszero() const noexcept { return (_block.size() == 0 || ((_block.size() == 1) && _block[0] == bt(0))); }
        constexpr bool isone()  const noexcept { return true; }
        constexpr bool isodd()  const noexcept { return (_block.size() > 0) ? (_block[0] & 0x1) : false; }
        constexpr bool iseven() const noexcept { return !isodd(); }
        constexpr bool ispos()  const noexcept { return !_sign; }
        constexpr bool isneg()  const noexcept { return _sign; }

        constexpr bool test(unsigned index) const noexcept {
                if (index < nbits()) {
                        unsigned blockIndex = index / bitsInBlock;
                        unsigned bitIndexInBlock = index % bitsInBlock;
                        BlockType data = _block[blockIndex];
                        BlockType mask = (0x1u << bitIndexInBlock);
                        if (data & mask) return true;
                }
                return false;
        }
        constexpr bool sign()   const noexcept { return _sign; }
        constexpr int scale()   const noexcept { return findMsb(); } // TODO: when value = 0, scale returns -1 which is incorrect

        constexpr BlockType block(unsigned b) const noexcept {
                if (b < _block.size()) return _block[b];
                return static_cast<BlockType>(0u);
        }
        constexpr unsigned limbs() const noexcept { return static_cast<unsigned>(_block.size()); }

        constexpr unsigned nbits() const noexcept { return static_cast<unsigned>(_block.size() * sizeof(BlockType) * 8); }

        // findMsb takes an einteger reference and returns the position of the most significant bit, -1 if v == 0
        constexpr int findMsb() const noexcept {
                int nrBlocks = static_cast<int>(_block.size());
                if (nrBlocks == 0) return -1; // no significant bit found, all bits are zero
                int msb = nrBlocks * static_cast<int>(bitsInBlock);
                for (int b = nrBlocks - 1; b >= 0; --b) {
                        // derive mask from the actual block width so this works for
                        // uint8_t / uint16_t / uint32_t instantiations (was hardcoded
                        // to 0x8000'0000ul, which only worked for uint32_t)
                        std::uint64_t segment = static_cast<std::uint64_t>(_block[static_cast<size_t>(b)]);
                        std::uint64_t mask = (std::uint64_t(1) << (bitsInBlock - 1));
                        for (int i = bitsInBlock - 1; i >= 0; --i) {
                                --msb;
                                if (segment & mask) return msb;
                                mask >>= 1;
                        }
                }
                return -1; // no significant bit found, all bits are zero
        }

        // convert to string containing digits number of digits
        std::string str(size_t nrDigits = 0) const {
                return std::string("tbd");
        }

        // show the binary encodings of the limbs
        std::string showLimbs() const {
                if (_block.empty()) return "no limbs";
                std::stringstream s;
                size_t i = _block.size() - 1;
                while (i > 0) {
                        s << to_binary(_block[i], sizeof(BlockType) * 8, true) << ' ';
                        --i;
                }
                s << to_binary(_block[0], sizeof(BlockType) * 8, true);
                return s.str();
        }
        // show the values of the limbs as a radix-BlockType number
        std::string showLimbValues() const {
                if (_block.empty()) return "no limbs";
                std::stringstream s;
                size_t i = _block.size() - 1;
                while (i > 0) {
                        s << std::setw(5) << unsigned(_block[i]) << ", ";
                        --i;
                }
                s << std::setw(5) << unsigned(_block[0]);
                return s.str();
        }

protected:
        bool                   _sign;   // sign of the number: -1 if true, +1 if false, zero is positive
        std::vector<BlockType> _block;  // building blocks representing a 1's complement magnitude

        // HELPER methods
        // compare_magnitude returns 1 if a > b, 0 if they are equal, and -1 if a < b
        int compare_magnitude(const einteger& a, const einteger& b) {
                unsigned aLimbs = a.limbs();
                unsigned bLimbs = b.limbs();
                if (aLimbs != bLimbs) {
                        return (aLimbs > bLimbs ? 1 : -1);  // return 1 if a > b, otherwise -1
                }
                for (int i = static_cast<int>(aLimbs) - 1; i >= 0; --i) {
                        BlockType _a = a._block[static_cast<size_t>(i)];
                        BlockType _b = b._block[static_cast<size_t>(i)];
                        if ( _a != _b) {
                                return (_a > _b ? 1 : -1);
                        }
                }
                return 0;
        }
        void remove_leading_zeros() {
                unsigned leadingZeroBlocks{ 0 };
                typename std::vector<BlockType>::reverse_iterator rit = _block.rbegin();
                while (rit != _block.rend()) {
                        if (*rit == 0) {
                                ++leadingZeroBlocks;
                        }
                        else {
                                break;
                        }
                        ++rit;
                }
                _block.resize(_block.size() - leadingZeroBlocks);
        }
        
        template<typename SignedInt>
        einteger& convert_signed(SignedInt v) {
                clear();
                if (v != 0) {
                        if (v < 0) {
                                setbits(static_cast<unsigned long long>(-v));
                                setsign(true);
                        }
                        else {
                                setbits(static_cast<unsigned long long>(v)); // TODO: what about -2^63
                        }
                }
                return *this;
        }

        template<typename UnsignedInt>
        einteger& convert_unsigned(UnsignedInt v) {
                if (0 == v) {
                        setzero();
                }
                else {
                        setbits(static_cast<unsigned long long>(v));
                }
                return *this;
        }

        template<typename Real>
        einteger& convert_ieee754(Real& rhs) {
                clear();
                bool s{ false };
                std::uint64_t rawExponent{ 0 };
                std::uint64_t rawFraction{ 0 };
                uint64_t bits{ 0 };
                extractFields(rhs, s, rawExponent, rawFraction, bits);
                if (rawExponent == ieee754_parameter<Real>::eallset) { // nan and inf
                        // we can't represent NaNs or Infinities
                        return *this;
                }
                int exponent = static_cast<int>(rawExponent) - ieee754_parameter<Real>::bias;
                if (exponent < 0) {
                        return *this; // we are zero
                }
                // normal and subnormal handling
                constexpr size_t fbits = ieee754_parameter<Real>::fbits;
                std::uint64_t hiddenBit = (0x1ull << fbits);
                rawFraction |= hiddenBit;
                setbits(rawFraction);
                setsign(s);
                // scale the fraction bits
                *this <<= static_cast<int>(exponent - fbits);
                return *this;
        }

        template<typename Integer>
        Integer convert_to_native_integer() const noexcept {
                using Unsigned = std::make_unsigned_t<Integer>;
                Unsigned v{ 0 };
                Unsigned m{ 1 };
                const bool neg = sign();
                constexpr Integer kMax = std::numeric_limits<Integer>::max();
                constexpr Integer kMin = std::numeric_limits<Integer>::min();
                constexpr unsigned kDigits = std::numeric_limits<Unsigned>::digits;
                for (unsigned i = 0; i < nbits(); ++i) {
                        if (i >= kDigits) {
                                if (test(i)) return neg ? kMin : kMax;
                                continue;
                        }
                        if (test(i)) {
                                if (v > (static_cast<Unsigned>(kMax) - m)) {
                                        return neg ? kMin : kMax;
                                }
                                v += m;
                        }
                        m <<= 1;
                }
                return (neg ? -static_cast<Integer>(v) : static_cast<Integer>(v));
        }
        template<typename Real>
        Real convert_to_native_ieee() const noexcept {
                Real v{ 0 };
                Real m{ 1.0 };
                for (unsigned i = 0; i < nbits(); ++i) {
                        if (test(i)) {
                                v += m;
                        }
                        m *= Real(2.0);
                }
                return (sign() ? -v : v);
        }

private:

        template<typename BBlockType>
        friend constexpr bool operator==(const einteger<BBlockType>&, const einteger<BBlockType>&) noexcept;
};

////////////////////////    einteger functions   /////////////////////////////////

template<typename BlockType>
inline einteger<BlockType> abs(const einteger<BlockType>& a) {
        return (a.isneg()  ? -a : a);
}

////////////////////////    INTEGER operators   /////////////////////////////////

/// stream operators

// read a einteger ASCII format and make a binary einteger out of it
template<typename BlockType>
bool parse(const std::string& number, einteger<BlockType>& value) {
        using Integer = einteger<BlockType>;
        bool bSuccess = false;
        value.clear();
        std::regex binary_regex("^[-+]*0b[01']+");
        // check if the txt is an integer form: [0123456789]+
        std::regex decimal_regex("^[-+]*[0-9]+");
        std::regex octal_regex("^[-+]*0[1-7][0-7]*$");
        std::regex hex_regex("^[-+]*0[xX][0-9a-fA-F']+");
        // setup associative array to map chars to nibbles
        std::map<char, int> charLookup{
                { '0', 0 },
                { '1', 1 },
                { '2', 2 },
                { '3', 3 },
                { '4', 4 },
                { '5', 5 },
                { '6', 6 },
                { '7', 7 },
                { '8', 8 },
                { '9', 9 },
                { 'a', 10 },
                { 'b', 11 },
                { 'c', 12 },
                { 'd', 13 },
                { 'e', 14 },
                { 'f', 15 },
                { 'A', 10 },
                { 'B', 11 },
                { 'C', 12 },
                { 'D', 13 },
                { 'E', 14 },
                { 'F', 15 },
        };
        if (std::regex_match(number, octal_regex)) {
                std::cout << "found an octal representation\n";
                for (std::string::const_reverse_iterator r = number.rbegin();
                        r != number.rend();
                        ++r) {
                        std::cout << "char = " << *r << std::endl;
                }
                bSuccess = false; // TODO
        }
        else if (std::regex_match(number, hex_regex)) {
                //std::cout << "found a hexadecimal representation\n";
                // each char is a nibble
                int byte = 0;
                int byteIndex = 0;
                bool odd = false;
                for (std::string::const_reverse_iterator r = number.rbegin();
                        r != number.rend();
                        ++r) {
                        if (*r == '\'') {
                                // ignore
                        }
                        else if (*r == 'x' || *r == 'X') {
                                if (odd) {
                                        // complete the most significant byte
                                        value.setbyte(static_cast<size_t>(byteIndex), static_cast<uint8_t>(byte));
                                }
                                // check that we have [-+]0[xX] format
                                ++r;
                                if (r != number.rend()) {
                                        if (*r == '0') {
                                                // check if we have a sign
                                                ++r;
                                                if (r == number.rend()) {
                                                        // no sign, thus by definition positive
                                                        bSuccess = true;
                                                }
                                                else if (*r == '+') {
                                                        // optional positive sign, no further action necessary
                                                        bSuccess = true;
                                                }
                                                else if (*r == '-') {
                                                        // negative sign, invert
                                                        value = -value;
                                                        bSuccess = true;
                                                }
                                                else {
                                                        // the regex will have filtered this out
                                                        bSuccess = false;
                                                }
                                        }
                                        else {
                                                // we didn't find the obligatory '0', the regex should have filtered this out
                                                bSuccess = false;
                                        }
                                }
                                else {
                                        // we are missing the obligatory '0', the regex should have filtered this out
                                        bSuccess = false;
                                }
                                // we have reached the end of our parse
                                break;
                        }
                        else {
                                if (odd) {
                                        byte += charLookup.at(*r) << 4;
                                        value.setbyte(static_cast<size_t>(byteIndex), static_cast<uint8_t>(byte));
                                        ++byteIndex;
                                }
                                else {
                                        byte = charLookup.at(*r);
                                }
                                odd = !odd;
                        }
                }
        }
        else if (std::regex_match(number, decimal_regex)) {
                //std::cout << "found a decimal integer representation\n";
                Integer scale = 1;
                bool sign{ false };
                for (std::string::const_reverse_iterator r = number.rbegin();
                        r != number.rend();
                        ++r) {
                        if (*r == '-') {
                                sign = true;;
                        }
                        else if (*r == '+') {
                                break;
                        }
                        else {
                                Integer digit = charLookup.at(*r);
                                value += scale * digit;
                                scale *= 10;
                        }
                }
                value.setsign(sign);
                bSuccess = true;
        }
        else if (std::regex_match(number, binary_regex)) {
                //std::cout << "found a binary integer representation\n";
                Integer scale = 1;
                //bool sign{ false };
                unsigned byte{ 0 }; // using an unsigned to simplify accumulation, but accumulating 8-bit byte values
                unsigned bitIndex = 0;
                for (std::string::const_reverse_iterator r = number.rbegin();
                        r != number.rend();
                        ++r) {
                        if (*r == '-') {
                                //sign = true;;
                        }
                        else if (*r == '+') {
                                break;
                        }
                        else if (*r == '\'') {
                                // ignore separator
                        }
                        else {
                                if (*r == '1') {
                                        byte |= (1u << (bitIndex % 8));
                                }
                                if (bitIndex == 7) {
                                        value += scale * byte;
                                        scale *= 256;
                                        byte = 0;
                                }
                                ++bitIndex;
                        }
                }
                if (bitIndex % 8) {
                        value += scale * byte;
                }
                bSuccess = true;
        }
        return bSuccess;
}

template<typename BlockType>
std::string convert_to_string(std::ios_base::fmtflags flags, const einteger<BlockType>& n) {
        using AdaptiveInteger = einteger<BlockType>;

        if (n.limbs() == 0) return std::string("0");

        // set the base of the target number system to convert to
        int base = 10;
        if ((flags & std::ios_base::oct) == std::ios_base::oct) base = 8;
        if ((flags & std::ios_base::hex) == std::ios_base::hex) base = 16;

        unsigned nbits = n.limbs() * sizeof(BlockType) * 8;

        std::string result;
        if (base == 8 || base == 16) {
                if (n.sign()) return std::string("negative value: ignored");

                size_t shift = (base == 8 ? 3ull : 4ull);
                BlockType mask = static_cast<BlockType>((1u << shift) - 1);
                AdaptiveInteger t(n);
                result.assign(nbits / shift + ((nbits % shift) ? 1 : 0), '0');
                size_t pos = result.size() - 1ull;
                for (size_t i = 0; i < nbits / shift; ++i) {
                        char c = '0' + static_cast<char>(t.block(0) & mask);
                        if (c > '9')
                                c += 'A' - '9' - 1;
                        result[pos--] = c;
                        t >>= static_cast<int>(shift);
                }
                if (nbits % shift) {
                        mask = static_cast<BlockType>((1u << (nbits % shift)) - 1);
                        char c = '0' + static_cast<char>(t.block(0) & mask);
                        if (c > '9')
                                c += 'A' - '9';
                        result[pos] = c;
                }
                //
                // Get rid of leading zeros:
                //
                std::string::size_type fnz = result.find_first_not_of('0');
                if (!result.empty() && (fnz == std::string::npos)) fnz = result.size() - 1;
                result.erase(0, fnz);
                if (flags & std::ios_base::showbase) {
                        const char* pp = base == 8 ? "0" : "0x";
                        result.insert(static_cast<std::string::size_type>(0), pp);
                }
        }
        else {
                unsigned block10;
                unsigned digits_in_block10;
                if constexpr (AdaptiveInteger::bitsInBlock == 8) {
                        block10 = 100u;
                        digits_in_block10 = 2;
                }
                else if constexpr (AdaptiveInteger::bitsInBlock == 16) {
                        block10 = 10'000ul;
                        digits_in_block10 = 4;
                }
                else if constexpr (AdaptiveInteger::bitsInBlock == 32) {
                        block10 = 1'000'000'000ul;
                        digits_in_block10 = 9;
                }
                else if constexpr (AdaptiveInteger::bitsInBlock == 64) {
                        // not allowed as the whole multi-digit arithmetic
                        // requires that there is a 'larger' type that
                        // can receive carries and borrows.
                        // If your platform does have a native 128bit
                        // integer, this could be enabled
                        //block10 = 1'000'000'000'000'000'000ull;
                        //digits_in_block10 = 18;
                }
                result.assign(nbits / 3 + 1ull, '0');
                size_t pos = result.size() - 1ull;
                AdaptiveInteger t(n);
                while (!t.iszero()) {
                        AdaptiveInteger q,r;
                        q.reduce(t, block10, r);
                        BlockType v = r.block(0);
//                        std::cout << "v  " << uint32_t(v) << '\n';
                        for (unsigned i = 0; i < digits_in_block10; ++i) {
                                char c = '0' + static_cast<char>(v % 10);
                                v /= 10;
                                result[pos] = c;
//                                std::cout << result << " pos: " << pos << '\n';
                                if (pos-- == 0)        break;
                        }
                        t = q;
                }

                std::string::size_type firstDigit = result.find_first_not_of('0');
                result.erase(0, firstDigit);
                if (result.empty())
                        result = "0";
                if (n.isneg())
                        result.insert(0ull, 1ull, '-');
                else if (flags & std::ios_base::showpos)
                        result.insert(0ull, 1ull, '+');
        }
        return result;
}

// generate an einteger format ASCII format
template<typename BlockType>
inline std::ostream& operator<<(std::ostream& ostr, const einteger<BlockType>& i) {
        std::string s = convert_to_string(ostr.flags(), i);
        std::streamsize width = ostr.width();
        if (width > static_cast<std::streamsize>(s.size())) {
                char fill = ostr.fill();
                if ((ostr.flags() & std::ios_base::left) == std::ios_base::left)
                        s.append(static_cast<std::string::size_type>(width - s.size()), fill);
                else
                        s.insert(static_cast<std::string::size_type>(0), static_cast<std::string::size_type>(width - s.size()), fill);
        }
        return ostr << s;
}

// read an ASCII einteger format

template<typename BlockType>
inline std::istream& operator>>(std::istream& istr, einteger<BlockType>& p) {
        std::string txt;
        if (!(istr >> txt)) {
                // extraction failed (already-bad stream or EOF); failbit set by >>.
                return istr;
        }
        if (!parse(txt, p)) {
                std::cerr << "unable to parse -" << txt << "- into an einteger value\n";
                istr.setstate(std::ios::failbit);
        }
        return istr;
}

////////////////// string operators

template<typename BlockType>
inline std::string to_binary(const einteger<BlockType>& a, bool nibbleMarker = true) {
        if (a.limbs() == 0) return std::string("0b0");

        std::stringstream s;
        s << "0b";
        for (int b = static_cast<int>(a.limbs()) - 1; b >= 0; --b) {
                BlockType segment = a.block(static_cast<size_t>(b));
                BlockType mask = (0x1u << (a.bitsInBlock - 1));
                for (int i = a.bitsInBlock - 1; i >= 0; --i) {
                        s << ((segment & mask) ? '1' : '0');
                        if (i > 0 && (i % 4) == 0 && nibbleMarker) s << '\'';
                        if (b > 0 && i == 0 && nibbleMarker) s << '\'';
                        mask >>= 1;
                }
        }

        return s.str();
}

template<typename BlockType>
inline std::string to_hex(const einteger<BlockType>& a, bool wordMarker = true) {
        if (a.limbs() == 0) return std::string("0x0");

        std::vector<char> nibbleLookup = { '0', '1', '2', '3', '4', '5', '6', '7', '8', '9', 'A', 'B', 'C', 'D', 'E', 'F' };
        std::stringstream s;
        s << "0x";
        unsigned bitIndex = a.limbs() * a.bitsInBlock - 1u;
        for (int b = static_cast<int>(a.limbs()) - 1; b >= 0; --b) {
                BlockType limb = a.block(static_cast<size_t>(b));
                BlockType mask = (0x1u << (a.bitsInBlock - 1));
                unsigned nibble{ 0 };
                unsigned rightShift = a.bitsInBlock - 4u;
                for (int i = a.bitsInBlock - 1; i >= 0; --i) {

                        nibble |= (limb & mask);
                        if ((i % 4) == 0) {
                                nibble >>= rightShift;
                                s << nibbleLookup[nibble];
                                nibble = 0;
                                rightShift -= 4u;
                        }
                        if (bitIndex > 0 && ((bitIndex % 16) == 0) && wordMarker) s << '\'';
                        mask >>= 1;
                        --bitIndex;
                }
        }

        return s.str();

}

//////////////////////////////////////////////////////////////////////////////////////////////////////
// einteger - einteger binary logic operators

// equal: sign-aware (treats +0 == -0 by canonicalizing zero); pre-existing
// implementation ignored sign entirely, so +n == -n returned true.
// Constexpr-callable: pure reads of the digit vector, no allocation.

template<typename BlockType>
constexpr bool operator==(const einteger<BlockType>& lhs, const einteger<BlockType>& rhs) noexcept {
        if (lhs.limbs() != rhs.limbs()) {
                return false;
        }
        if (lhs.sign() != rhs.sign()) {
                // keep +0 == -0, but require sign match for non-zero values
                return lhs.iszero() && rhs.iszero();
        }
        for (unsigned i = 0; i < lhs.limbs(); ++i) {
                if (lhs._block[i] != rhs._block[i]) return false;
        }
        return true;
}

template<typename BlockType>
constexpr bool operator!=(const einteger<BlockType>& lhs, const einteger<BlockType>& rhs) noexcept {
        return !operator==(lhs, rhs);
}

template<typename BlockType>
constexpr bool operator< (const einteger<BlockType>& lhs, const einteger<BlockType>& rhs) noexcept {
        const bool ls = lhs.sign();
        const bool rs = rhs.sign();
        if (ls != rs) {
                // +0 and -0 are equal, not ordered
                if (lhs.iszero() && rhs.iszero()) return false;
                return ls; // negative < positive
        }
        unsigned ll = lhs.limbs();
        unsigned rl = rhs.limbs();
        // for negatives, larger magnitude means smaller value -- swap the sense
        if (ll != rl) return ls ? (ll > rl) : (ll < rl);
        if (ll == 0) return false; // both empty: equal in magnitude
        for (unsigned b = ll; b-- > 0;) {
                BlockType l = lhs.block(b);
                BlockType r = rhs.block(b);
                if (l == r) continue;
                return ls ? (l > r) : (l < r);
        }
        return false; // lhs and rhs are the same
}

template<typename BlockType>
constexpr bool operator> (const einteger<BlockType>& lhs, const einteger<BlockType>& rhs) noexcept {
        return operator< (rhs, lhs);
}

template<typename BlockType>
constexpr bool operator<=(const einteger<BlockType>& lhs, const einteger<BlockType>& rhs) noexcept {
        return operator< (lhs, rhs) || operator==(lhs, rhs);
}

template<typename BlockType>
constexpr bool operator>=(const einteger<BlockType>& lhs, const einteger<BlockType>& rhs) noexcept {
        return !operator< (lhs, rhs);
}

//////////////////////////////////////////////////////////////////////////////////////////////////////
// einteger - literal binary logic operators
// equal: precondition is that the byte-storage is properly nulled in all arithmetic paths

template<typename BlockType>
inline bool operator==(const einteger<BlockType>& lhs, long long rhs) {
        return operator==(lhs, einteger(rhs));
}

template<typename BlockType>
inline bool operator!=(const einteger<BlockType>& lhs, long long rhs) {
        return !operator==(lhs, rhs);
}

template<typename BlockType>
inline bool operator< (const einteger<BlockType>& lhs, long long rhs) {
        return operator<(lhs, einteger<BlockType>(rhs));
}

template<typename BlockType>
inline bool operator> (const einteger<BlockType>& lhs, long long rhs) {
        return operator< (einteger<BlockType>(rhs), lhs);
}

template<typename BlockType>
inline bool operator<=(const einteger<BlockType>& lhs, long long rhs) {
        return operator< (lhs, rhs) || operator==(lhs, rhs);
}

template<typename BlockType>
inline bool operator>=(const einteger<BlockType>& lhs, long long rhs) {
        return !operator< (lhs, rhs);
}

//////////////////////////////////////////////////////////////////////////////////////////////////////
// literal - einteger binary logic operators
// precondition is that the byte-storage is properly nulled in all arithmetic paths


template<typename BlockType>
inline bool operator==(long long lhs, const einteger<BlockType>& rhs) {
        return operator==(einteger<BlockType>(lhs), rhs);
}

template<typename BlockType>
inline bool operator!=(long long lhs, const einteger<BlockType>& rhs) {
        return !operator==(lhs, rhs);
}

template<typename BlockType>
inline bool operator< (long long lhs, const einteger<BlockType>& rhs) {
        return operator<(einteger<BlockType>(lhs), rhs);
}

template<typename BlockType>
inline bool operator> (long long lhs, const einteger<BlockType>& rhs) {
        return operator< (rhs, lhs);
}

template<typename BlockType>
inline bool operator<=(long long lhs, const einteger<BlockType>& rhs) {
        return operator< (lhs, rhs) || operator==(lhs, rhs);
}

template<typename BlockType>
inline bool operator>=(long long lhs, const einteger<BlockType>& rhs) {
        return !operator< (lhs, rhs);
}

//////////////////////////////////////////////////////////////////////////////////////////////////////

//////////////////////////////////////////////////////////////////////////////////////////////////////
// einteger - einteger binary arithmetic operators

template<typename BlockType>
inline einteger<BlockType> operator+(const einteger<BlockType>& lhs, const einteger<BlockType>& rhs) {
        einteger sum = lhs;
        sum += rhs;
        return sum;
}

template<typename BlockType>
inline einteger<BlockType> operator-(const einteger<BlockType>& lhs, const einteger<BlockType>& rhs) {
        einteger diff = lhs;
        diff -= rhs;
        return diff;
}

template<typename BlockType>
inline einteger<BlockType> operator*(const einteger<BlockType>& lhs, const einteger<BlockType>& rhs) {
        einteger product = lhs;
        product *= rhs;
        return product;
}

template<typename BlockType>
inline einteger<BlockType> operator/(const einteger<BlockType>& lhs, const einteger<BlockType>& rhs) {
        einteger ratio = lhs;
        ratio /= rhs;
        return ratio;
}

template<typename BlockType>
inline einteger<BlockType> operator%(const einteger<BlockType>& lhs, const einteger<BlockType>& rhs) {
        einteger remainder = lhs;
        remainder %= rhs;
        return remainder;
}

//////////////////////////////////////////////////////////////////////////////////////////////////////
// einteger - literal binary arithmetic operators

template<typename BlockType>
inline einteger<BlockType> operator+(const einteger<BlockType>& lhs, long long rhs) {
        return operator+(lhs, einteger<BlockType>(rhs));
}

template<typename BlockType>
inline einteger<BlockType> operator-(const einteger<BlockType>& lhs, long long rhs) {
        return operator-(lhs, einteger<BlockType>(rhs));
}

template<typename BlockType>
inline einteger<BlockType> operator*(const einteger<BlockType>& lhs, long long rhs) {
        return operator*(lhs, einteger<BlockType>(rhs));
}

template<typename BlockType>
inline einteger<BlockType> operator/(const einteger<BlockType>& lhs, long long rhs) {
        return operator/(lhs, einteger<BlockType>(rhs));
}

template<typename BlockType>
inline einteger<BlockType> operator%(const einteger<BlockType>& lhs, long long rhs) {
        return operator/(lhs, einteger<BlockType>(rhs));
}

template<typename BlockType>
inline einteger<BlockType> operator/(const einteger<BlockType>& lhs, unsigned long long rhs) {
        return operator/(lhs, einteger<BlockType>(rhs));
}

//////////////////////////////////////////////////////////////////////////////////////////////////////
// literal - einteger binary arithmetic operators

template<typename BlockType>
inline einteger<BlockType> operator+(long long lhs, const einteger<BlockType>& rhs) {
        return operator+(einteger<BlockType>(lhs), rhs);
}

template<typename BlockType>
inline einteger<BlockType> operator-(long long lhs, const einteger<BlockType>& rhs) {
        return operator-(einteger<BlockType>(lhs), rhs);
}

template<typename BlockType>
inline einteger<BlockType> operator*(long long lhs, const einteger<BlockType>& rhs) {
        return operator*(einteger<BlockType>(lhs), rhs);
}

template<typename BlockType>
inline einteger<BlockType> operator/(long long lhs, const einteger<BlockType>& rhs) {
        return operator/(einteger<BlockType>(lhs), rhs);
}

template<typename BlockType>
inline einteger<BlockType> operator%(long long lhs, const einteger<BlockType>& rhs) {
        return operator/(einteger<BlockType>(lhs), rhs);
}

}} // namespace sw::universal

stillwater-sc / universal / 26002766134

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