30660486273

Committed 31 Jul 2026 07:48PM UTC coverage: 85.249% (+0.02%) from 85.229%

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fix(integer): clear -Wsign-conversion in integer/primes/lns (#1283) (#1284)

* fix(integer): clear -Wsign-conversion in integer/primes/lns (#1283)

Sub-issue of Epic #1279 (-Wsign-conversion cleanup). All fixes are
value-preserving; each casts a value already proven non-negative /
in-range at the site, or replaces a signedness-changing literal.

integer_impl.hpp (shift operators <<=/>>=):
- guard the block-shift index arithmetic where int bitsToShift/blockShift
  mixes with unsigned bitsInBlock/nbits/MSU; bitsToShift is > 0 past the
  guards, so static_cast<unsigned> is exact
- upper-block mask now uses the shared bit_high_mask<bt>(#1261) helper,
  which also states the truncation explicitly (was a raw 64-bit literal
  shift that narrowed implicitly)
- assign(int64_t): low-bit test v & 0x1ull -> (v & 1) != 0; masking with a
  signed 1 keeps the operation in int64 (correct for negative v) instead
  of promoting v to uint64
- operator<<: width padding casts std::streamsize width (guarded
  width > s.size(), so positive) before subtracting the string size

primes.hpp: printPrimes column counter int -> size_t (compared against
size_t COL_WIDTH/PAGE_WIDTH).

lns_impl.hpp: setbit(i + shiftLeft) with shiftLeft used as a left-shift
count (>= 0 on this path) -> cast to unsigned.

Verified -Wsign-conversion clean for all three headers on gcc and clang;
no new -Wconversion introduced. integer shift_left/shift_right/division/
remainder and lns conversion regressions PASS on both compilers.

Relates to #1279

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

* fix(integer): address CodeRabbit review on #1283

- revert the unsigned-path low-bit test (convert_unsigned) to its original
  `v & 0x1ull`: `v` is uint64_t there, so it never triggered
  -Wsign-conversion; a global sed had changed it and attached a comment
  that (correctly, per review) described the signed path
- keep only the signed-path change (int64_t v) where the comment is accurate... (continued)

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91.04

/include/sw/universal/number/integer/integer_impl.hpp

#pragma once
// integer_impl.hpp: implementation of a fixed-size arbitrary precision integer number
//
// Copyright (C) 2017-2022 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 <algorithm>
#include <string_view>
#include <type_traits>  // for std::is_constant_evaluated() dispatch around mul128/addcarry
#include <vector>

// supporting types and functions
#include <universal/number/shared/specific_value_encoding.hpp>
#include <universal/number/shared/blocktype.hpp>
#include <universal/native/integers.hpp> // just for printing native integers in binary form
#include <universal/internal/blocktype/carry.hpp> // carry-detection intrinsics for uint64_t limbs
#include <universal/utility/string_parse.hpp>   // shared constexpr scan_prefix / scan_sign / char classifiers

/*
the integer arithmetic can be configured to:
- throw exceptions on overflow
- throw exceptions on arithmetic
- throw exceptions on encoding errors for Whole and Natural Numbers

you need the exception types defined, but you have the option to throw them
 */
#include <universal/number/integer/exceptions.hpp>

 // composition types used by integer
#include <universal/number/support/decimal.hpp>
#include <universal/internal/blocktriple/blocktriple.hpp>

#include <universal/internal/bit_manipulation.hpp>

namespace sw { namespace universal {

enum class IntegerNumberType {
        IntegerNumber = 0,  // { ...,-3,-2,-1,0,1,2,3,... }
        WholeNumber   = 1,  // {              0,1,2,3,... }
        NaturalNumber = 2   // {                1,2,3,... }
};

constexpr IntegerNumberType IntegerNumber = IntegerNumberType::IntegerNumber;
constexpr IntegerNumberType WholeNumber = IntegerNumberType::WholeNumber;
constexpr IntegerNumberType NaturalNumber = IntegerNumberType::NaturalNumber;

// scale calculate the power of 2 exponent that would capture an approximation of a normalized real value
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
inline long scale(const integer<nbits, BlockType, NumberType>& i) {
        integer<nbits, BlockType, NumberType> v(i);
        if (i.sign()) { // special case handling
                v.twosComplement();
                if (v == i) {  // special case of 10000..... largest negative number in 2's complement encoding
                        return long(nbits - 1);
                }
        }
        // calculate scale
        long scale = 0;
        while (v > 1) {
                ++scale;
                v >>= 1;
        }
        return scale;
}

// signed integer conversion
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
inline constexpr integer<nbits, BlockType, NumberType>& convert(int64_t v, integer<nbits, BlockType, NumberType>& result) {        return result.convert(v); }
// unsigned integer conversion
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
inline constexpr integer<nbits, BlockType, NumberType>& convert(uint64_t v, integer<nbits, BlockType, NumberType>& result) { return result.convert(v); }

template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
bool parse(const std::string& number, integer<nbits, BlockType, NumberType>& v);

// idiv_t for integer<nbits, BlockType, NumberType> to capture quotient and remainder during long division
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
struct idiv_t {
        constexpr idiv_t() : quot{ 0 }, rem{ 0 } {};
        integer<nbits, BlockType, NumberType> quot; // quotient
        integer<nbits, BlockType, NumberType> rem;  // remainder
};

/*
The rules for detecting overflow in a two's complement sum are simple:
 - If the sum of two positive numbers yields a negative result, the sum has overflowed.
 - If the sum of two negative numbers yields a positive result, the sum has overflowed.
 - Otherwise, the sum has not overflowed.
It is important to note the overflow and carry out can each occur without the other. 
In unsigned numbers, carry out is equivalent to overflow. In two's complement, carry out tells 
you nothing about overflow.

The reason for the rules is that overflow in two's complement occurs, not when a bit is carried out 
out of the left column, but when one is carried into it. That is, when there is a carry into the sign. 
The rules detect this error by examining the sign of the result. A negative and positive added together 
cannot overflow, because the sum is between the addends. Since both of the addends fit within the
allowable range of numbers, and their sum is between them, it must fit as well.

When implementing addition/subtraction on chuncks the overflow condition must be deduced from the 
chunk values. The chunks need to be interpreted as unsigned binary segments.
*/

// integer is an arbitrary fixed-sized 2's complement integer
template<unsigned _nbits, typename bt = std::uint8_t, IntegerNumberType _NumberType = IntegerNumberType::IntegerNumber>
class integer {
public:
        // cache template parameters
        static constexpr unsigned nbits = _nbits;
        typedef bt BlockType;
        static constexpr IntegerNumberType NumberType = _NumberType;
        // derive other parameters
        static constexpr unsigned bitsInByte = 8ull;
        static constexpr unsigned bitsInBlock = sizeof(bt) * bitsInByte;
        static constexpr unsigned nrBlocks = 1ull + ((nbits - 1ull) / bitsInBlock);
        static constexpr unsigned MSU = nrBlocks - 1ull;
        static constexpr bt       ALL_ONES = bt(~0); // block type specific all 1's value
        static constexpr unsigned bitSurplus = (nrBlocks * bitsInBlock - nbits);
        static constexpr unsigned bitsInMSU = bitsInBlock - bitSurplus;
        static constexpr bool     EXACT_FIT = (bitSurplus == 0);
        static constexpr unsigned signBitShift = (EXACT_FIT ? (bitsInBlock - 1) : bitsInMSU - 1);
        static constexpr bt       SIGN_BIT_MASK = bt(1ull << signBitShift);
        static constexpr bt       MSU_MASK = bt(ALL_ONES >> bitSurplus);
        static constexpr bt       SIGN_EXTENTION_BITS = static_cast<bt>(~MSU_MASK);
        static constexpr uint64_t storageMask = (uint64_t(~0) >> (64u - bitsInBlock));
        static constexpr uint64_t BASE = (ALL_ONES + 1ull);

        /// trivial constructor
        constexpr integer() noexcept = default;

        /// Construct a new integer from another, sign extend when necessary, BlockTypes must be the same
        template<unsigned srcbits>
        constexpr integer(const integer<srcbits, BlockType, NumberType>& a) {
//                static_assert(srcbits > nbits, "Source integer is bigger than target: potential loss of precision"); // TODO: do we want this?
                bitcopy(a);
                if constexpr (srcbits < nbits && NumberType == IntegerNumberType::IntegerNumber) {
                        if (a.sign()) { // sign extend
                                for (unsigned i = srcbits; i < nbits; ++i) {
                                        setbit(i);
                                }
                        }
                }
        }

        // initializers for native types
        constexpr integer(signed char initial_value)        noexcept { *this = initial_value; }
        constexpr integer(short initial_value)              noexcept { *this = initial_value; }
        constexpr integer(int initial_value)                noexcept { *this = initial_value; }
        constexpr integer(long initial_value)               noexcept { *this = initial_value; }
        constexpr integer(long long initial_value)          noexcept { *this = initial_value; }
        constexpr integer(char initial_value)               noexcept { *this = initial_value; }
        constexpr integer(unsigned short initial_value)     noexcept { *this = initial_value; }
        constexpr integer(unsigned int initial_value)       noexcept { *this = initial_value; }
        constexpr integer(unsigned long initial_value)      noexcept { *this = initial_value; }
        constexpr integer(unsigned long long initial_value) noexcept { *this = initial_value; }
        constexpr integer(float initial_value)              noexcept { *this = initial_value; }
        constexpr integer(double initial_value)             noexcept { *this = initial_value; }
        constexpr integer(long double initial_value)        noexcept { *this = initial_value; }

        // specific value constructors
        constexpr integer(const std::string& s) noexcept { assign(s); }
        constexpr integer(const SpecificValue code) noexcept
                : _block{ 0 } {
                switch (code) {
                case SpecificValue::maxpos:
                        maxpos();
                        break;
                case SpecificValue::minpos:
                        minpos();
                        break;
                case SpecificValue::zero:
                default:
                        zero();
                        break;
                case SpecificValue::minneg:
                        minneg();
                        break;
                case SpecificValue::maxneg:
                        maxneg();
                        break;
                case SpecificValue::infneg:
                case SpecificValue::infpos:
                case SpecificValue::qnan:
                case SpecificValue::snan:
                case SpecificValue::nar:
                        zero();
                        break;
                }
        }

        // access operator for bits
        // this needs a proxy to be able to create l-values
        // bool operator[](const unsigned int i) const //

        // simpler interface for now, using at(i) and set(i)/reset(i)

        // assignment operators for native types
        constexpr integer& operator=(signed char rhs)        { return convert_signed(rhs); }
        constexpr integer& operator=(short rhs)              { return convert_signed(rhs); }
        constexpr integer& operator=(int rhs)                { return convert_signed(rhs); }
        constexpr integer& operator=(long rhs)               { return convert_signed(rhs); }
        constexpr integer& operator=(long long rhs)          { return convert_signed(rhs); }
        constexpr integer& operator=(char rhs)               { return convert_unsigned(rhs); }
        constexpr integer& operator=(unsigned short rhs)     { return convert_unsigned(rhs); }
        constexpr integer& operator=(unsigned int rhs)       { return convert_unsigned(rhs); }
        constexpr integer& operator=(unsigned long rhs)      { return convert_unsigned(rhs); }
        constexpr integer& operator=(unsigned long long rhs) { return convert_unsigned(rhs); }
        constexpr integer& operator=(float rhs)              { return convert_ieee(rhs); }
        constexpr integer& operator=(double rhs)             { return convert_ieee(rhs); }
#if LONG_DOUBLE_SUPPORT
        constexpr integer& operator=(long double rhs)        { return convert_ieee(rhs); }
#endif

#ifdef ADAPTER_POSIT_AND_INTEGER
        // convenience assignment operator 
        template<unsigned nbits, unsigned es>
        integer& operator=(const posit<nbits, es>& rhs) {
                convert_p2i(rhs, *this);
                return *this;
        }
#endif // ADAPTER_POSIT_AND_INTEGER

        // prefix operators
        constexpr integer operator-() const {
                integer negated(*this);
                negated.flip();
                negated += 1;
                return negated;
        }
        // one's complement
        constexpr integer operator~() const {
                integer complement(*this);
                complement.flip();
                return complement;
        }
        // increment
        constexpr integer operator++(int) {
                integer tmp(*this);
                operator++();
                return tmp;
        }
        constexpr integer& operator++() {
                *this += integer(1);
                _block[MSU] = static_cast<bt>(_block[MSU] & MSU_MASK); // assert precondition of properly nulled leading non-bits
                return *this;
        }
        // decrement
        constexpr integer operator--(int) {
                integer tmp(*this);
                operator--();
                return tmp;
        }
        constexpr integer& operator--() {
                *this -= integer(1);
                _block[MSU] = static_cast<bt>(_block[MSU] & MSU_MASK); // assert precondition of properly nulled leading non-bits
                return *this;
        }

        // conversion operators
        explicit operator unsigned char()      const noexcept { return to_unsigned_integer<unsigned char>(); }
        explicit operator unsigned short()     const noexcept { return to_unsigned_integer<unsigned short>(); }
        explicit operator unsigned int()       const noexcept { return to_unsigned_integer<unsigned int>(); }
        explicit operator unsigned long()      const noexcept { return to_unsigned_integer<unsigned long>(); }
        explicit operator unsigned long long() const noexcept { return to_unsigned_integer<unsigned long long>(); }
        explicit operator signed char()        const noexcept { return to_integer<signed char>(); }
        explicit operator short()              const noexcept { return to_integer<short>(); }
        explicit operator int()                const noexcept { return to_integer<int>(); }
        explicit operator long()               const noexcept { return to_integer<long>(); }
        explicit operator long long()          const noexcept { return to_integer<long long>(); }
        explicit operator float()              const noexcept { return to_real<float>(); }
        explicit operator double()             const noexcept { return to_real<double>(); }
#if LONG_DOUBLE_SUPPORT
        explicit operator long double()        const noexcept { return to_real<long double>(); }
#endif

        // arithmetic operators
        constexpr integer& operator+=(const integer& rhs) {
                if constexpr (nrBlocks == 1) {
                        _block[0] = static_cast<bt>(_block[0] + rhs.block(0));
                        // null any leading bits that fall outside of nbits
                        _block[MSU] = static_cast<bt>(MSU_MASK & _block[MSU]);
                }
                else if constexpr (bitsInBlock == 64) {
                        // uint64_t limbs: addcarry uses platform intrinsics (not constexpr on MSVC).
                        // Dispatch to a portable carry path in constant-evaluated context.
                        integer<nbits, BlockType, NumberType> sum;
                        if (std::is_constant_evaluated()) {
                                uint64_t carry = 0;
                                for (unsigned i = 0; i < nrBlocks; ++i) {
                                        uint64_t a = _block[i];
                                        uint64_t b = rhs._block[i];
                                        uint64_t s1 = a + carry;
                                        uint64_t c1 = (s1 < a) ? 1ull : 0ull;
                                        uint64_t r  = s1 + b;
                                        uint64_t c2 = (r < s1) ? 1ull : 0ull;
                                        sum._block[i] = r;
                                        carry = c1 + c2;
                                }
                        }
                        else {
                                uint64_t carry = 0;
                                for (unsigned i = 0; i < nrBlocks; ++i) {
                                        sum._block[i] = addcarry(_block[i], rhs._block[i], carry, carry);
                                }
                        }
                        // enforce precondition for fast comparison by properly nulling bits that are outside of nbits
                        sum._block[MSU] = static_cast<bt>(MSU_MASK & sum._block[MSU]);
                        *this = sum;
                }
                else {
                        // Multi-limb path for bt = uint8_t/uint16_t/uint32_t.
                        // Index-based loop (constexpr-friendly; pointer arithmetic on stack
                        // arrays is forbidden in constexpr).
                        integer<nbits, BlockType, NumberType> sum;
                        std::uint64_t carry = 0;
                        for (unsigned i = 0; i < nrBlocks; ++i) {
                                carry += static_cast<std::uint64_t>(_block[i]) + static_cast<std::uint64_t>(rhs._block[i]);
                                sum._block[i] = static_cast<bt>(carry);
                                carry >>= bitsInBlock;
                        }
                        // enforce precondition for fast comparison by properly nulling bits that are outside of nbits
                        sum._block[MSU] = static_cast<bt>(MSU_MASK & sum._block[MSU]);
        #if INTEGER_THROW_ARITHMETIC_EXCEPTION
                        // TODO: what is the real overflow condition?
                        // it is not carry == 1 as  say 1 + -1 sets the carry but is 0
                //                if (carry) throw integer_overflow();
        #endif
                        *this = sum;
                }
                return *this;
        }
        constexpr integer& operator-=(const integer& rhs) {
                if constexpr (NumberType == WholeNumber || NumberType == NaturalNumber) {
                        // Both modes are unsigned-domain. The semantic preconditions
                        // (WholeNumber: result must be >= 1; NaturalNumber: result >= 0)
                        // are checked only when INTEGER_THROW_ARITHMETIC_EXCEPTION is on
                        // -- matching the convention used by operator/= and convert_*.
                        //
                        // Under THROW=0 (the default) the magnitude subtractor runs
                        // unconditionally. This is necessary because internal algorithms
                        // (long division at line 1598/1632, decimal conversion via the
                        // quotient/remainder loop) call -= with operands that may be
                        // equal or that may produce an out-of-domain modular result;
                        // throwing in those paths would break printing and division.
                        // Out-of-domain user code under THROW=0 gets defined modular
                        // wraparound, the same convention IntegerNumber uses.
#if INTEGER_THROW_ARITHMETIC_EXCEPTION
                        if (*this < rhs) {
                                throw integer_wholenumber_cannot_be_negative{};
                        }
                        if constexpr (NumberType == WholeNumber) {
                                if (*this == rhs) {
                                        throw integer_wholenumber_cannot_be_zero{};
                                }
                        }
#endif
                        // Magnitude subtraction with limb-wise borrow propagation.
                        // constexpr-safe: only bt-typed integer arithmetic.
                        //
                        // Borrow-out logic per limb:
                        //   incoming borrow == 0: borrow-out iff a < b
                        //   incoming borrow == 1: borrow-out iff a <= b  (a - b - 1 wraps when a == b too)
                        // This formulation works uniformly for bt = uint8/16/32/64 without
                        // overflow concerns from widening b + borrow at the largest limb size.
                        unsigned borrow = 0;
                        for (unsigned i = 0; i < nrBlocks; ++i) {
                                bt a = _block[i];
                                bt b = rhs._block[i];
                                bt diff = static_cast<bt>(a - b - borrow);
                                borrow = (borrow == 0) ? (a < b ? 1u : 0u)
                                                       : (a <= b ? 1u : 0u);
                                _block[i] = diff;
                        }
                        // Clear bits above nbits in the most significant unit.
                        _block[MSU] = static_cast<bt>(_block[MSU] & MSU_MASK);
                }
                else {
                        integer twos(rhs);
                        operator+=(twos.twosComplement());
                }
                return *this;
        }
        constexpr integer& operator*=(const integer& rhs) {
                if constexpr (NumberType == IntegerNumberType::IntegerNumber) {
                        if constexpr (nrBlocks == 1) {
                                _block[0] = static_cast<bt>(_block[0] * rhs.block(0));
                        }
                        else if constexpr (bitsInBlock == 64) {
                                // uint64_t limbs: mul128/addcarry use platform intrinsics that are
                                // not constexpr on MSVC. In constant-evaluated context, fall back
                                // to a portable double-loop using the index-loop formulation
                                // (same as the bt < 64 branch below). At runtime we keep the
                                // intrinsic path for performance.
                                integer<nbits + 1, BlockType, NumberType> base(*this);
                                integer<nbits + 1, BlockType, NumberType> multiplicant(rhs);
                                bool resultIsNeg = (base.isneg() ^ multiplicant.isneg());
                                if (base.isneg()) {
                                        base.twosComplement();
                                }
                                if (multiplicant.isneg()) {
                                        multiplicant.twosComplement();
                                }
                                clear();
                                if (std::is_constant_evaluated()) {
                                        // Constexpr-safe 64x64 -> 128 mul + carry propagation. Uses
                                        // __int128 on gcc/clang (where it is constexpr-friendly).
                                        // On MSVC (no __int128), constexpr eval of integer<N, uint64_t>
                                        // is unsupported -- callers can use uint32_t limbs there.
#if defined(__SIZEOF_INT128__)
                                        for (unsigned i = 0; i < nrBlocks; ++i) {
                                                uint128_t carry = 0;
                                                for (unsigned j = 0; j < nrBlocks; ++j) {
                                                        if (i + j < nrBlocks) {
                                                                uint128_t product = static_cast<uint128_t>(base.block(i)) * multiplicant.block(j);
                                                                uint128_t sum = product + _block[i + j] + carry;
                                                                _block[i + j] = static_cast<bt>(sum);
                                                                carry = sum >> 64;
                                                        }
                                                }
                                        }
#else
                                        // MSVC fallback (or any platform without __int128): truncating
                                        // fallback. Documented limitation: uint64-limb integer<>
                                        // constexpr arithmetic loses high-half of partial products.
                                        for (unsigned i = 0; i < static_cast<unsigned>(nrBlocks); ++i) {
                                                std::uint64_t segment(0);
                                                for (unsigned j = 0; j < static_cast<unsigned>(nrBlocks); ++j) {
                                                        segment += static_cast<std::uint64_t>(base.block(i)) * static_cast<std::uint64_t>(multiplicant.block(j));
                                                        if (i + j < static_cast<unsigned>(nrBlocks)) {
                                                                segment += _block[i + j];
                                                                _block[i + j] = static_cast<bt>(segment);
                                                                segment = 0;
                                                        }
                                                }
                                        }
#endif
                                }
                                else {
                                        for (unsigned i = 0; i < nrBlocks; ++i) {
                                                uint64_t carry = 0;
                                                for (unsigned j = 0; j < nrBlocks; ++j) {
                                                        if (i + j < nrBlocks) {
                                                                uint64_t lo, hi;
                                                                mul128(base.block(i), multiplicant.block(j), lo, hi);
                                                                uint64_t c1 = 0;
                                                                uint64_t sum = addcarry(_block[i + j], lo, carry, c1);
                                                                _block[i + j] = sum;
                                                                carry = hi + c1;
                                                        }
                                                }
                                        }
                                }
                                if (resultIsNeg) twosComplement();
                        }
                        else {
                                // is there a better way than upconverting to deal with maxneg in a 2's complement encoding?
                                integer<nbits + 1, BlockType, NumberType> base(*this);
                                integer<nbits + 1, BlockType, NumberType> multiplicant(rhs);
                                bool resultIsNeg = (base.isneg() ^ multiplicant.isneg());
                                if (base.isneg()) {
                                        base.twosComplement();
                                }
                                if (multiplicant.isneg()) {
                                        multiplicant.twosComplement();
                                }
                                clear();
                                for (unsigned i = 0; i < static_cast<unsigned>(nrBlocks); ++i) {
                                        std::uint64_t segment(0);
                                        for (unsigned j = 0; j < static_cast<unsigned>(nrBlocks); ++j) {
                                                segment += static_cast<std::uint64_t>(base.block(i)) * static_cast<std::uint64_t>(multiplicant.block(j));

                                                if (i + j < static_cast<unsigned>(nrBlocks)) {
                                                        segment += _block[i + j];
                                                        _block[i + j] = static_cast<bt>(segment);
                                                        segment >>= bitsInBlock;
                                                }
                                        }
                                }
                                if (resultIsNeg) twosComplement();
                        }
                }
                else {  // whole and natural numbers are closed under multiplication (modulo)
                        if constexpr (nrBlocks == 1) {
                                _block[0] = static_cast<bt>(_block[0] * rhs.block(0));
                        }
                        else if constexpr (bitsInBlock == 64) {
                                // Same is_constant_evaluated dispatch as the IntegerNumber branch
                                // above to keep mul128/addcarry intrinsics out of constexpr context.
                                integer<nbits, BlockType, NumberType> base(*this), multiplicant(rhs);
                                clear();
                                if (std::is_constant_evaluated()) {
                                        // Same __int128 carry propagation as the IntegerNumber branch above
#if defined(__SIZEOF_INT128__)
                                        for (unsigned i = 0; i < nrBlocks; ++i) {
                                                uint128_t carry = 0;
                                                for (unsigned j = 0; j < nrBlocks; ++j) {
                                                        if (i + j < nrBlocks) {
                                                                uint128_t product = static_cast<uint128_t>(base.block(i)) * multiplicant.block(j);
                                                                uint128_t sum = product + _block[i + j] + carry;
                                                                _block[i + j] = static_cast<bt>(sum);
                                                                carry = sum >> 64;
                                                        }
                                                }
                                        }
#else
                                        for (unsigned i = 0; i < static_cast<unsigned>(nrBlocks); ++i) {
                                                std::uint64_t segment(0);
                                                for (unsigned j = 0; j < static_cast<unsigned>(nrBlocks); ++j) {
                                                        segment += static_cast<std::uint64_t>(base.block(i)) * static_cast<std::uint64_t>(multiplicant.block(j));
                                                        if (i + j < static_cast<unsigned>(nrBlocks)) {
                                                                segment += _block[i + j];
                                                                _block[i + j] = static_cast<bt>(segment);
                                                                segment = 0;
                                                        }
                                                }
                                        }
#endif
                                }
                                else {
                                        for (unsigned i = 0; i < nrBlocks; ++i) {
                                                uint64_t carry = 0;
                                                for (unsigned j = 0; j < nrBlocks; ++j) {
                                                        if (i + j < nrBlocks) {
                                                                uint64_t lo, hi;
                                                                mul128(base.block(i), multiplicant.block(j), lo, hi);
                                                                uint64_t c1 = 0;
                                                                uint64_t sum = addcarry(_block[i + j], lo, carry, c1);
                                                                _block[i + j] = sum;
                                                                carry = hi + c1;
                                                        }
                                                }
                                        }
                                }
                        }
                        else {
                                integer<nbits, BlockType, NumberType> base(*this), multiplicant(rhs);
                                clear();
                                for (unsigned i = 0; i < static_cast<unsigned>(nrBlocks); ++i) {
                                        std::uint64_t segment(0);
                                        for (unsigned j = 0; j < static_cast<unsigned>(nrBlocks); ++j) {
                                                segment += static_cast<std::uint64_t>(base.block(i)) * static_cast<std::uint64_t>(multiplicant.block(j));

                                                if (i + j < static_cast<unsigned>(nrBlocks)) {
                                                        segment += _block[i + j];
                                                        _block[i + j] = static_cast<bt>(segment);
                                                        segment >>= bitsInBlock;
                                                }
                                        }
                                }
                        }
                }
                // null any leading bits that fall outside of nbits
                _block[MSU] = static_cast<bt>(MSU_MASK & _block[MSU]);
                return *this;
        }
        constexpr integer& operator*=(const BlockType& scale) noexcept {
                if constexpr (bitsInBlock == 64) {
                        // uint64_t limbs: same mul128/addcarry dispatch as operator*=
                        if (std::is_constant_evaluated()) {
#if defined(__SIZEOF_INT128__)
                                uint128_t carry = 0;
                                for (unsigned i = 0; i < nrBlocks; ++i) {
                                        uint128_t product = static_cast<uint128_t>(_block[i]) * scale + carry;
                                        _block[i] = static_cast<BlockType>(product);
                                        carry = product >> 64;
                                }
#else
                                std::uint64_t scaleFactor(scale), segment(0);
                                for (unsigned i = 0; i < nrBlocks; ++i) {
                                        segment += static_cast<std::uint64_t>(_block[i]) * scaleFactor;
                                        _block[i] = static_cast<BlockType>(segment);
                                        segment = 0;
                                }
#endif
                        }
                        else {
                                uint64_t carry = 0;
                                for (unsigned i = 0; i < nrBlocks; ++i) {
                                        uint64_t lo, hi;
                                        mul128(_block[i], static_cast<uint64_t>(scale), lo, hi);
                                        uint64_t c1 = 0;
                                        _block[i] = addcarry(lo, carry, uint64_t(0), c1);
                                        carry = hi + c1;
                                }
                        }
                }
                else {
                        std::uint64_t scaleFactor(scale), segment(0);
                        for (unsigned i = 0; i < nrBlocks; ++i) {
                                segment += static_cast<std::uint64_t>(_block[i]) * scaleFactor;
                                _block[i] = static_cast<BlockType>(segment);
                                segment >>= bitsInBlock;
                        }
                }
                return *this;
        }
        constexpr integer& operator/=(const integer& rhs) {
                if constexpr (EXACT_FIT && 1 == nrBlocks) {
                        if (rhs._block[0] == 0) {
#if INTEGER_THROW_ARITHMETIC_EXCEPTION
                                throw integer_divide_by_zero{};
#else
                                // When exceptions are disabled we still must avoid the actual
                                // divide-by-zero below (UB at runtime; constexpr eval would
                                // be ill-formed). Set the result to 0 and bail out.
                                if (!std::is_constant_evaluated()) {
                                        std::cerr << "integer_divide_by_zero\n";
                                }
                                clear();
                                return *this;
#endif // INTEGER_THROW_ARITHMETIC_EXCEPTION
                        }
                        if constexpr (NumberType == WholeNumber) {
                                if (*this < rhs) {
#if INTEGER_THROW_ARITHMETIC_EXCEPTION
                                        throw integer_wholenumber_cannot_be_zero{};
#else
                                        if (!std::is_constant_evaluated()) {
                                                std::cerr << "whole number cannot be zero but division would yield 0\n";
                                        }
                                        clear();
                                        return *this;
#endif // INTEGER_THROW_ARITHMETIC_EXCEPTION
                                }
                        }
                        if constexpr (sizeof(BlockType) == 1) {
                                _block[0] = static_cast<bt>(std::int8_t(_block[0]) / std::int8_t(rhs._block[0]));
                        }
                        else if constexpr (sizeof(BlockType) == 2) {
                                _block[0] = static_cast<bt>(std::int16_t(_block[0]) / std::int16_t(rhs._block[0]));
                        }
                        else if constexpr (sizeof(BlockType) == 4) {
                                _block[0] = static_cast<bt>(std::int32_t(_block[0]) / std::int32_t(rhs._block[0]));
                        }
                        else if constexpr (sizeof(BlockType) == 8) {
                                _block[0] = static_cast<bt>(std::int64_t(_block[0]) / std::int64_t(rhs._block[0]));                
                        }
                        _block[0] = static_cast<bt>(MSU_MASK & _block[0]);
                }
                else {
                        idiv_t<nbits, BlockType, NumberType> divresult = idiv<nbits, BlockType, NumberType>(*this, rhs);
                        *this = divresult.quot;
                }
                return *this;
        }
        constexpr integer& operator%=(const integer& rhs) {
                if constexpr (nbits == (sizeof(BlockType) * 8)) {
                        if (rhs._block[0] == 0) {
#if INTEGER_THROW_ARITHMETIC_EXCEPTION
                                throw integer_divide_by_zero{};
#else
                                // When exceptions are disabled we still must avoid the actual
                                // modulo-by-zero below (UB at runtime; constexpr would be
                                // ill-formed). Set the result to 0 and bail out.
                                if (!std::is_constant_evaluated()) {
                                        std::cerr << "integer_divide_by_zero\n";
                                }
                                clear();
                                return *this;
#endif // INTEGER_THROW_ARITHMETIC_EXCEPTION
                        }
                        if constexpr (sizeof(BlockType) == 1) {
                                _block[0] = static_cast<bt>(std::int8_t(_block[0]) % std::int8_t(rhs._block[0]));
                        }
                        else if constexpr (sizeof(BlockType) == 2) {
                                _block[0] = static_cast<bt>(std::int16_t(_block[0]) % std::int16_t(rhs._block[0]));
                        }
                        else if constexpr (sizeof(BlockType) == 4) {
                                _block[0] = static_cast<bt>(std::int32_t(_block[0]) % std::int32_t(rhs._block[0]));
                        }
                        else if constexpr (sizeof(BlockType) == 8) {
                                _block[0] = static_cast<bt>(std::int64_t(_block[0]) % std::int64_t(rhs._block[0]));
                        }
                        _block[0] = static_cast<bt>(MSU_MASK & _block[0]);
                }
                else {
                        idiv_t<nbits, BlockType, NumberType> divresult = idiv<nbits, BlockType, NumberType>(*this, rhs);
                        *this = divresult.rem;
                }

                return *this;
        }

        // arithmetic shift right operator
        constexpr integer& operator<<=(int bitsToShift) {
                if (bitsToShift == 0) return *this;
                if (bitsToShift < 0) return operator>>=(-bitsToShift);
                if (bitsToShift > static_cast<int>(nbits)) {
                        setzero();
                        return *this;
                }
                if (bitsToShift >= static_cast<int>(bitsInBlock)) {
                        int blockShift = bitsToShift / static_cast<int>(bitsInBlock);
                        for (int i = static_cast<int>(MSU); i >= blockShift; --i) {
                                _block[i] = bt(_block[i - blockShift]);
                        }
                        for (int i = blockShift - 1; i >= 0; --i) {
                                _block[i] = bt(0);
                        }
                        // adjust the shift
                        bitsToShift -= static_cast<int>(static_cast<unsigned>(blockShift) * bitsInBlock);
                        if (bitsToShift == 0) {
                                _block[MSU] &= MSU_MASK;
                                return *this;
                        }
                }
                if constexpr (MSU > 0) {
                        // construct the mask for the upper bits in the block that needs to move to the higher word
                        // bitsToShift is > 0 here (guarded above); the shared helper also states the
                        // mask truncation explicitly (bit_high_mask, #1261) and the cast removes the
                        // unsigned/int mixing in the index arithmetic (#1283).
                        bt mask = bit_high_mask<bt>(static_cast<unsigned>(bitsToShift), bitsInBlock);
                        for (unsigned i = MSU; i > 0; --i) {
                                _block[i] <<= bitsToShift;
                                // mix in the bits from the right
                                bt bits = bt(mask & _block[i - 1]);
                                _block[i] |= (bits >> (bitsInBlock - static_cast<unsigned>(bitsToShift)));
                        }
                }
                _block[0] <<= bitsToShift;        
                _block[MSU] &= MSU_MASK; // null any leading bits that fall outside of nbits
                return *this;
        }
        constexpr integer& operator>>=(int bitsToShift) {
                if (bitsToShift == 0) return *this;
                if (bitsToShift < 0) return operator<<=(-bitsToShift);
                if (bitsToShift >= static_cast<int>(nbits)) {
                        setzero();
                        return *this;
                }
                bool signext = sign();
                unsigned blockShift = 0;
                if (bitsToShift >= static_cast<int>(bitsInBlock)) {
                        blockShift = static_cast<unsigned>(bitsToShift) / bitsInBlock;
                        if (MSU >= blockShift) {
                                // shift by blocks
                                for (unsigned i = 0; i <= MSU - blockShift; ++i) {
                                        _block[i] = bt(_block[i + blockShift]);
                                }
                        }
                        // adjust the shift
                        bitsToShift -= static_cast<int>(blockShift * bitsInBlock);
                        if (bitsToShift == 0) {
                                // fix up the leading zeros if we have a negative number
                                if (signext) {
                                        // bitsToShift is guaranteed to be less than nbits
                                        bitsToShift += static_cast<int>(blockShift * bitsInBlock);
                                        for (unsigned i = nbits - static_cast<unsigned>(bitsToShift); i < nbits; ++i) {
                                                setbit(i);
                                        }
                                }
                                else {
                                        // clean up the blocks we have shifted clean
                                        bitsToShift += static_cast<int>(blockShift * bitsInBlock);
                                        for (unsigned i = nbits - static_cast<unsigned>(bitsToShift); i < nbits; ++i) {
                                                setbit(i, false);
                                        }
                                }
                                return *this;
                        }
                }
                if constexpr (MSU > 0) {
                        bt mask = ALL_ONES;
                        // mask for the lower bits in the block that need to move to the lower word
                        mask >>= (bitsInBlock - static_cast<unsigned>(bitsToShift));
                        // TODO: can this be improved? we should not have to work on the upper
                        // blocks in case we block shifted
                        for (unsigned i = 0; i < MSU; ++i) {
                                _block[i] >>= bitsToShift;
                                // mix in the bits from the left
                                bt bits = bt(mask & _block[i + 1]);
                                _block[i] |= (bits << (bitsInBlock - static_cast<unsigned>(bitsToShift)));
                        }
                }
                _block[MSU] >>= bitsToShift;

                // fix up the leading zeros if we have a negative number
                if (signext) {
                        // bitsToShift is guaranteed to be less than nbits
                        bitsToShift += static_cast<int>(blockShift * bitsInBlock);
                        for (unsigned i = nbits - static_cast<unsigned>(bitsToShift); i < nbits; ++i) {
                                setbit(i);
                        }
                }
                else {
                        // clean up the blocks we have shifted clean
                        bitsToShift += static_cast<int>(blockShift * bitsInBlock);
                        for (unsigned i = nbits - static_cast<unsigned>(bitsToShift); i < nbits; ++i) {
                                setbit(i, false);
                        }
                }

                // enforce precondition for fast comparison by properly nulling bits that are outside of nbits
                _block[MSU] &= MSU_MASK;
                return *this;
        }
        constexpr integer& logicShiftRight(int shift) {
                if (shift == 0) return *this;
                if (shift < 0) {
                        return operator<<=(-shift);
                }
                if (nbits <= unsigned(shift)) {
                        clear();
                        return *this;
                }
                integer<nbits, BlockType, NumberType> target{};
                for (int i = nbits - 1; i >= shift; --i) {  // TODO: inefficient as it works at the bit level
                        target.setbit(static_cast<unsigned>(i - shift), at(static_cast<unsigned>(i)));
                }
                *this = target;
                return *this;
        }
        constexpr integer& operator&=(const integer& rhs) {
                for (unsigned i = 0; i < nrBlocks; ++i) {
                        _block[i] &= rhs._block[i];
                }
                _block[MSU] &= MSU_MASK;
                return *this;
        }
        constexpr integer& operator|=(const integer& rhs) {
                for (unsigned i = 0; i < nrBlocks; ++i) {
                        _block[i] |= rhs._block[i];
                }
                _block[MSU] &= MSU_MASK;
                return *this;
        }
        constexpr integer& operator^=(const integer& rhs) {
                for (unsigned i = 0; i < nrBlocks; ++i) {
                        _block[i] ^= rhs._block[i];
                }
                _block[MSU] &= MSU_MASK;
                return *this;
        }

        // modifiers
        constexpr void clear() noexcept { 
                bt* p = _block;
                if constexpr (0 == nrBlocks) {
                        return;
                }
                else if constexpr (1 == nrBlocks) {
                        *p = bt(0);
                }
                else if constexpr (2 == nrBlocks) {
                        *p++ = bt(0);
                        *p = bt(0);
                }
                else if constexpr (3 == nrBlocks) {
                        *p++ = bt(0);
                        *p++ = bt(0);
                        *p   = bt(0);
                }
                else if constexpr (4 == nrBlocks) {
                        *p++ = bt(0);
                        *p++ = bt(0);
                        *p++ = bt(0);
                        *p   = bt(0);
                }
                else {
                        for (unsigned i = 0; i < nrBlocks; ++i) {
                                *p++ = bt(0);
                        }
                }
        }
        constexpr void setzero() noexcept { clear(); }
        constexpr integer& maxpos() noexcept {
                clear();
                setbit(nbits - 1ull, true);
                flip();
                return *this;
        }
        constexpr integer& minpos() noexcept {
                clear();
                setbit(0, true);
                return *this;
        }
        constexpr integer& zero() noexcept {
                clear();
                return *this;
        }
        constexpr integer& minneg() noexcept {
                clear();
                flip();
                return *this;
        }
        constexpr integer& maxneg() noexcept {
                clear();
                setbit(nbits - 1ull, true);
                return *this;
        }
        constexpr void setbit(unsigned i, bool v = true) noexcept {
                unsigned blockIndex = i / bitsInBlock;
                if (blockIndex < nrBlocks) {
                        bt block = _block[blockIndex];
                        bt null = bit_clear_mask<bt>(i, bitsInBlock);
                        bt bit = bt(v ? 1 : 0);
                        bt mask = bt(bit << (i % bitsInBlock));
                        _block[blockIndex] = bt((block & null) | mask);
                }
                // nop if out of bounds
        }
        constexpr void setbyte(unsigned byteIndex, uint8_t data) {
                uint8_t mask = 0x1u;
                unsigned start = byteIndex * 8;
                unsigned end = start + 8;
                for (unsigned i = start; i < end; ++i) {
                        setbit(i, static_cast<bool>(mask & data));
                        mask <<= 1;
                }
        }
        constexpr void setblock(unsigned i, bt value) noexcept {
                if (i < nrBlocks) _block[i] = value;
        }
        // use un-interpreted raw bits to set the bits of the integer
        constexpr integer& setbits(uint64_t raw_bits) noexcept {
                if constexpr (0 == nrBlocks) {
                        return *this;
                }
                else if constexpr (1 == nrBlocks) {
                        _block[0] = raw_bits & storageMask;
                }
                else if constexpr (2 == nrBlocks) {
                        if constexpr (bitsInBlock < 64) {
                                _block[0] = raw_bits & storageMask;
                                raw_bits >>= bitsInBlock;
                                _block[1] = raw_bits & storageMask;
                        }
                        else {
                                _block[0] = raw_bits & storageMask;
                                _block[1] = 0;
                        }
                }
                else if constexpr (3 == nrBlocks) {
                        if constexpr (bitsInBlock < 64) {
                                _block[0] = raw_bits & storageMask;
                                raw_bits >>= bitsInBlock;
                                _block[1] = raw_bits & storageMask;
                                raw_bits >>= bitsInBlock;
                                _block[2] = raw_bits & storageMask;
                        }
                        else {
                                _block[0] = raw_bits & storageMask;
                                _block[1] = 0;
                                _block[2] = 0;
                        }
                }
                else if constexpr (4 == nrBlocks) {
                        if constexpr (bitsInBlock < 64) {
                                _block[0] = raw_bits & storageMask;
                                raw_bits >>= bitsInBlock;
                                _block[1] = raw_bits & storageMask;
                                raw_bits >>= bitsInBlock;
                                _block[2] = raw_bits & storageMask;
                                raw_bits >>= bitsInBlock;
                                _block[3] = raw_bits & storageMask;
                        }
                        else {
                                _block[0] = raw_bits & storageMask;
                                _block[1] = 0;
                                _block[2] = 0;
                                _block[3] = 0;
                        }
                }
                else {
                        if constexpr (bitsInBlock < 64) {
                                for (unsigned i = 0; i < nrBlocks; ++i) {
                                        _block[i] = raw_bits & storageMask;
                                        raw_bits >>= bitsInBlock;
                                }
                        }
                        else {
                                _block[0] = raw_bits & storageMask;
                                for (unsigned i = 1; i < nrBlocks; ++i) {
                                        _block[i] = 0;
                                }
                        }
                }
                _block[MSU] &= MSU_MASK; // enforce precondition for fast comparison by properly nulling bits that are outside of nbits
                return *this;
        }
        integer& assign(const std::string& txt) noexcept {
                if (!parse(txt, *this)) {
                        std::cerr << "Unable to parse: " << txt << std::endl;
                }
                // enforce precondition for fast comparison by properly nulling bits that are outside of nbits
                _block[MSU] = static_cast<BlockType>(MSU_MASK & _block[MSU]);
                return *this;
        }
        // pure bit copy of source integer, no sign extension
        template<unsigned src_nbits>
        constexpr void bitcopy(const integer<src_nbits, BlockType, NumberType>& src) noexcept {
                // no need to clear as we are going to overwrite all blocks
                for (unsigned i = 0; i < nrBlocks; ++i) { // use nrBlocks of receiver even when src is smaller, src.block() will return 0 for blocks it doesn't have, nulling the receiver's blocks
                        _block[i] = src.block(i);
                }
                _block[MSU] &= MSU_MASK; // assert precondition of properly nulled leading non-bits
        }
        // in-place one's complement
        constexpr integer& flip() {
                for (unsigned i = 0; i < nrBlocks; ++i) {
                        _block[i] = static_cast<bt>(~_block[i]);
                }
                _block[MSU] = static_cast<bt>(_block[MSU] & MSU_MASK); // assert precondition of properly nulled leading non-bits
                return *this;
        }
        // in-place 2's complement
        constexpr integer& twosComplement() {
                flip();
                return ++(*this);
        }

        // selectors
        constexpr bool iszero() const noexcept {
                for (unsigned i = 0; i < nrBlocks; ++i) {
                        if (_block[i] != 0) return false;
                }
                return true;
        }
        constexpr bool ispos()  const noexcept { if constexpr (NumberType == IntegerNumberType::IntegerNumber) return *this > 0; else return true; }
        constexpr bool isneg()  const noexcept { if constexpr (NumberType == IntegerNumberType::IntegerNumber) return *this < 0; else return false; }
        constexpr bool isone()  const noexcept {
                for (unsigned i = 0; i < nrBlocks; ++i) {
                        if (i == 0) {
                                if (_block[0] != BlockType(1u)) return false;
                        }
                        else {
                                if (_block[i] != BlockType(0u)) return false;
                        }
                }
                return true;
        }
        constexpr bool isodd()  const noexcept  { return bool(_block[0] & 0x01); }
        constexpr bool iseven() const noexcept { return !isodd(); }
        constexpr bool sign()   const noexcept { return at(nbits - 1); }
        constexpr bool at(unsigned bitIndex) const noexcept {
                if (bitIndex < nbits) {
                        bt word = _block[bitIndex / bitsInBlock];
                        bt mask = bt(1ull << (bitIndex % bitsInBlock));
                        return (word & mask);
                }
                return false;
        }
        constexpr bool test(unsigned i)  const noexcept { return at(i); }
        constexpr bt   block(unsigned i) const noexcept { if (i < nrBlocks) return _block[i]; else return bt(0u); }
        constexpr uint8_t nibble(unsigned n) const noexcept {
                if (n < (1 + ((nbits - 1) >> 2))) {
                        bt word = _block[(n * 4) / bitsInBlock];
                        int nibbleIndexInWord = int(n % (bitsInBlock >> 2ull));
                        bt mask = bt(0xF << (nibbleIndexInWord * 4));
                        bt nibblebits = bt(mask & word);
                        return uint8_t(nibblebits >> (nibbleIndexInWord * 4));
                }
                return 0;
        }
        // normalize: decompose integer value into a blocktriple<nbits-1, REP> for quire accumulation
        template<typename TargetBlockType = BlockType>
        void normalize(blocktriple<nbits - 1, BlockTripleOperator::REP, TargetBlockType>& tgt) const {
                if (iszero()) { tgt.setzero(); return; }
                tgt.setzero();
                tgt.setnormal();
                // For WholeNumber/NaturalNumber, sign() returns the raw MSB which is a data bit,
                // not a sign indicator. Only IntegerNumber has a true sign bit.
                const bool negative = (NumberType == IntegerNumberType::IntegerNumber) && sign();
                tgt.setsign(negative);
                // get magnitude
                integer mag = negative ? -(*this) : *this;
                signed msb = findMsb(mag);
                tgt.setscale(msb);  // integer: scale = position of MSB
                // copy magnitude bits into significand
                constexpr unsigned f = nbits - 1;
                tgt.setbit(f);  // hidden bit
                for (signed i = msb - 1; i >= 0 && (msb - 1 - i) < static_cast<signed>(f); --i) {
                        tgt.setbit(static_cast<unsigned>(f - 1 - (msb - 1 - i)), mag.at(static_cast<unsigned>(i)));
                }
        }

        // operators
        // reduce returns the ratio and remainder of a and b in *this and r
        void reduce(const integer& a, const integer& b, integer& r) {
                if (b.iszero()) {
#if INTEGER_THROW_ARITHMETIC_EXCEPTION
                        throw integer_divide_by_zero{};
#else
                        std::cerr << "integer_divide_by_zero\n";
                        return;
#endif // INTEGER_THROW_ARITHMETIC_EXCEPTION
                }

                if (a.iszero()) {
                        clear();
                        r.clear();
                        return;
                }
                if constexpr (nrBlocks == 1) { // completely reduce this to native div and rem
                        BlockType _a = a._block[0];
                        BlockType _b = b._block[0];
                        if constexpr (NumberType == IntegerNumber) {
                                bool sign_a = _a & SIGN_BIT_MASK;
                                bool sign_b = _b & SIGN_BIT_MASK;
                                if constexpr (8 == bitsInBlock) {
                                        std::int8_t a0 = (sign_a ? SIGN_EXTENTION_BITS | _a : _a);
                                        std::int8_t b0 = (sign_b ? SIGN_EXTENTION_BITS | _b : _b);
                                        *this = static_cast<BlockType>(a0 / b0);
                                        r = static_cast<BlockType>(a0 % b0);
                                }
                                else if constexpr (16 == bitsInBlock) {
                                        std::int16_t a0 = (sign_a ? SIGN_EXTENTION_BITS | _a : _a);
                                        std::int16_t b0 = (sign_b ? SIGN_EXTENTION_BITS | _b : _b);
                                        *this = static_cast<BlockType>(a0 / b0);
                                        r = static_cast<BlockType>(a0 % b0);
                                }
                                else if constexpr (32 == bitsInBlock) {
                                        std::int32_t a0 = (sign_a ? SIGN_EXTENTION_BITS | _a : _a);
                                        std::int32_t b0 = (sign_b ? SIGN_EXTENTION_BITS | _b : _b);
                                        *this = static_cast<BlockType>(a0 / b0);
                                        r = static_cast<BlockType>(a0 % b0);
                                }
                                else {
                                        std::int64_t a0 = (sign_a ? SIGN_EXTENTION_BITS | _a : _a);
                                        std::int64_t b0 = (sign_b ? SIGN_EXTENTION_BITS | _b : _b);
                                        *this = static_cast<BlockType>(a0 / b0);
                                        r = static_cast<BlockType>(a0 % b0);
                                }
                        }
                        else {
                                *this = static_cast<BlockType>(_a / _b);
                                r = static_cast<BlockType>(_a % _b);
                        }
                }
                else {
                        clear();
                        // no need to constexpr guard this for IntegerNumber as sign() will return false for Whole and Natural Numbers
                        bool sign_a = a.sign();
                        bool sign_b = b.sign();
                        bool sign_q = sign_a ^ sign_b;
                        using Integer = integer<nbits+1, BlockType, NumberType>; // nbits+1 to deal with maxneg
                        Integer _a(a), _b(b);
                        if (sign_a) _a.twosComplement();
                        if (sign_b) _b.twosComplement();                        
                        
                        // filter out the easy stuff
                        if (_a < _b) { r = a; clear(); return; }

                        // determine first non-zero limbs
                        unsigned m{ 0 }, n{ 0 };
                        for (unsigned i = nrBlocks; i > 0; --i) {
                                if (_a.block(i - 1) != 0) {
                                        m = static_cast<unsigned>(i);
                                        break;
                                }
                        }
                        for (unsigned i = nrBlocks; i > 0; --i) {
                                if (_b.block(i - 1) != 0) {
                                        n = static_cast<unsigned>(i);
                                        break;
                                }
                        }

                        // single limb divisor
                        if (n == 1) {
                                std::uint64_t remainder{ 0 };
                                auto divisor = _b.block(0);
                                for (unsigned j = m; j > 0; --j) {
                                        std::uint64_t dividend = remainder * BASE + _a.block(j - 1);
                                        std::uint64_t limbQuotient = dividend / divisor;
                                        _block[j - 1] = static_cast<BlockType>(limbQuotient);
                                        remainder = dividend - limbQuotient * divisor;
                                }
                                r._block[0] = static_cast<BlockType>(remainder);
                                if (sign_q) twosComplement();
                                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

                        using OriginalInteger = integer<nbits, BlockType, NumberType>;
                        using ExpandedInteger = integer<nbits + sizeof(BlockType) * 8, BlockType, NumberType>; // need room for overflow to receive the normalization bits

                        int shift = nlz(b.block(n - 1));
                        ExpandedInteger 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
                        OriginalInteger 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;
                                }
                                std::uint64_t borrow{ 0 };
                                std::uint64_t diff{ 0 };
                                for (unsigned i = 0; i < n; ++i) {
                                        std::uint64_t p = qhat * normalized_b.block(i);
                                        diff = normalized_a.block(i + j) - static_cast<BlockType>(p) - borrow;
                                        normalized_a.setblock(i + j, static_cast<BlockType>(diff));
                                        borrow = (p >> bitsInBlock) - (diff >> bitsInBlock);
                                }
                                std::int64_t signedBorrow = static_cast<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
                                        std::cout << "subtracted too much, add back\n";
                                        _block[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 >>= 32;
                                        }
                                        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';
                        }
                        if (sign_q) twosComplement();

                        // remainder needs to be normalized
                        for (unsigned i = 0; i < n - 1; ++i) {
                                std::uint64_t remainder = static_cast<uint64_t>(normalized_a.block(i) >> shift);
                                remainder |= static_cast<uint64_t>(normalized_a.block(i + 1) << (32 - shift));
                                r.setblock(i, static_cast<BlockType>(remainder));
                        }
                        r.setblock(n - 1, static_cast<BlockType>(normalized_a.block(n - 1) >> shift));
                }
        }
        // signed integer conversion
        template<typename SignedInt>
        constexpr integer& convert_signed(SignedInt rhs) {
                clear();
                if (0 == rhs) {
                        if constexpr (NumberType == WholeNumber) {
#if INTEGER_THROW_ARITHMETIC_EXCEPTION
                                throw integer_wholenumber_cannot_be_zero();
#else
                                return *this;
#endif
                        }
                        else {
                                return *this;
                        }
                }
                if constexpr (NumberType == WholeNumber) {
#if INTEGER_THROW_ARITHMETIC_EXCEPTION
                        if (rhs < 0) throw integer_wholenumber_cannot_be_negative();
#else
                        if (rhs < 0) return *this;
#endif
                }
                if constexpr (NumberType == NaturalNumber) {
#if INTEGER_THROW_ARITHMETIC_EXCEPTION
                        if (rhs < 0) throw integer_naturalnumber_cannot_be_negative();
#else
                        if (rhs < 0) return *this;
#endif
                }

                int64_t v = rhs;
                for (unsigned i = 0; i < nbits && v != 0; ++i) {
                        if ((v & 1) != 0) setbit(i);   // low-bit test; & with signed 1 avoids int64->uint64
                        v >>= 1;
                }
                if constexpr (nbits > 64) {
                        if (rhs < 0) {        // sign extend if negative
                                for (unsigned i = 64; i < nbits; ++i) {
                                        setbit(i);
                                }
                        }
                }
                return *this;
        }
        // unsigned integer conversion
        template<typename UnsignedInt>
        constexpr integer& convert_unsigned(UnsignedInt rhs) {
                clear();
                if (0 == rhs) {
                        if constexpr (NumberType == WholeNumber) {
#if INTEGER_THROW_ARITHMETIC_EXCEPTION
                                throw integer_wholenumber_cannot_be_zero();
#else
                                return *this;
#endif
                        }
                        else {
                                return *this;
                        }
                }
                uint64_t v = rhs;
                constexpr unsigned argbits = sizeof(rhs);
                unsigned upper = (nbits <= _nbits ? nbits : argbits);
                for (unsigned i = 0; i < upper; ++i) {
                        if (v & 0x1ull) setbit(i);
                        v >>= 1;
                }
                return *this;
        }
        // native IEEE-754 conversion
        // TODO: currently only supports integer values of 64bits or less
        template<typename Real>
        constexpr integer& convert_ieee(Real rhs) noexcept {
                clear();
                return *this = static_cast<long long>(rhs); // TODO: this clamps the IEEE range to +-2^63
        }

        // show the binary encodings of the limbs
        std::string showLimbs() const {
                using Integer = sw::universal::integer<nbits, BlockType, NumberType>;
                std::stringstream s;
                unsigned i = Integer::MSU;
                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 {
                using Integer = sw::universal::integer<nbits, BlockType, NumberType>;
                std::stringstream s;
                unsigned i = Integer::MSU;
                while (i > 0) {
                        s << std::setw(5) << unsigned(_block[i]) << ", ";
                        --i;
                }
                s << std::setw(5) << unsigned(_block[0]);
                return s.str();
        }

protected:
        // HELPER methods

        // to_integer converts to native signed integer
        // TODO: enable_if this for integral types only
        template<typename TargetInt>
        TargetInt to_integer() const noexcept {
                TargetInt v{ 0 };
                if (iszero()) return v;  // this should only occur for Integer and Natural Numbers

                constexpr unsigned sizeoftarget   = 8 * sizeof(TargetInt);
                constexpr unsigned upperTargetBlock = (sizeoftarget - 1ul) / bitsInBlock;
                unsigned upperBlock = static_cast<unsigned>(std::min(MSU, upperTargetBlock));
                if constexpr (NumberType == IntegerNumberType::IntegerNumber) {
                        for (unsigned b = 0; b <= upperBlock; ++b) {
                                std::uint64_t data = _block[b];
                                v |= (data << (b * bitsInBlock));
                        }
                        unsigned upper = nbits;
                        if (sign() && upper < sizeoftarget) { // sign extend
                                uint64_t mask = (1ull << upper);
                                for (unsigned i = upper; i < sizeoftarget; ++i) {
                                        v |= mask;
                                        mask <<= 1;
                                }
                        }
                }
                else {
                        for (unsigned b = 0; b <= upperBlock; ++b) {
                                std::uint64_t data = _block[b];
                                v |= (data << (b * bitsInBlock));
                        }
                }

                return v;
        }

        // to_unsigned_integer converts to native unsigned integer
    // TODO: enable_if this for integral types only
        template<typename TargetInt>
        TargetInt to_unsigned_integer() const noexcept {
                TargetInt v{ 0 };
                if (iszero()) return v;  // this should only occur for Integer and Natural Numbers

                constexpr unsigned sizeoftarget = 8 * sizeof(TargetInt);
                constexpr unsigned upperTargetBlock = (sizeoftarget - 1ul) / bitsInBlock;
                unsigned upperBlock = std::min(MSU, upperTargetBlock);
                for (unsigned b = 0; b <= upperBlock; ++b) {
                        std::uint64_t data = _block[b];
                        v |= (data << (b * bitsInBlock));
                }

                return v;
        }

        // TODO: enable_if this for native floating-point types only
        template<typename Real>
        constexpr Real to_real() const noexcept {
                Real r = 0.0;
                Real bitValue = static_cast<Real>(1.0);
                if constexpr (NumberType == IntegerNumberType::IntegerNumber) {
                        integer<nbits + 1, bt, NumberType> v{ *this }; // deal with maxneg in 2's complement
                        if (isneg()) v = -v;
                        for (unsigned i = 0; i < nbits; ++i) { // upper bound is nbits + 1 - 1 == nbits
                                if (v.at(i)) r += bitValue;
                                bitValue *= static_cast<Real>(2.0);
                        }
                        if (isneg()) r = -r;
                }
                else if constexpr (NumberType == IntegerNumberType::WholeNumber) {
                        for (unsigned i = 0; i < nbits; ++i) {
                                if (at(i)) r += bitValue;
                                bitValue *= static_cast<Real>(2.0);
                        }
                }
                else { // NaturalNumber
                        if (iszero()) std::cerr << "internal error: natural number is set to 0\n";
                        for (unsigned i = 0; i < nbits; ++i) {
                                if (at(i)) r += bitValue;
                                bitValue *= static_cast<Real>(2.0);
                        }
                }

                return r;
        }

private:
        bt _block[nrBlocks];

        // convert
        template<unsigned nnbits, typename BBlockType, IntegerNumberType NNumberType>
        friend std::string convert_to_decimal_string(const integer<nnbits, BBlockType, NNumberType>& value);

        // integer - integer logic comparisons
        template<unsigned nnbits, typename BBlockType, IntegerNumberType NNumberType>
        friend constexpr bool operator==(const integer<nnbits, BBlockType, NNumberType>& lhs, const integer<nnbits, BBlockType, NNumberType>& rhs);

        // find the most significant bit set
        template<unsigned nnbits, typename BBlockType, IntegerNumberType NNumberType>
        friend constexpr signed findMsb(const integer<nnbits, BBlockType, NNumberType>& v);
};

////////////////////////    INTEGER functions   /////////////////////////////////

template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> abs(const integer<nbits, BlockType, NumberType>& a) {
        integer<nbits, BlockType, NumberType> b(a);
        return (a >= 0 ? b : b.twosComplement());
}

// free function to create a 1's complement copy of an integer
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> onesComplement(const integer<nbits, BlockType, NumberType>& value) {
        integer<nbits, BlockType, NumberType> ones(value);
        return ones.flip();
}
// free function to create the 2's complement of an integer
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> twosComplement(const integer<nbits, BlockType, NumberType>& value) {
        integer<nbits, BlockType, NumberType> twos(value);
        return twos.twosComplement();;
}

// convert integer to decimal string
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
std::string convert_to_decimal_string(const integer<nbits, BlockType, NumberType>& value) {
        if (value.iszero()) {
                return std::string("0");
        }
        integer<nbits, BlockType, NumberType> number = value.sign() ? twosComplement(value) : value;
        support::decimal partial, multiplier;
        partial.setzero();
        multiplier.setdigit(1);
        // convert integer to decimal by adding and doubling multipliers
        for (unsigned i = 0; i < nbits; ++i) {
                if (number.at(i)) {
                        support::add(partial, multiplier);
                        // std::cout << partial << std::endl;
                }
                support::add(multiplier, multiplier);
        }
        std::stringstream str;
        if (value.sign()) str << '-';
        for (support::decimal::const_reverse_iterator rit = partial.rbegin(); rit != partial.rend(); ++rit) {
                str << (int)*rit;
        }
        return str.str();
}

// findMsb takes an integer<nbits, BlockType, NumberType> reference and returns the 0-based position of the most significant bit, -1 if v == 0
// Index-based loop (constexpr-clean; pointer arithmetic on stack arrays is forbidden in constexpr).
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr signed findMsb(const integer<nbits, BlockType, NumberType>& v) {
        using IntegerT = integer<nbits, BlockType, NumberType>;
        // Access static members via the type (clang requires this in constexpr context).
        constexpr unsigned bitsInBlk = IntegerT::bitsInBlock;
        constexpr unsigned nBlks = IntegerT::nrBlocks;
        constexpr BlockType BlockMsb = BlockType(BlockType(1u) << (bitsInBlk - 1));
        signed msb = static_cast<signed>(IntegerT::nbits - 1ull); // the case for an aligned MSB
        constexpr unsigned rem = nbits % bitsInBlk;

        // little-endian: most significant block is at index nBlks - 1
        unsigned blockIdx = nBlks - 1;

        // check if the blocks are aligned with the representation
        if constexpr (rem > 0) {
                // the top bits are unaligned: construct the right mask
                BlockType mask = BlockType(BlockType(1u) << (rem - 1u));
                while (mask != 0) {
                        if (v._block[blockIdx] & mask) return msb;
                        --msb;
                        mask >>= 1;
                }
                if (msb < 0) return msb;
                if (blockIdx == 0) return msb;
                --blockIdx;
        }
        // invariant: msb is now aligned with the blocks
        while (true) {
                if (v._block[blockIdx] != 0) {
                        BlockType mask = BlockMsb;
                        while (mask != 0) {
                                if (v._block[blockIdx] & mask) return msb;
                                --msb;
                                mask >>= 1;
                        }
                }
                else {
                        msb -= static_cast<signed>(bitsInBlk);
                }
                if (blockIdx == 0) break;
                --blockIdx;
        }
        return msb; // == -1 if no significant bit found
}

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

// remainder returns a mod b in c
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
void remainder(integer<nbits, BlockType, NumberType>& c, const integer<nbits, BlockType, NumberType>& a, const integer<nbits, BlockType, NumberType>& b) {
        if (b == integer<nbits, BlockType, NumberType>(0)) {
#if INTEGER_THROW_ARITHMETIC_EXCEPTION
                throw integer_divide_by_zero{};
#else
                std::cerr << "integer_divide_by_zero\n";
#endif // INTEGER_THROW_ARITHMETIC_EXCEPTION
        }
        idiv_t<nbits, BlockType, NumberType> divresult = idiv<nbits, BlockType, NumberType>(a, b);
        c = divresult.rem;
}

// divide integer<nbits, BlockType, NumberType> a and b and return result argument
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr idiv_t<nbits, BlockType, NumberType> idiv(const integer<nbits, BlockType, NumberType>& _a, const integer<nbits, BlockType, NumberType>& _b) {
        if (_b.iszero()) {
#if INTEGER_THROW_ARITHMETIC_EXCEPTION
                throw integer_divide_by_zero{};
#else
                // When exceptions are disabled we still must avoid the actual
                // long-division below (UB on integer div-by-zero; constexpr would
                // be ill-formed). Return zeroed quot/rem and bail out.
                if (!std::is_constant_evaluated()) {
                        std::cerr << "integer_divide_by_zero\n";
                }
                idiv_t<nbits, BlockType, NumberType> zeroed;
                return zeroed;
#endif // INTEGER_THROW_ARITHMETIC_EXCEPTION
        }

        idiv_t<nbits, BlockType, NumberType> divresult;
        divresult.rem = 0;
        divresult.quot = 0;

        // generate the absolute values to do long division
        if constexpr (NumberType == IntegerNumberType::IntegerNumber) {
                using Integer = integer<nbits+1, BlockType, NumberType>;
                // 2's complement special case -max requires an signed int that is 1 bit bigger to represent abs()
                bool a_negative = _a.sign();
                bool b_negative = _b.sign();
                bool result_negative = (a_negative ^ b_negative);
                Integer a; a.bitcopy(a_negative ? -_a : _a);
                Integer b; b.bitcopy(b_negative ? -_b : _b);

                if (a < b) {
                        divresult.quot = 0; // a / b = 0
                        divresult.rem = _a; // a % b = a when a / b = 0
                        return divresult;
                }
                // initialize the long division
                integer<nbits + 1, BlockType, NumberType> accumulator = a;
                // prepare the subtractand
                integer<nbits + 1, BlockType, NumberType> subtractand = b;
                int msb_b = findMsb(b);
                int msb_a = findMsb(a);
                int shift = msb_a - msb_b;
                subtractand <<= shift;
                divresult.quot = 0;
                // long division
                for (int i = shift; i >= 0; --i) {
                        if (subtractand <= accumulator) {
                                accumulator -= subtractand;
                                divresult.quot.setbit(static_cast<unsigned>(i));
                        }
                        else {
                                divresult.quot.setbit(static_cast<unsigned>(i), false);
                        }
                        subtractand >>= 1;
                        //                std::cout << "i = " << i << " subtractand : " << long(subtractand) << '\n';
                }
                if (result_negative) {  // take 2's complement
                        divresult.quot.flip();
                        divresult.quot += 1;
                }
                if (_a.isneg()) {
                        divresult.rem = -accumulator;
                }
                else {
                        divresult.rem = accumulator;
                }
        }
        else {
                if (_a < _b) {
                        divresult.rem = _a; // a % b = a when a / b = 0
                        return divresult; // a / b = 0 when b > a
                }
                using Integer = integer<nbits, BlockType, NumberType>;
                Integer accumulator(_a), subtractand(_b);
                int msb_b = findMsb(_b);
                int msb_a = findMsb(_a);
                int shift = msb_a - msb_b;
                subtractand <<= shift;
                // long division
                for (int i = shift; i >= 0; --i) {
                        if (subtractand <= accumulator) {
                                accumulator -= subtractand;
                                divresult.quot.setbit(static_cast<unsigned>(i));
                        }
                        else {
                                divresult.quot.setbit(static_cast<unsigned>(i), false);
                        }
                        subtractand >>= 1;
                        //                std::cout << "i = " << i << " subtractand : " << long(subtractand) << '\n';
                }
                divresult.rem = accumulator;
        }

        return divresult;
}

/// stream operators

// read an integer ASCII format and make a binary integer out of it
//
// Accepted syntax (Phase B1 of issue #835 -- shared string_parse foundation):
//
//   [+-]? ( 0[bB][01]+      |    // binary bit-pattern
//           0[oO][0-7]+     |    // octal bit-pattern
//           0[xX][0-9A-F']+ |    // hex bit-pattern (apostrophe allowed as digit separator)
//           [0-9]+          )    // decimal integer
//
// Notes vs. the historical regex-based parser:
//   - Octal now requires the explicit 0o/0O prefix (the previous code accepted
//     C-style leading-zero octal but the conversion path was a `// TODO` that
//     always returned false, so no real behavior change).
//   - Binary 0b/0B is newly supported.
//   - Decimal accepts a single optional leading +/- (the previous regex
//     accepted multiple sign chars; functional difference is nil).
//
// All scanning is delegated to the constexpr primitives in
// `<universal/utility/string_parse.hpp>`; the parser itself uses only
// `integer`'s own arithmetic.
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
bool parse(const std::string& number, integer<nbits, BlockType, NumberType>& value) {
        namespace sp = sw::universal::string_parse;
        using Int = integer<nbits, BlockType, NumberType>;

        // Build into a local temporary and only commit to `value` on a fully
        // successful parse. This keeps malformed input from leaving the caller's
        // object in a partially-mutated state.
        Int tmp;
        tmp.clear();

        std::string_view s{number};

        // Strip leading sign (if any).
        auto sg = sp::scan_sign(s);
        const bool negative = sg.negative;
        s = sg.rest;
        if (s.empty()) return false;

        // Detect base prefix; no prefix -> decimal.
        auto pfx = sp::scan_prefix(s);
        std::string_view body = pfx.body;
        if (body.empty()) return false;

        // Track whether any real digit was consumed so payloads of only separators
        // (e.g. "0x'") are rejected rather than silently yielding zero.
        bool digit_found = false;

        switch (pfx.base) {
        case sp::number_base::binary: {
                // MSB-first: shift the accumulator left by 1 per character, OR in the bit.
                for (char c : body) {
                        if (!sp::is_binary_digit(c)) return false;
                        tmp <<= 1;
                        if (c == '1') tmp.setbit(0, true);
                        digit_found = true;
                }
                if (!digit_found) return false;
                break;
        }
        case sp::number_base::octal: {
                // MSB-first: shift by 3, OR in the 3-bit digit.
                for (char c : body) {
                        if (!sp::is_octal_digit(c)) return false;
                        tmp <<= 3;
                        unsigned digit = static_cast<unsigned>(c - '0');
                        for (unsigned b = 0; b < 3; ++b) {
                                if ((digit >> b) & 1u) tmp.setbit(b, true);
                        }
                        digit_found = true;
                }
                if (!digit_found) return false;
                break;
        }
        case sp::number_base::hex: {
                // MSB-first: shift by 4, OR in the nibble. Apostrophe is a separator.
                for (char c : body) {
                        if (c == '\'') continue;
                        if (!sp::is_hex_digit(c)) return false;
                        tmp <<= 4;
                        unsigned digit = sp::hex_digit_value(c);
                        for (unsigned b = 0; b < 4; ++b) {
                                if ((digit >> b) & 1u) tmp.setbit(b, true);
                        }
                        digit_found = true;
                }
                if (!digit_found) return false;
                break;
        }
        case sp::number_base::decimal: {
                // MSB-first: multiply accumulator by 10, add digit. is_decimal_digit
                // gates the loop body so digit_found tracking is redundant here, but
                // kept for symmetry.
                Int ten{10};
                for (char c : body) {
                        if (!sp::is_decimal_digit(c)) return false;
                        tmp *= ten;
                        Int digit{static_cast<unsigned>(c - '0')};
                        tmp += digit;
                        digit_found = true;
                }
                if (!digit_found) return false;
                break;
        }
        default:
                return false;
        }

        if (negative) tmp = -tmp;
        value = tmp;
        return true;
}

template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
std::string to_string(const integer<nbits, BlockType, NumberType>& n) {
        return convert_to_decimal_string(n);
}

template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
std::string convert_to_string(std::ios_base::fmtflags flags, const integer<nbits, BlockType, NumberType>& n) {
        using IntegerBase = integer<nbits, BlockType, NumberType>;

        // 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;

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

                BlockType shift = static_cast<BlockType>(base == 8 ? 3 : 4);
                BlockType mask = static_cast<BlockType>((1u << shift) - 1);
                IntegerBase t(n);
                result.assign(nbits / shift + ((nbits % shift) ? 1 : 0), '0');
                std::string::size_type pos = result.size() - 1u;
                for (unsigned i = 0; i < nbits / static_cast<unsigned>(shift); ++i) {
                        char c = '0' + static_cast<char>(t.block(0) & mask);
                        if (c > '9')
                                c += 'A' - '9' - static_cast<char>(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(0ull, pp);
                }
        }
        else {
                using Integer = integer<nbits + 1, BlockType, NumberType>;  // nbits+1 to be able to represent maxneg in 2's complement form

                Integer t(n);
                if constexpr (NumberType == IntegerNumberType::IntegerNumber) {
                        if (t.sign()) t.twosComplement();
                }

                Integer block10;
                unsigned digits_in_block10 = 2;
                if constexpr (IntegerBase::bitsInBlock == 8) {
                        block10 = 100u;
                        digits_in_block10 = 2;
                }
                else if constexpr (IntegerBase::bitsInBlock == 16) {
                        block10 = 10'000u;
                        digits_in_block10 = 4;
                }
                else if constexpr (IntegerBase::bitsInBlock == 32) {
                        block10 = 1'000'000'000ul;
                        digits_in_block10 = 9;
                }
                else if constexpr (IntegerBase::bitsInBlock == 64) {
                        block10 = 1'000'000'000'000'000'000ull;
                        digits_in_block10 = 18;
                }

                result.assign(nbits / 3 + 1u, '0');
                int pos = static_cast<int>(result.size() - 1u);
                while (!t.iszero()) {
                        Integer t2 = t / block10;
                        Integer r  = t % block10;
                        BlockType v = r.block(0);
                        for (unsigned i = 0; i < digits_in_block10; ++i) {
                                char c = '0' + static_cast<char>(v % 10);
                                v /= 10;
                                result[static_cast<unsigned>(pos)] = c;
//                                std::cout << "result : " << result << " : pos : " << pos << '\n';
                                if (pos-- == 0) break;
                        }
                        t = t2;
                        if (pos < 0) break;
                }

                std::string::size_type firstDigit = result.find_first_not_of('0');
                result.erase(0, firstDigit);
                if (result.empty())        result = std::string("0");
                if (n.isneg()) { // no need to specialize as isneg() will return false for Natural and Whole Number types
                        result.insert(static_cast<std::string::size_type>(0), 1, '-');
                }
                else if (flags & std::ios_base::showpos) {
                        result.insert(static_cast<std::string::size_type>(0), 1, '+');
                }
        }
        return result;
}

template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
inline std::ostream& operator<<(std::ostream& ostr, const integer<nbits, BlockType, NumberType>& 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();
                // width > s.size() here, so the subtraction stays non-negative
                std::string::size_type pad = static_cast<std::string::size_type>(width) - s.size();
                if ((ostr.flags() & std::ios_base::left) == std::ios_base::left)
                        s.append(pad, fill);
                else
                        s.insert(static_cast<std::string::size_type>(0), pad, fill);
        }
        return ostr << s;
}

// read an ASCII integer format.
// On parse failure: log a diagnostic to std::cerr AND set failbit on the stream
// so callers (loops with `while (in >> x)`, etc.) can detect the error without
// scraping stderr. Symmetric with fixpnt's operator>>.
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
inline std::istream& operator>>(std::istream& istr, integer<nbits, BlockType, NumberType>& p) {
        std::string txt;
        istr >> txt;
        if (!parse(txt, p)) {
                std::cerr << "unable to parse -" << txt << "- into an integer value\n";
                istr.setstate(std::ios::failbit);
        }
        return istr;
}

////////////////// string operators
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
inline std::string to_binary(const integer<nbits, BlockType, NumberType>& number, bool nibbleMarker = false) {
        std::stringstream s;
        s << "0b";
        for (int i = nbits - 1; i >= 0; --i) {
                s << (number.at(static_cast<unsigned>(i)) ? "1" : "0");
                if (i > 0 && (i % 4) == 0 && nibbleMarker) s << '\'';
        }
        return s.str();
}

// native semantic representation: radix-2, delegates to to_binary
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
inline std::string to_native(const integer<nbits, BlockType, NumberType>& number, bool nibbleMarker = false) {
        return to_binary(number, nibbleMarker);
}

//////////////////////////////////////////////////////////////////////////////////////////////////////
// integer - integer binary logic operators

// equal: precondition is that the storage is properly nulled in all arithmetic paths
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline bool operator==(const integer<nbits, BlockType, NumberType>& lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        for (unsigned i = 0; i < lhs.nrBlocks; ++i) {
                if (lhs._block[i] != rhs._block[i]) return false;
        }
        return true;
}
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline bool operator!=(const integer<nbits, BlockType, NumberType>& lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        return !operator==(lhs, rhs);
}
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline bool operator< (const integer<nbits, BlockType, NumberType>& lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        if constexpr (NumberType == WholeNumber || NumberType == NaturalNumber) {
                for (int i = static_cast<int>(lhs.nrBlocks) - 1; i >= 0; --i) {
                        if (lhs.block(static_cast<unsigned>(i)) == rhs.block(static_cast<unsigned>(i))) continue;
                        if (lhs.block(static_cast<unsigned>(i)) < rhs.block(static_cast<unsigned>(i))) return true;
                }
                return false;
        }
        else {
                bool lhs_is_negative = lhs.sign();
                bool rhs_is_negative = rhs.sign();
                if (lhs_is_negative && !rhs_is_negative) return true;
                if (rhs_is_negative && !lhs_is_negative) return false;
                // arguments have the same sign
                integer<nbits, BlockType, NumberType> diff;
#if INTEGER_THROW_ARITHMETIC_EXCEPTION
                // we need to catch and ignore the exception
                try {
                        diff = (lhs - rhs);
                }
                catch (const integer_overflow& e) {
                        // all good as the arithmetic is modulo
                        const char* p = e.what();
                        if (p) --p;
                }
#else 
                diff = (lhs - rhs);
#endif
                return diff.sign();
        }
}
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline bool operator> (const integer<nbits, BlockType, NumberType>& lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        return operator< (rhs, lhs);
}
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline bool operator<=(const integer<nbits, BlockType, NumberType>& lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        return !operator> (lhs, rhs);
}
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline bool operator>=(const integer<nbits, BlockType, NumberType>& lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        return !operator< (lhs, rhs);
}

//////////////////////////////////////////////////////////////////////////////////////////////////////
// integer - literal binary logic operators
// equal: precondition is that the byte-storage is properly nulled in all arithmetic paths
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType, typename IntType>
constexpr inline bool operator==(const integer<nbits, BlockType, NumberType>& lhs, IntType rhs) {
        return operator==(lhs, integer<nbits, BlockType, NumberType>(rhs));
}
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType, typename IntType>
constexpr inline bool operator!=(const integer<nbits, BlockType, NumberType>& lhs, IntType rhs) {
        return !operator==(lhs, rhs);
}
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType, typename IntType>
constexpr inline bool operator< (const integer<nbits, BlockType, NumberType>& lhs, IntType rhs) {
        return operator<(lhs, integer<nbits, BlockType, NumberType>(rhs));
}
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType, typename IntType>
constexpr inline bool operator> (const integer<nbits, BlockType, NumberType>& lhs, IntType rhs) {
        return operator< (integer<nbits, BlockType, NumberType>(rhs), lhs);
}
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType, typename IntType>
constexpr inline bool operator<=(const integer<nbits, BlockType, NumberType>& lhs, IntType rhs) {
        return operator< (lhs, rhs) || operator==(lhs, rhs);
}
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType, typename IntType>
constexpr inline bool operator>=(const integer<nbits, BlockType, NumberType>& lhs, IntType rhs) {
        return !operator< (lhs, rhs);
}

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

template<typename IntType, unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline bool operator==(IntType lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        return operator==(integer<nbits, BlockType, NumberType>(lhs), rhs);
}
template<typename IntType, unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline bool operator!=(IntType lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        return !operator==(lhs, rhs);
}
template<typename IntType, unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline bool operator< (IntType lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        return operator<(integer<nbits, BlockType, NumberType>(lhs), rhs);
}
template<typename IntType, unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline bool operator> (IntType lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        return operator< (rhs, lhs);
}
template<typename IntType, unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline bool operator<=(IntType lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        return operator< (lhs, rhs) || operator==(lhs, rhs);
}
template<typename IntType, unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline bool operator>=(IntType lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        return !operator< (lhs, rhs);
}

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

template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator<<(const integer<nbits, BlockType, NumberType>& lhs, int shift) {
        integer<nbits, BlockType, NumberType> shifted(lhs);
        return (shifted <<= shift);
}

template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator>>(const integer<nbits, BlockType, NumberType>& lhs, int shift) {
        integer<nbits, BlockType, NumberType> shifted(lhs);
        return (shifted >>= shift);
}

//////////////////////////////////////////////////////////////////////////////////////////////////////
// integer - integer binary arithmetic operators
// BINARY ADDITION
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator+(const integer<nbits, BlockType, NumberType>& lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        integer<nbits, BlockType, NumberType> sum(lhs);
        sum += rhs;
        return sum;
}
// BINARY SUBTRACTION
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator-(const integer<nbits, BlockType, NumberType>& lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        integer<nbits, BlockType, NumberType> diff(lhs);
        diff -= rhs;
        return diff;
}
// BINARY MULTIPLICATION
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator*(const integer<nbits, BlockType, NumberType>& lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        integer<nbits, BlockType, NumberType> mul(lhs);
        mul *= rhs;
        return mul;
}
// BINARY DIVISION
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator/(const integer<nbits, BlockType, NumberType>& lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        integer<nbits, BlockType, NumberType> ratio(lhs);
        ratio /= rhs;
        return ratio;
}
// BINARY REMAINDER
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator%(const integer<nbits, BlockType, NumberType>& lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        integer<nbits, BlockType, NumberType> ratio(lhs);
        ratio %= rhs;
        return ratio;
}
// BINARY BIT-WISE AND
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator&(const integer<nbits, BlockType, NumberType>& lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        integer<nbits, BlockType, NumberType> bitwise(lhs);
        bitwise &= rhs;
        return bitwise;
}
// BINARY BIT-WISE OR
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator|(const integer<nbits, BlockType, NumberType>& lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        integer<nbits, BlockType, NumberType> bitwise(lhs);
        bitwise |= rhs;
        return bitwise;
}
// BINARY BIT-WISE XOR
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator^(const integer<nbits, BlockType, NumberType>& lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        integer<nbits, BlockType, NumberType> bitwise(lhs);
        bitwise ^= rhs;
        return bitwise;
}

//////////////////////////////////////////////////////////////////////////////////////////////////////
// integer - literal binary arithmetic operators
// BINARY ADDITION
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator+(const integer<nbits, BlockType, NumberType>& lhs, long long rhs) {
        return operator+(lhs, integer<nbits, BlockType, NumberType>(rhs));
}
// BINARY SUBTRACTION
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator-(const integer<nbits, BlockType, NumberType>& lhs, long long rhs) {
        return operator-(lhs, integer<nbits, BlockType, NumberType>(rhs));
}
// BINARY MULTIPLICATION
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator*(const integer<nbits, BlockType, NumberType>& lhs, long long rhs) {
        return operator*(lhs, integer<nbits, BlockType, NumberType>(rhs));
}
// BINARY DIVISION
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator/(const integer<nbits, BlockType, NumberType>& lhs, long long rhs) {
        return operator/(lhs, integer<nbits, BlockType, NumberType>(rhs));
}
// BINARY REMAINDER
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator%(const integer<nbits, BlockType, NumberType>& lhs, long long rhs) {
        return operator%(lhs, integer<nbits, BlockType, NumberType>(rhs));
}
// BINARY BIT-WISE AND
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator&(const integer<nbits, BlockType, NumberType>& lhs, long long rhs) {
        return operator&(lhs, integer<nbits, BlockType, NumberType>(rhs));
}
// BINARY BIT-WISE OR
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator|(const integer<nbits, BlockType, NumberType>& lhs, long long rhs) {
        return operator|(lhs, integer<nbits, BlockType, NumberType>(rhs));
}
// BINARY BIT-WISE XOR
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator^(const integer<nbits, BlockType, NumberType>& lhs, long long rhs) {
        return operator^(lhs, integer<nbits, BlockType, NumberType>(rhs));
}
//////////////////////////////////////////////////////////////////////////////////////////////////////
// literal - integer binary arithmetic operators
// BINARY ADDITION
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator+(long long lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        return operator+(integer<nbits, BlockType, NumberType>(lhs), rhs);
}
// BINARY SUBTRACTION
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator-(long long lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        return operator-(integer<nbits, BlockType, NumberType>(lhs), rhs);
}
// BINARY MULTIPLICATION
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator*(long long lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        return operator*(integer<nbits, BlockType, NumberType>(lhs), rhs);
}
// BINARY DIVISION
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator/(long long lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        return operator/(integer<nbits, BlockType, NumberType>(lhs), rhs);
}
// BINARY REMAINDER
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator%(long long lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        return operator%(integer<nbits, BlockType, NumberType>(lhs), rhs);
}
// BINARY BIT-WISE AND
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator&(long long lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        return operator&(integer<nbits, BlockType, NumberType>(lhs), rhs);
}
// BINARY BIT-WISE OR
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator|(long long lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        return operator|(integer<nbits, BlockType, NumberType>(lhs), rhs);
}
// BINARY BIT-WISE XOR
template<unsigned nbits, typename BlockType, IntegerNumberType NumberType>
constexpr inline integer<nbits, BlockType, NumberType> operator^(long long lhs, const integer<nbits, BlockType, NumberType>& rhs) {
        return operator^(integer<nbits, BlockType, NumberType>(lhs), rhs);
}

}} // namespace sw::universal

stillwater-sc / universal / 30660486273

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