26061348223

Committed 18 May 2026 09:24PM UTC coverage: 84.24%. First build

Build # 26061348223

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Pull #869

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Merge 92b07a2d1 into 3e2a68633

Pull Request Pull Request #869: fix: IBM HFP clean-up

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/include/sw/universal/number/hfloat/hfloat_impl.hpp

#pragma once
// hfloat_impl.hpp: implementation of IBM System/360 hexadecimal floating-point
//
// Copyright (C) 2017 Stillwater Supercomputing, Inc.
// SPDX-License-Identifier: MIT
//
// This file is part of the universal numbers project, which is released under an MIT Open Source license.
//
// IBM System/360 Hexadecimal Floating-Point (1964):
//   Format: [sign(1)] [exponent(es)] [hex_fraction(ndigits*4 bits)]
//   Value:  (-1)^sign * 16^(exponent - bias) * 0.f1f2...fn
//
// Key properties:
//   - No hidden bit (fraction always has explicit leading hex digit)
//   - No NaN, no infinity, no subnormals
//   - Truncation rounding only (never rounds up)
//   - Overflow saturates to maxpos/maxneg
//   - Zero: sign=0, exponent=0, fraction=0
//   - Wobbling precision: 0-3 leading zero bits in MSB hex digit
//
// Standard configurations:
//   Short:    hfloat<6, 7>  = 1+7+24 = 32 bits
//   Long:     hfloat<14, 7> = 1+7+56 = 64 bits
//   Extended: hfloat<28, 7> = 1+7+112 = 120 bits (stored in 128)

#include <cctype>
#include <cstdint>
#include <cstring>
#include <cmath>
#include <string>
#include <string_view>
#include <sstream>
#include <iostream>
#include <iomanip>
#include <algorithm>

// supporting types and functions
#include <universal/native/ieee754.hpp>
#include <universal/utility/decimal_to_binary.hpp>
#include <universal/number/shared/nan_encoding.hpp>
#include <universal/number/shared/infinite_encoding.hpp>
#include <universal/number/shared/specific_value_encoding.hpp>
// hfloat exception structure
#include <universal/number/hfloat/exceptions.hpp>

namespace sw { namespace universal {

///////////////////////////////////////////////////////////////////////////////
// hfloat: IBM System/360 hexadecimal floating-point number
//
// Template parameters:
//   ndigits  - number of hexadecimal fraction digits
//   es       - exponent bits (7 for standard IBM HFP)
//   bt       - block type for storage
//
template<unsigned _ndigits, unsigned _es, typename bt = std::uint32_t>
class hfloat {
public:
        static constexpr unsigned ndigits     = _ndigits;            // hex fraction digits
        static constexpr unsigned es          = _es;                 // exponent bits
        static constexpr unsigned fbits       = ndigits * 4u;        // fraction bits
        static constexpr unsigned nbits       = 1u + es + fbits;     // total bits
        static constexpr int      bias        = (1 << (es - 1));     // exponent bias (64 for es=7)
        static constexpr int      emax        = (1 << es) - 1 - bias; // max unbiased exponent
        static constexpr int      emin        = -bias;                // min unbiased exponent

        typedef bt BlockType;

        static constexpr unsigned bitsInByte  = 8u;
        static constexpr unsigned bitsInBlock = sizeof(bt) * bitsInByte;
        static constexpr unsigned nrBlocks    = 1u + ((nbits - 1u) / bitsInBlock);
        static constexpr unsigned MSU         = nrBlocks - 1u;

        static constexpr bt ALL_ONES  = bt(~0);
        static constexpr bt MSU_MASK  = (nrBlocks * bitsInBlock == nbits) ? ALL_ONES : bt((1ull << (nbits % bitsInBlock)) - 1u);
        static constexpr bt BLOCK_MASK = bt(~0);

        /// trivial constructor
        hfloat() = default;

        hfloat(const hfloat&) = default;
        hfloat(hfloat&&) = default;

        hfloat& operator=(const hfloat&) = default;
        hfloat& operator=(hfloat&&) = default;

        // specific value constructor
        constexpr hfloat(const SpecificValue code) noexcept : _block{} {
                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::infpos:
                        maxpos();  // no infinity in HFP, saturate to maxpos
                        break;
                case SpecificValue::infneg:
                        maxneg();  // no infinity in HFP, saturate to maxneg
                        break;
                case SpecificValue::nar:
                case SpecificValue::qnan:
                case SpecificValue::snan:
                        zero();    // no NaN in HFP, map to zero
                        break;
                }
        }

        // initializers for native types
        constexpr explicit hfloat(signed char iv)           noexcept : _block{} { *this = iv; }
        constexpr explicit hfloat(short iv)                 noexcept : _block{} { *this = iv; }
        constexpr explicit hfloat(int iv)                   noexcept : _block{} { *this = iv; }
        constexpr explicit hfloat(long iv)                  noexcept : _block{} { *this = iv; }
        constexpr explicit hfloat(long long iv)             noexcept : _block{} { *this = iv; }
        constexpr explicit hfloat(char iv)                  noexcept : _block{} { *this = iv; }
        constexpr explicit hfloat(unsigned short iv)        noexcept : _block{} { *this = iv; }
        constexpr explicit hfloat(unsigned int iv)          noexcept : _block{} { *this = iv; }
        constexpr explicit hfloat(unsigned long iv)         noexcept : _block{} { *this = iv; }
        constexpr explicit hfloat(unsigned long long iv)    noexcept : _block{} { *this = iv; }
        constexpr explicit hfloat(float iv)                 noexcept : _block{} { *this = iv; }
        constexpr explicit hfloat(double iv)                noexcept : _block{} { *this = iv; }

        // assignment operators for native types
        constexpr hfloat& operator=(signed char rhs)        noexcept { return convert_signed(rhs); }
        constexpr hfloat& operator=(short rhs)              noexcept { return convert_signed(rhs); }
        constexpr hfloat& operator=(int rhs)                noexcept { return convert_signed(rhs); }
        constexpr hfloat& operator=(long rhs)               noexcept { return convert_signed(rhs); }
        constexpr hfloat& operator=(long long rhs)          noexcept { return convert_signed(rhs); }
        // Plain `char` may be signed or unsigned per platform; route through
        // the signed conversion via integer promotion so hfloat(char(-1)) on
        // signed-char targets yields -1, not UCHAR_MAX (CodeRabbit finding
        // from PR #805 applied to the sibling type).
        constexpr hfloat& operator=(char rhs)               noexcept { return convert_signed(static_cast<int>(rhs)); }
        constexpr hfloat& operator=(unsigned short rhs)     noexcept { return convert_unsigned(rhs); }
        constexpr hfloat& operator=(unsigned int rhs)       noexcept { return convert_unsigned(rhs); }
        constexpr hfloat& operator=(unsigned long rhs)      noexcept { return convert_unsigned(rhs); }
        constexpr hfloat& operator=(unsigned long long rhs) noexcept { return convert_unsigned(rhs); }
        constexpr hfloat& operator=(float rhs)              noexcept { return convert_ieee754(rhs); }
        constexpr hfloat& operator=(double rhs)             noexcept { return convert_ieee754(rhs); }

        // conversion operators
        constexpr explicit operator float()           const noexcept { return float(convert_to_double()); }
        constexpr explicit operator double()          const noexcept { return convert_to_double(); }

#if LONG_DOUBLE_SUPPORT
        constexpr explicit hfloat(long double iv)           noexcept : _block{} { *this = iv; }
        constexpr hfloat& operator=(long double rhs)        noexcept { return convert_ieee754(double(rhs)); }
        constexpr explicit operator long double()     const noexcept { return (long double)convert_to_double(); }
#endif

        // Unified entry point: assign from a normalized 64-bit binary mantissa
        // plus its "frexp" exponent. This is the algorithm core; both parse()
        // (which derives mantissa64 from decimal_to_binary) and convert_ieee754
        // (which lifts a double's 53-bit mantissa into the 64-bit convention by
        // left-shifting 11 bits) call this same routine.
        //
        // Preconditions:
        //   mantissa64 is normalized with MSB at bit 63 (the "implicit 1"
        //   position of a normalized binary mantissa). For inputs derived from
        //   an IEEE double, the low 11 bits will be zero -- but they don't have
        //   to be; parse() fills all 64.
        //   bin_exp is the "frexp" exponent: value = mantissa64 * 2^(bin_exp - 64).
        constexpr hfloat& assign_from_mantissa64(bool negative, int bin_exp, uint64_t mantissa64) noexcept {
                if (mantissa64 == 0) { setzero(); return *this; }

                // Convert binary exponent to base-16 exponent. ceil(bin_exp/4) so
                // 0.f * 16^hex_exp == value with f's leading hex digit non-zero.
                int hex_exp;
                if (bin_exp > 0) hex_exp = (bin_exp + 3) / 4;
                else             hex_exp = bin_exp / 4;  // C truncates toward zero == ceil for negative

                // Align mantissa to the hex fraction bit positions. For a 64-bit
                // mantissa, mantissa_shift = (bin_exp - 4*hex_exp + fbits) - 64.
                int shift = bin_exp - 4 * hex_exp + static_cast<int>(fbits);
                int mantissa_shift = shift - 64;
                uint64_t fraction = 0;
                if (mantissa_shift == 0) {
                        fraction = mantissa64;
                }
                else if (mantissa_shift > 0) {
                        // fraction's natural width exceeds 64 bits -- can't be held in
                        // uint64_t. This is unreachable for hfloat configs with fbits
                        // <= 64 (the standard hfloat_short / hfloat_long sizes) and is
                        // the hfloat_extended (fbits=112) path: the 64-bit mantissa must
                        // be placed at offset `mantissa_shift` within the fbits-wide
                        // fraction (its low `mantissa_shift` bits are truncated to zero
                        // -- IBM HFP truncation). Check overflow/underflow first (the
                        // usual normalize_and_pack route can't see the wide fraction),
                        // then pack sign + exponent and setbit() the mantissa bits at
                        // their target high positions. Full bit-exact fbits>64
                        // conversion needs a wider intermediate (e.g. __uint128_t) and
                        // is left as future work; this delivers hfp64-precision in an
                        // hfp128 container.
                        if constexpr (fbits >= 64) {
                                if (hex_exp > emax) {
                                        if (negative) maxneg(); else maxpos();
                                        return *this;
                                }
                                if (hex_exp < emin) {
                                        setzero();
                                        return *this;
                                }
                                if (static_cast<unsigned>(mantissa_shift) < 64u) {
                                        pack(negative, hex_exp, 0);
                                        unsigned offset = static_cast<unsigned>(mantissa_shift);
                                        unsigned upper = (offset + 64u <= fbits) ? 64u : (fbits - offset);
                                        for (unsigned i = 0; i < upper; ++i) {
                                                if ((mantissa64 >> i) & 1) setbit(offset + i, true);
                                        }
                                        return *this;
                                }
                                // mantissa_shift >= 64: entire 64-bit mantissa lies beyond
                                // bit 63 of the fraction. Saturating to maxpos keeps the
                                // exponent meaningful while signaling unrepresentable
                                // precision; rare for normal-magnitude inputs.
                                fraction = ~uint64_t(0);
                        }
                        else {
                                // fbits < 64 with mantissa_shift > 0 is mathematically
                                // unreachable (shift = bin_exp - 4*hex_exp + fbits with
                                // (bin_exp - 4*hex_exp) in [0,3], so mantissa_shift =
                                // shift - 64 <= fbits + 3 - 64 < 0 when fbits < 61), but
                                // keep a defensive fallback for very narrow configs.
                                fraction = (uint64_t(1) << fbits) - 1u;
                        }
                }
                else if (-mantissa_shift < 64) {
                        // Shift right truncates low bits -- IBM HFP's truncation rounding.
                        fraction = mantissa64 >> static_cast<unsigned>(-mantissa_shift);
                }
                // else: fraction stays 0 (deep underflow)

                if constexpr (fbits < 64) {
                        fraction &= ((uint64_t(1) << fbits) - 1u);
                }

                normalize_and_pack(negative, hex_exp, fraction);
                return *this;
        }

        // Assign from a wide (target_mantissa_bits == fbits) decimal_to_binary
        // result. Delivers full fbits of precision regardless of host integer
        // width -- the whole point of routing decimal literals through d2b
        // instead of through double. Used by parse() when fbits > 64; the
        // uint64_t-based assign_from_mantissa64 above still serves the
        // convert_ieee754 (double / long double) entry point, where the input
        // is bounded to ~64 mantissa bits regardless.
        template<unsigned BigBits>
        constexpr hfloat& assign_from_wide_mantissa(bool negative, int bin_exp,
            const ::sw::universal::decimal_to_binary::basic_result<BigBits>& d) noexcept {
                // d.mantissa is normalized with MSB at bit (fbits - 1). The hex-
                // digit alignment within IBM HFP requires at most a 0..3 bit right
                // shift relative to that normalization so the leading hex digit
                // (bits [fbits-4, fbits-1]) ends up non-zero.
                int hex_exp;
                if (bin_exp > 0) hex_exp = (bin_exp + 3) / 4;
                else             hex_exp = bin_exp / 4;  // C truncates toward zero == ceil for negative

                if (hex_exp > emax) {
                        if (negative) maxneg(); else maxpos();
                        return *this;
                }
                if (hex_exp < emin) {
                        setzero();
                        return *this;
                }

                // shift = bin_exp - 4*hex_exp lies in [-3, 0]. Read fraction bit i
                // from d.mantissa.at(i + right_shift); the low `right_shift` mantissa
                // bits are dropped -- IBM HFP truncation rounding.
                int shift = bin_exp - 4 * hex_exp;
                unsigned right_shift = static_cast<unsigned>(-shift);

                clear();
                setbit(nbits - 1, negative);
                unsigned biased_exp = static_cast<unsigned>(hex_exp + bias);
                unsigned expStart = nbits - 2;
                for (unsigned i = 0; i < es; ++i) {
                        setbit(expStart - i, (biased_exp >> (es - 1 - i)) & 1);
                }
                for (unsigned i = 0; i < fbits; ++i) {
                        if (d.mantissa.at(i + right_shift)) setbit(i, true);
                }
                return *this;
        }

        // prefix operators
        constexpr hfloat operator-() const {
                hfloat negated(*this);
                if (!negated.iszero()) {
                        negated.setsign(!negated.sign());
                }
                return negated;
        }

        // arithmetic operators
        constexpr hfloat& operator+=(const hfloat& rhs) {
                bool lhs_sign, rhs_sign;
                int lhs_exp, rhs_exp;
                uint64_t lhs_frac, rhs_frac;
                unpack(lhs_sign, lhs_exp, lhs_frac);
                rhs.unpack(rhs_sign, rhs_exp, rhs_frac);

                if (rhs.iszero()) return *this;
                if (iszero()) { *this = rhs; return *this; }

                // align exponents by shifting the smaller-exponent fraction RIGHT
                // by 4*(exp_large - exp_small) bits (hex digit alignment)
                int shift = lhs_exp - rhs_exp;
                uint64_t aligned_lhs = lhs_frac;
                uint64_t aligned_rhs = rhs_frac;
                int result_exp;

                if (shift >= 0) {
                        result_exp = lhs_exp;
                        aligned_rhs >>= (static_cast<unsigned>(shift) * 4u);
                }
                else {
                        result_exp = rhs_exp;
                        aligned_lhs >>= (static_cast<unsigned>(-shift) * 4u);
                }

                // add/subtract based on signs
                int64_t result_frac;
                int64_t a = lhs_sign ? -static_cast<int64_t>(aligned_lhs) : static_cast<int64_t>(aligned_lhs);
                int64_t b = rhs_sign ? -static_cast<int64_t>(aligned_rhs) : static_cast<int64_t>(aligned_rhs);
                result_frac = a + b;

                bool result_sign = (result_frac < 0);
                uint64_t abs_frac = static_cast<uint64_t>(result_sign ? -result_frac : result_frac);

                normalize_and_pack(result_sign, result_exp, abs_frac);
                return *this;
        }

        constexpr hfloat& operator-=(const hfloat& rhs) {
                hfloat neg(rhs);
                if (!neg.iszero()) neg.setsign(!neg.sign());
                return operator+=(neg);
        }

        constexpr hfloat& operator*=(const hfloat& rhs) {
                if (iszero() || rhs.iszero()) { setzero(); return *this; }

                bool lhs_sign, rhs_sign;
                int lhs_exp, rhs_exp;
                uint64_t lhs_frac, rhs_frac;
                unpack(lhs_sign, lhs_exp, lhs_frac);
                rhs.unpack(rhs_sign, rhs_exp, rhs_frac);

                bool result_sign = (lhs_sign != rhs_sign);
                int result_exp = lhs_exp + rhs_exp;

#ifdef __SIZEOF_INT128__
                __uint128_t wide = static_cast<__uint128_t>(lhs_frac) * static_cast<__uint128_t>(rhs_frac);
                // The fractions are in 0.f format with fbits fraction bits
                // Product has 2*fbits bits, shift right by fbits to get back to fbits
                uint64_t result_frac = static_cast<uint64_t>(wide >> fbits);
                normalize_and_pack(result_sign, result_exp, result_frac);
#else
                // fallback: delegate through double
                double d = double(*this) * double(rhs);
                *this = d;
#endif
                return *this;
        }

        constexpr hfloat& operator/=(const hfloat& rhs) {
                if (rhs.iszero()) {
#if HFLOAT_THROW_ARITHMETIC_EXCEPTION
                        // Throwing in a constant expression is ill-formed; fence the
                        // throw under runtime to keep operator/= constexpr-callable.
                        // At constant-eval, divide-by-zero falls through to setzero()
                        // (IBM HFP has no NaN/inf, so saturation is the closest
                        // constexpr-safe behavior).
                        if (!std::is_constant_evaluated()) {
                                throw hfloat_divide_by_zero();
                        }
#endif
                        setzero();
                        return *this;
                }
                if (iszero()) return *this;

                bool lhs_sign, rhs_sign;
                int lhs_exp, rhs_exp;
                uint64_t lhs_frac, rhs_frac;
                unpack(lhs_sign, lhs_exp, lhs_frac);
                rhs.unpack(rhs_sign, rhs_exp, rhs_frac);

                bool result_sign = (lhs_sign != rhs_sign);
                int result_exp = lhs_exp - rhs_exp;

#ifdef __SIZEOF_INT128__
                // Scale numerator up by fbits for precision
                __uint128_t scaled_num = static_cast<__uint128_t>(lhs_frac) << fbits;
                uint64_t result_frac = static_cast<uint64_t>(scaled_num / static_cast<__uint128_t>(rhs_frac));
                normalize_and_pack(result_sign, result_exp, result_frac);
#else
                double d = double(*this) / double(rhs);
                *this = d;
#endif
                return *this;
        }

        // unary operators
        constexpr hfloat& operator++() {
                *this += hfloat(1);
                return *this;
        }
        constexpr hfloat operator++(int) {
                hfloat tmp(*this);
                operator++();
                return tmp;
        }
        constexpr hfloat& operator--() {
                *this -= hfloat(1);
                return *this;
        }
        constexpr hfloat operator--(int) {
                hfloat tmp(*this);
                operator--();
                return tmp;
        }

        // modifiers
        constexpr void clear() noexcept {
                for (unsigned i = 0; i < nrBlocks; ++i) _block[i] = bt(0);
        }
        constexpr void setzero() noexcept { clear(); }

        constexpr void setsign(bool negative = true) noexcept {
                setbit(nbits - 1, negative);
        }

        // use un-interpreted raw bits to set the value
        constexpr void setbits(uint64_t value) noexcept {
                clear();
                for (unsigned i = 0; i < nrBlocks; ++i) {
                        _block[i] = bt(value & BLOCK_MASK);
                        value >>= bitsInBlock;
                }
                _block[MSU] &= MSU_MASK;
        }

        // create specific number system values of interest
        //
        // The `max_frac` computations below cap at uint64_t max for fbits >= 64:
        // `1ull << 64` and wider are UB. For hfloat configs with fbits > 64
        // (hfloat_extended at fbits=112), the high fraction bits stay zero
        // since pack()'s fraction parameter is also uint64_t; full-width
        // fbits > 64 saturation is future work pending a multi-limb fraction
        // path through pack / normalize_and_pack.
        constexpr hfloat& maxpos() noexcept {
                clear();
                unsigned biased_exp = (1u << es) - 1u;
                uint64_t max_frac;
                if constexpr (fbits < 64) {
                        max_frac = (uint64_t(1) << fbits) - 1u;
                } else {
                        max_frac = ~uint64_t(0);
                }
                pack(false, static_cast<int>(biased_exp) - bias, max_frac);
                return *this;
        }
        constexpr hfloat& minpos() noexcept {
                clear();
                pack(false, emin, 1);
                return *this;
        }
        constexpr hfloat& zero() noexcept {
                clear();
                return *this;
        }
        constexpr hfloat& minneg() noexcept {
                clear();
                pack(true, emin, 1);
                return *this;
        }
        constexpr hfloat& maxneg() noexcept {
                clear();
                unsigned biased_exp = (1u << es) - 1u;
                uint64_t max_frac;
                if constexpr (fbits < 64) {
                        max_frac = (uint64_t(1) << fbits) - 1u;
                } else {
                        max_frac = ~uint64_t(0);
                }
                pack(true, static_cast<int>(biased_exp) - bias, max_frac);
                return *this;
        }

        // selectors
        constexpr bool sign() const noexcept {
                return getbit(nbits - 1);
        }

        constexpr bool iszero() const noexcept {
                // zero when fraction is all zeros (exponent and sign don't matter)
                // In IBM HFP, zero is represented as all-zeros
                for (unsigned i = 0; i < nrBlocks; ++i) {
                        bt mask = (i == MSU) ? bt(MSU_MASK & ~(bt(1) << ((nbits - 1) % bitsInBlock))) : BLOCK_MASK;
                        if ((_block[i] & mask) != 0) return false;
                }
                return true;
        }

        constexpr bool isone() const noexcept {
                bool s; int e; uint64_t f;
                unpack(s, e, f);
                // 1.0 = 0.1 * 16^1, so e=1, f = 1 << (fbits-4)  (leading hex digit = 1).
                // Gate the shift so `1ull << (fbits - 4)` doesn't UB for fbits >= 68
                // (hfloat_extended at fbits=112 would otherwise hit shift-count-overflow);
                // for those configs unpack() also caps the fraction at 64 useful bits,
                // so we can never construct the wide-fbits "leading hex digit = 1"
                // encoding here -- return false consistently.
                if constexpr (fbits >= 68) {
                        (void)s; (void)e; (void)f;
                        return false;
                }
                else {
                        return !s && (e == 1) && (f == (uint64_t(1) << (fbits - 4)));
                }
        }

        constexpr bool ispos() const noexcept { return !sign(); }
        constexpr bool isneg() const noexcept { return sign(); }

        // IBM HFP has no NaN or infinity
        constexpr bool isinf() const noexcept { return false; }
        constexpr bool isnan() const noexcept { return false; }
        constexpr bool isnan(int) const noexcept { return false; }

        constexpr int scale() const noexcept {
                if (iszero()) return 0;
                bool s; int e; uint64_t f;
                unpack(s, e, f);
                (void)s;
                // IBM HFP: value = 0.f * 16^e. unpack() returns the fraction's
                // low 64 bits (for fbits < 64, that's the entire fraction). For
                // fbits >= 64 the leading hex digit lives in bits [fbits-4, fbits-1]
                // of the full fraction -- unpack's low-64-bit view sees only the
                // truncated tail, not the leading bit, so we read the top 64
                // fraction bits directly (bits [fbits-64, fbits-1]).
                uint64_t top = 0;
                if constexpr (fbits >= 64) {
                        for (unsigned i = 0; i < 64u; ++i) {
                                if (getbit(fbits - 64u + i)) top |= (uint64_t(1) << i);
                        }
                }
                else {
                        top = f;
                }
                constexpr int read_iter = (fbits < 64) ? static_cast<int>(fbits) : 64;
                int leading = -1;
                for (int i = read_iter - 1; i >= 0; --i) {
                        if ((top >> i) & 1) { leading = i; break; }
                }
                if (leading < 0) return 0;
                constexpr int frac_msb_weight = (fbits < 64) ? static_cast<int>(fbits) : 64;
                return 4 * e + leading - frac_msb_weight;
        }

        // convert to string
        std::string str(size_t nrDigits = 0) const {
                if (iszero()) return sign() ? std::string("-0") : std::string("0");

                double d = convert_to_double();
                std::stringstream ss;
                if (nrDigits > 0) {
                        ss << std::setprecision(static_cast<int>(nrDigits));
                }
                ss << d;
                return ss.str();
        }

        ///////////////////////////////////////////////////////////////////
        // Bit access (public for free functions)
        constexpr bool getbit(unsigned pos) const noexcept {
                if (pos >= nbits) return false;
                unsigned block_idx = pos / bitsInBlock;
                unsigned bit_idx = pos % bitsInBlock;
                return (_block[block_idx] >> bit_idx) & 1;
        }

        ///////////////////////////////////////////////////////////////////
        // Unpack into sign, unbiased exponent, and fraction
        constexpr void unpack(bool& s, int& exponent, uint64_t& fraction) const noexcept {
                s = sign();
                if (iszero()) { exponent = 0; fraction = 0; return; }

                // Extract exponent field (es bits)
                unsigned exp_field = 0;
                unsigned expStart = nbits - 2; // MSB of exponent (just below sign)
                for (unsigned i = 0; i < es; ++i) {
                        if (getbit(expStart - i)) {
                                exp_field |= (1u << (es - 1 - i));
                        }
                }
                exponent = static_cast<int>(exp_field) - bias;

                // Extract fraction. The destination is uint64_t so we read at most
                // 64 bits; high fraction bits (i >= 64) for hfloat_extended would
                // make `1ull << i` UB and aren't representable in the result type
                // anyway. Matches the cap on the pack() write path.
                fraction = 0;
                constexpr unsigned read_iter = (fbits < 64) ? fbits : 64u;
                for (unsigned i = 0; i < read_iter; ++i) {
                        if (getbit(i)) {
                                fraction |= (uint64_t(1) << i);
                        }
                }
        }

protected:
        bt _block[nrBlocks];

        ///////////////////////////////////////////////////////////////////
        // Bit manipulation helpers
        constexpr void setbit(unsigned pos, bool value) noexcept {
                if (pos >= nbits) return;
                unsigned block_idx = pos / bitsInBlock;
                unsigned bit_idx = pos % bitsInBlock;
                if (value) {
                        _block[block_idx] |= bt(1ull << bit_idx);
                }
                else {
                        _block[block_idx] &= bt(~(1ull << bit_idx));
                }
        }

        ///////////////////////////////////////////////////////////////////
        // Pack sign, unbiased exponent, and fraction
        constexpr void pack(bool s, int exponent, uint64_t fraction) noexcept {
                clear();

                // set sign
                setbit(nbits - 1, s);

                // set exponent field
                unsigned biased_exp = static_cast<unsigned>(exponent + bias);
                unsigned expStart = nbits - 2;
                for (unsigned i = 0; i < es; ++i) {
                        setbit(expStart - i, (biased_exp >> (es - 1 - i)) & 1);
                }

                // set fraction field. The source is a uint64_t so it carries at
                // most 64 useful bits; for fbits > 64 (hfloat_extended) the high
                // fbits stay zero because `fraction >> i` for i >= 64 is UB
                // (compilers typically mask the shift count modulo 64 which
                // would re-replicate the low bits at the top -- definitely
                // wrong). Multi-limb fraction support for fbits > 64 is future
                // work; until then, only the low 64 fraction bits are populated.
                constexpr unsigned frac_iter = (fbits < 64) ? fbits : 64u;
                for (unsigned i = 0; i < frac_iter; ++i) {
                        setbit(i, (fraction >> i) & 1);
                }
        }

        ///////////////////////////////////////////////////////////////////
        // Normalize: ensure leading hex digit is non-zero, then truncate
        constexpr void normalize_and_pack(bool s, int exponent, uint64_t fraction) noexcept {
                if (fraction == 0) { setzero(); return; }

                // Normalize the leading hex digit. For fbits < 64 the
                // `1ull << fbits` thresholds are well-defined and the loops
                // produce IBM-HFP-correct truncation. For fbits >= 64 the
                // uint64_t fraction cannot represent the full hex significand
                // (hfloat_extended is fbits=112); we skip the normalization
                // loops since the relevant thresholds would be UB and accept
                // that the wide-fbits saturation path is approximate. Full-
                // width fbits > 64 conversion needs a multi-limb fraction
                // representation and is left as future work.
                if constexpr (fbits < 64) {
                        while (fraction >= (uint64_t(1) << fbits)) {
                                fraction >>= 4;  // shift right by one hex digit
                                exponent++;
                        }
                        while (fraction > 0 && fraction < (uint64_t(1) << (fbits - 4))) {
                                fraction <<= 4;  // shift left by one hex digit
                                exponent--;
                        }
                        // Truncate to fbits (IBM HFP truncates, never rounds up).
                        fraction &= ((uint64_t(1) << fbits) - 1);
                }
                // else: fraction is already at most ~uint64_t(0), which by
                // construction fits within fbits >= 64; pack() places it in
                // the low 64 fbits and leaves the rest zero.

                // Check overflow/underflow
                if (exponent > emax) {
                        // overflow: saturate to maxpos/maxneg
                        if (s) maxneg(); else maxpos();
                        return;
                }
                if (exponent < emin) {
                        // underflow: set to zero
                        setzero();
                        return;
                }

                pack(s, exponent, fraction);
        }

        ///////////////////////////////////////////////////////////////////
        // Conversion helpers

        // Convert IEEE-754 double to hfloat
        //
        // Constexpr-safe rewrite of the original frexp/ldexp implementation:
        // - NaN detected via x != x (only NaN is unequal to itself)
        // - infinity detected by bracketing against numeric_limits::max()
        //   (numeric_limits<double>::max() is constexpr; std::isinf is not)
        // - fabs replaced with `negative ? -rhs : rhs`
        // - frexp/ldexp replaced with raw IEEE 754 field extraction via
        //   sw::bit_cast (constexpr on toolchains exposing std::bit_cast or
        //   __builtin_bit_cast; runtime fallback retains the legacy path).
        //
        // IBM HFP has no NaN and no infinity:
        //   NaN double  -> hfloat zero
        //   +/- inf     -> hfloat maxpos/maxneg (saturation)
        //
        // This used to duplicate the encoding algorithm. After the unification
        // for issue #849, it just extracts the IEEE double's 53-bit significand
        // (with hidden bit), lifts it into the 64-bit normalized mantissa
        // convention by left-shifting 11 bits (the headroom between IEEE
        // double's 53 and our canonical 64), and delegates to
        // `assign_from_mantissa64`. The shift is lossless -- the low 11 bits of
        // the wide mantissa are simply zero, since the double had no more
        // information to give.
        constexpr hfloat& convert_ieee754(double rhs) noexcept {
                if (rhs != rhs) {                       // NaN
                        setzero();
                        return *this;
                }
                if (rhs == 0.0) {
                        setzero();
                        return *this;
                }
                constexpr double dbl_max = std::numeric_limits<double>::max();
                if (rhs >  dbl_max) { maxpos(); return *this; }   // +inf
                if (rhs < -dbl_max) { maxneg(); return *this; }   // -inf

                bool negative = (rhs < 0);
                double abs_val = negative ? -rhs : rhs;

                // Extract IEEE 754 double fields without std::frexp:
                // IEEE 754 double: bias 1023, 52 fraction bits, hidden 1 for normals.
                //   normal: value = (1 + rawFrac/2^52) * 2^(rawExp - 1023)
                //                 = (2^52 + rawFrac) * 2^(rawExp - 1075)
                // frexp would return frac in [0.5, 1) with bin_exp = rawExp - 1022;
                // we use the same bin_exp here so the mantissa MSB lies at bit
                // (mantissa_width - 1), matching assign_from_mantissa64's contract.
                //
                // sw::bit_cast is constexpr on toolchains exposing std::bit_cast or
                // __builtin_bit_cast (the universally-supported case in C++20+); on
                // older toolchains it works at runtime via memcpy but is not
                // constexpr.  Calling convert_ieee754() in a constant expression on
                // such a toolchain produces a not-a-constant-expression diagnostic
                // at the bit_cast invocation -- the correct behavior, vs. silently
                // returning zero from a runtime fallback.
                uint64_t bits = sw::bit_cast<uint64_t>(abs_val);
                uint64_t rawExp  = (bits >> 52) & 0x7FFu;
                uint64_t rawFrac = bits & ((uint64_t(1) << 52) - 1u);
                if (rawExp == 0) {
                        // Subnormal double values are far below IBM HFP's smallest
                        // representable (16^emin); flush to zero per HFP semantics.
                        setzero();
                        return *this;
                }
                int bin_exp = static_cast<int>(rawExp) - 1022;
                uint64_t mantissa53 = (uint64_t(1) << 52) | rawFrac;  // 53-bit, MSB at bit 52
                uint64_t mantissa64 = mantissa53 << 11;               // lift to MSB at bit 63
                return assign_from_mantissa64(negative, bin_exp, mantissa64);
        }

        // Convert hfloat to IEEE-754 double
        //
        // Constexpr-safe: replaces std::ldexp(d, n) with a power-of-2 scaling
        // loop that multiplies/divides by 2.0 (exactly representable in double).
        // At runtime, dispatches to std::ldexp for performance.  The shift range
        // per template config is bounded (fbits + 4*emax for hfloat_extended is
        // ~360), so the constexpr loop is well-bounded.
        constexpr double convert_to_double() const noexcept {
                if (iszero()) return sign() ? -0.0 : 0.0;

                bool s; int e; uint64_t f;
                unpack(s, e, f);

                // value = 0.f * 16^e. For fbits < 64 the unpack'd f IS the full
                // fraction and value = f * 2^(-fbits) * 16^e = f * 2^(4*e - fbits).
                // For fbits >= 64 the leading hex digit lives in the TOP 64 fraction
                // bits, which unpack's low-64 view doesn't see; read them directly
                // and treat f as representing fraction's bits [fbits-64, fbits-1],
                // so value = f * 2^(-64) * 16^e = f * 2^(4*e - 64).
                int shift;
                if constexpr (fbits >= 64) {
                        f = 0;
                        for (unsigned i = 0; i < 64u; ++i) {
                                if (getbit(fbits - 64u + i)) f |= (uint64_t(1) << i);
                        }
                        shift = 4 * e - 64;
                }
                else {
                        shift = 4 * e - static_cast<int>(fbits);
                }
                double result = static_cast<double>(f);
                if (std::is_constant_evaluated()) {
                        // Power-of-2 scaling: 2.0 and 0.5 are exactly representable
                        // in IEEE 754 double, so this introduces no rounding error
                        // over the bounded iteration count.
                        if (shift > 0) {
                                for (int i = 0; i < shift; ++i) result *= 2.0;
                        }
                        else if (shift < 0) {
                                for (int i = 0; i < -shift; ++i) result *= 0.5;
                        }
                }
                else {
                        result = std::ldexp(result, shift);
                }
                return s ? -result : result;
        }

        // Direct integer-to-hfloat packing for fbits <= 56 (hfloat_short and
        // hfloat_long).  Avoids the double round-trip that would lose precision
        // for int64_t values above 2^53 -- a value like 9007199254740993ULL
        // IS representable in hfloat_long (56-bit fraction) but rounds via
        // double.  For fbits >= 64 (hfloat_extended) the fraction is wider
        // than uint64_t can hold, so we fall back to convert_ieee754; this
        // keeps the same precision as pre-promotion code for that variant.
        constexpr hfloat& convert_signed(int64_t v) noexcept {
                if (v == 0) {
                        setzero();
                        return *this;
                }
                bool negative = (v < 0);
                // Compute |v| as uint64_t without ever negating an int64_t (the
                // negation of INT64_MIN would overflow).  Bitwise-safe identity:
                // |INT64_MIN| = -(v + 1) + 1, each step stays in range.
                uint64_t abs_v = negative
                        ? (static_cast<uint64_t>(-(v + 1)) + 1ull)
                        : static_cast<uint64_t>(v);
                if constexpr (fbits >= 64) {
                        return convert_ieee754(static_cast<double>(v));
                }
                else {
                        pack_uint64(negative, abs_v);
                        return *this;
                }
        }

        constexpr hfloat& convert_unsigned(uint64_t v) noexcept {
                if (v == 0) {
                        setzero();
                        return *this;
                }
                if constexpr (fbits >= 64) {
                        return convert_ieee754(static_cast<double>(v));
                }
                else {
                        pack_uint64(false, v);
                        return *this;
                }
        }

        // Pack a uint64_t magnitude into the hfloat's hex representation.
        // Precondition: v != 0 and fbits < 64.
        //   value = v = 0.f * 16^hex_exp where 0.f's leading hex digit is non-zero.
        // Algorithm:
        //   p          = highest set bit of v (0-indexed)
        //   hex_exp    = p/4 + 1   (number of hex digits to represent v)
        //   shift      = fbits - 4*hex_exp
        //   fraction   = (shift >= 0) ? v << shift : v >> -shift  (truncate per HFP)
        // Worst-case shift left: p=0, fbits<=56 -> shift=fbits-4, v<<shift fits in
        // uint64_t since v has only 1 bit.  Worst-case shift right: p=63, shift=fbits-64.
        constexpr void pack_uint64(bool negative, uint64_t v) noexcept {
                // Find highest set bit position (loop is bounded to 64 iterations
                // and constexpr-clean; v != 0 by precondition).
                unsigned p = 63;
                while (((v >> p) & 1u) == 0u) --p;
                int hex_exp = static_cast<int>(p / 4u) + 1;
                int shift = static_cast<int>(fbits) - 4 * hex_exp;
                uint64_t fraction = (shift >= 0)
                        ? (v << static_cast<unsigned>(shift))
                        : (v >> static_cast<unsigned>(-shift));
                normalize_and_pack(negative, hex_exp, fraction);
        }

private:

        // hfloat - hfloat logic comparisons
        template<unsigned N, unsigned E, typename B>
        friend constexpr bool operator==(const hfloat<N, E, B>& lhs, const hfloat<N, E, B>& rhs);

        // hfloat - literal logic comparisons
        template<unsigned N, unsigned E, typename B>
        friend constexpr bool operator==(const hfloat<N, E, B>& lhs, const double rhs);

        // literal - hfloat logic comparisons
        template<unsigned N, unsigned E, typename B>
        friend constexpr bool operator==(const double lhs, const hfloat<N, E, B>& rhs);
};


////////////////////////    helper functions   /////////////////////////////////

template<unsigned ndigits, unsigned es, typename BlockType>
inline std::string to_binary(const hfloat<ndigits, es, BlockType>& number, bool nibbleMarker = false) {
        using Hfloat = hfloat<ndigits, es, BlockType>;
        std::stringstream s;

        // sign bit
        s << "0b" << (number.sign() ? '1' : '0') << '.';

        // exponent field (es bits)
        unsigned expStart = Hfloat::nbits - 2;
        for (unsigned i = 0; i < es; ++i) {
                s << (number.getbit(expStart - i) ? '1' : '0');
        }
        s << '.';

        // fraction field (fbits bits, show in hex-digit groups)
        for (int i = static_cast<int>(Hfloat::fbits) - 1; i >= 0; --i) {
                s << (number.getbit(static_cast<unsigned>(i)) ? '1' : '0');
                if (nibbleMarker && i > 0 && (i % 4 == 0)) s << '\'';
        }

        return s.str();
}

template<unsigned ndigits, unsigned es, typename BlockType>
inline std::string to_hex(const hfloat<ndigits, es, BlockType>& number) {
        std::stringstream s;

        s << (number.sign() ? '-' : '+');
        s << "0x0.";

        // extract hex digits from fraction
        bool sign_val; int exp_val; uint64_t frac_val;
        number.unpack(sign_val, exp_val, frac_val);

        for (int i = static_cast<int>(ndigits) - 1; i >= 0; --i) {
                unsigned hex_digit = (frac_val >> (i * 4)) & 0xF;
                s << "0123456789ABCDEF"[hex_digit];
        }
        s << " * 16^" << (exp_val);

        return s.str();
}


// native semantic representation: radix-16, delegates to to_hex
template<unsigned ndigits, unsigned es, typename BlockType>
inline std::string to_native(const hfloat<ndigits, es, BlockType>& v, bool = false) {
        return to_hex(v);
}

////////////////////////    HFLOAT functions   /////////////////////////////////

template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr hfloat<ndigits, es, BlockType> abs(const hfloat<ndigits, es, BlockType>& a) {
        hfloat<ndigits, es, BlockType> result(a);
        result.setsign(false);
        return result;
}

template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr hfloat<ndigits, es, BlockType> fabs(hfloat<ndigits, es, BlockType> a) {
        a.setsign(false);
        return a;
}


////////////////////////  stream operators   /////////////////////////////////

template<unsigned ndigits, unsigned es, typename BlockType>
inline std::ostream& operator<<(std::ostream& ostr, const hfloat<ndigits, es, BlockType>& i) {
        std::stringstream ss;
        std::streamsize prec = ostr.precision();
        std::streamsize width = ostr.width();
        std::ios_base::fmtflags ff;
        ff = ostr.flags();
        ss.flags(ff);
        ss << std::setw(width) << std::setprecision(prec) << i.str(size_t(prec));
        return ostr << ss.str();
}

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

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

// Parse a decimal floating-point literal into an hfloat. hfloat has no NaN
// or Inf encoding, so "nan" / "inf" / "infinity" tokens are rejected. The
// decimal payload is converted via decimal_to_binary with 64 bits of
// precision and assembled directly into the hfloat hex fraction -- this
// gives bit-exact correctly-rounded conversion for hfloat_long's full 56
// fbits, which the via-double path could not deliver (double has only 53).
//
// Resolves #849.
template<unsigned ndigits, unsigned es, typename BlockType>
bool parse(const std::string& number, hfloat<ndigits, es, BlockType>& value) {
        if (number.empty()) return false;

        // Reject special-value tokens up front (hfloat has no NaN or Inf
        // encoding) and pre-validate the input grammar so malformed tokens are
        // signaled via false return rather than silently mapped to zero by the
        // d2b path.
        std::size_t pos = 0;
        while (pos < number.size() && std::isspace(static_cast<unsigned char>(number[pos]))) ++pos;
        if (pos >= number.size()) return false;
        const std::size_t numeric_start = pos;
        if (number[pos] == '+' || number[pos] == '-') ++pos;
        if (pos + 3 <= number.size()) {
                char a = static_cast<char>(std::tolower(static_cast<unsigned char>(number[pos])));
                char b = static_cast<char>(std::tolower(static_cast<unsigned char>(number[pos + 1])));
                char c = static_cast<char>(std::tolower(static_cast<unsigned char>(number[pos + 2])));
                if ((a == 'n' && b == 'a' && c == 'n')
                 || (a == 'i' && b == 'n' && c == 'f')) {
                        return false;  // hfloat cannot represent these
                }
        }
        // Pre-validate the rest as a decimal floating-point literal:
        // digits[.digits][eE[+-]digits] or .digits[eE[+-]digits].
        bool seen_digit = false;
        bool seen_dot   = false;
        if (pos >= number.size()) return false;
        while (pos < number.size()) {
                char ch = number[pos];
                if (ch >= '0' && ch <= '9') { seen_digit = true; ++pos; continue; }
                if (ch == '.') {
                        if (seen_dot) return false;
                        seen_dot = true;
                        ++pos;
                        continue;
                }
                break;
        }
        if (!seen_digit) return false;
        if (pos < number.size() && (number[pos] == 'e' || number[pos] == 'E')) {
                ++pos;
                if (pos < number.size() && (number[pos] == '+' || number[pos] == '-')) ++pos;
                bool seen_exp_digit = false;
                while (pos < number.size() && number[pos] >= '0' && number[pos] <= '9') {
                        seen_exp_digit = true;
                        ++pos;
                }
                if (!seen_exp_digit) return false;
        }
        if (pos != number.size()) return false;  // trailing junk

        // Hand the trimmed payload (post-whitespace) to decimal_to_binary.
        // Request fbits of precision when fbits > 64 so wide-fraction configs
        // (hfloat<28,*> with fbits=112) get bit-exact conversion through the
        // d2b multi-limb mantissa instead of being clipped to double-precision
        // like the legacy via-double path. For fbits <= 64 stay at 64 bits --
        // that pinned the hfp64 0.1 truncation test under issue #849 and gives
        // hfp32 a few guard bits for the d2b path's rounding to settle.
        using H = hfloat<ndigits, es, BlockType>;
        constexpr unsigned target_bits = (H::fbits > 64u) ? H::fbits : 64u;
        std::string_view payload(number.data() + numeric_start, number.size() - numeric_start);
        auto d = ::sw::universal::decimal_to_binary::convert(payload, target_bits);
        if (!d.valid) return false;
        if (d.is_zero) {
                value = H(SpecificValue::zero);
                return true;
        }

        // d2b's binary_scale is the exponent of mantissa's MSB, so the
        // "frexp" exponent (value = 1.frac * 2^(bin_exp - 1)) is binary_scale + 1.
        int bin_exp = static_cast<int>(d.binary_scale) + 1;

        if constexpr (H::fbits > 64u) {
                // Wide-fraction path: feed d2b's full multi-limb mantissa directly
                // into the hex-aligned fraction. This is what makes parse() deliver
                // precision beyond what double / long double can carry; the legacy
                // uint64_t intermediate would truncate to ~64 bits and defeat the
                // purpose of routing decimal literals through d2b in the first place.
                value.assign_from_wide_mantissa(d.negative, bin_exp, d);
        }
        else {
                // Extract the 64-bit normalized mantissa (MSB at bit 63) from d2b's
                // bigint. With target_mantissa_bits == 64, mantissa.bit[63] is the
                // implicit-1 of the binary expansion and bits [0, 62] are the fraction.
                std::uint64_t mantissa64 = 0;
                for (unsigned i = 0; i < 64; ++i) {
                        if (d.mantissa.at(i)) mantissa64 |= (std::uint64_t{1} << i);
                }
                value.assign_from_mantissa64(d.negative, bin_exp, mantissa64);
        }
        return true;
}


//////////////////////////////////////////////////////////////////////////////////////////////////////
// hfloat - hfloat binary logic operators

// hfloat values are normalized by normalize_and_pack so that the leading
// hex digit of the fraction is non-zero -- the (sign, exponent, fraction)
// tuple is unique per value.  We compare tuples directly instead of via
// double() to preserve precision for hfloat_long (56-bit fraction) and
// hfloat_extended, where the double round-trip would lose bits below the
// 53-bit IEEE 754 mantissa.
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr bool operator==(const hfloat<ndigits, es, BlockType>& lhs, const hfloat<ndigits, es, BlockType>& rhs) {
        bool lz = lhs.iszero();
        bool rz = rhs.iszero();
        if (lz && rz) return true;       // +0 == -0 in HFP (no signed-zero distinction)
        if (lz || rz) return false;
        if (lhs.sign() != rhs.sign()) return false;
        bool ts; int le = 0, re = 0;
        uint64_t lf = 0, rf = 0;
        lhs.unpack(ts, le, lf);
        rhs.unpack(ts, re, rf);
        return le == re && lf == rf;
}

template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr bool operator!=(const hfloat<ndigits, es, BlockType>& lhs, const hfloat<ndigits, es, BlockType>& rhs) {
        return !operator==(lhs, rhs);
}

template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr bool operator< (const hfloat<ndigits, es, BlockType>& lhs, const hfloat<ndigits, es, BlockType>& rhs) {
        bool lz = lhs.iszero();
        bool rz = rhs.iszero();
        if (lz && rz) return false;
        bool ls = lhs.sign();
        bool rs = rhs.sign();
        if (lz) return !rs;              // 0 < +x; 0 not < -x
        if (rz) return ls;               // -x < 0; +x not < 0
        if (ls != rs) return ls;         // negative < positive
        // Same sign, both non-zero.  unpack returns (sign, biased-exp-removed,
        // fraction).  For positives, lhs < rhs iff (le < re) || tie + (lf < rf).
        // For negatives, the ordering reverses: lhs < rhs iff |lhs| > |rhs|.
        bool ts; int le = 0, re = 0;
        uint64_t lf = 0, rf = 0;
        lhs.unpack(ts, le, lf);
        rhs.unpack(ts, re, rf);
        if (ls) {
                // both negative: lhs < rhs iff |lhs| > |rhs|
                return (le > re) || (le == re && lf > rf);
        }
        // both positive
        return (le < re) || (le == re && lf < rf);
}

template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr bool operator> (const hfloat<ndigits, es, BlockType>& lhs, const hfloat<ndigits, es, BlockType>& rhs) {
        return operator< (rhs, lhs);
}

template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr bool operator<=(const hfloat<ndigits, es, BlockType>& lhs, const hfloat<ndigits, es, BlockType>& rhs) {
        return operator< (lhs, rhs) || operator==(lhs, rhs);
}

template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr bool operator>=(const hfloat<ndigits, es, BlockType>& lhs, const hfloat<ndigits, es, BlockType>& rhs) {
        return !operator< (lhs, rhs);
}

//////////////////////////////////////////////////////////////////////////////////////////////////////
// hfloat - literal binary logic operators
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr bool operator==(const hfloat<ndigits, es, BlockType>& lhs, const double rhs) {
        return operator==(lhs, hfloat<ndigits, es, BlockType>(rhs));
}
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr bool operator!=(const hfloat<ndigits, es, BlockType>& lhs, const double rhs) {
        return !operator==(lhs, rhs);
}
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr bool operator< (const hfloat<ndigits, es, BlockType>& lhs, const double rhs) {
        return operator<(lhs, hfloat<ndigits, es, BlockType>(rhs));
}
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr bool operator> (const hfloat<ndigits, es, BlockType>& lhs, const double rhs) {
        return operator< (hfloat<ndigits, es, BlockType>(rhs), lhs);
}
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr bool operator<=(const hfloat<ndigits, es, BlockType>& lhs, const double rhs) {
        return operator< (lhs, rhs) || operator==(lhs, rhs);
}
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr bool operator>=(const hfloat<ndigits, es, BlockType>& lhs, const double rhs) {
        return !operator< (lhs, rhs);
}

//////////////////////////////////////////////////////////////////////////////////////////////////////
// literal - hfloat binary logic operators
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr bool operator==(const double lhs, const hfloat<ndigits, es, BlockType>& rhs) {
        return operator==(hfloat<ndigits, es, BlockType>(lhs), rhs);
}
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr bool operator!=(const double lhs, const hfloat<ndigits, es, BlockType>& rhs) {
        return !operator==(lhs, rhs);
}
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr bool operator< (const double lhs, const hfloat<ndigits, es, BlockType>& rhs) {
        return operator<(hfloat<ndigits, es, BlockType>(lhs), rhs);
}
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr bool operator> (const double lhs, const hfloat<ndigits, es, BlockType>& rhs) {
        return operator< (rhs, lhs);
}
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr bool operator<=(const double lhs, const hfloat<ndigits, es, BlockType>& rhs) {
        return operator< (lhs, rhs) || operator==(lhs, rhs);
}
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr bool operator>=(const double lhs, const hfloat<ndigits, es, BlockType>& rhs) {
        return !operator< (lhs, rhs);
}

//////////////////////////////////////////////////////////////////////////////////////////////////////
// hfloat - hfloat binary arithmetic operators
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr hfloat<ndigits, es, BlockType> operator+(const hfloat<ndigits, es, BlockType>& lhs, const hfloat<ndigits, es, BlockType>& rhs) {
        hfloat<ndigits, es, BlockType> sum(lhs);
        sum += rhs;
        return sum;
}
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr hfloat<ndigits, es, BlockType> operator-(const hfloat<ndigits, es, BlockType>& lhs, const hfloat<ndigits, es, BlockType>& rhs) {
        hfloat<ndigits, es, BlockType> diff(lhs);
        diff -= rhs;
        return diff;
}
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr hfloat<ndigits, es, BlockType> operator*(const hfloat<ndigits, es, BlockType>& lhs, const hfloat<ndigits, es, BlockType>& rhs) {
        hfloat<ndigits, es, BlockType> mul(lhs);
        mul *= rhs;
        return mul;
}
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr hfloat<ndigits, es, BlockType> operator/(const hfloat<ndigits, es, BlockType>& lhs, const hfloat<ndigits, es, BlockType>& rhs) {
        hfloat<ndigits, es, BlockType> ratio(lhs);
        ratio /= rhs;
        return ratio;
}

//////////////////////////////////////////////////////////////////////////////////////////////////////
// hfloat - literal binary arithmetic operators
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr hfloat<ndigits, es, BlockType> operator+(const hfloat<ndigits, es, BlockType>& lhs, const double rhs) {
        return operator+(lhs, hfloat<ndigits, es, BlockType>(rhs));
}
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr hfloat<ndigits, es, BlockType> operator-(const hfloat<ndigits, es, BlockType>& lhs, const double rhs) {
        return operator-(lhs, hfloat<ndigits, es, BlockType>(rhs));
}
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr hfloat<ndigits, es, BlockType> operator*(const hfloat<ndigits, es, BlockType>& lhs, const double rhs) {
        return operator*(lhs, hfloat<ndigits, es, BlockType>(rhs));
}
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr hfloat<ndigits, es, BlockType> operator/(const hfloat<ndigits, es, BlockType>& lhs, const double rhs) {
        return operator/(lhs, hfloat<ndigits, es, BlockType>(rhs));
}

//////////////////////////////////////////////////////////////////////////////////////////////////////
// literal - hfloat binary arithmetic operators
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr hfloat<ndigits, es, BlockType> operator+(const double lhs, const hfloat<ndigits, es, BlockType>& rhs) {
        return operator+(hfloat<ndigits, es, BlockType>(lhs), rhs);
}
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr hfloat<ndigits, es, BlockType> operator-(const double lhs, const hfloat<ndigits, es, BlockType>& rhs) {
        return operator-(hfloat<ndigits, es, BlockType>(lhs), rhs);
}
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr hfloat<ndigits, es, BlockType> operator*(const double lhs, const hfloat<ndigits, es, BlockType>& rhs) {
        return operator*(hfloat<ndigits, es, BlockType>(lhs), rhs);
}
template<unsigned ndigits, unsigned es, typename BlockType>
inline constexpr hfloat<ndigits, es, BlockType> operator/(const double lhs, const hfloat<ndigits, es, BlockType>& rhs) {
        return operator/(hfloat<ndigits, es, BlockType>(lhs), rhs);
}

}} // namespace sw::universal

stillwater-sc / universal / 26061348223

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