detail/conversions/to_chars.hpp

#pragma once

#include <array> // array
#include <cassert> // assert
#include <ciso646> // or, and, not
#include <cmath>   // signbit, isfinite
#include <cstdint> // intN_t, uintN_t
#include <cstring> // memcpy, memmove
#include <limits> // numeric_limits
#include <type_traits> // conditional
#include <nlohmann/detail/macro_scope.hpp>

namespace nlohmann
{
namespace detail
{

/*!
@brief implements the Grisu2 algorithm for binary to decimal floating-point
conversion.

This implementation is a slightly modified version of the reference
implementation which may be obtained from
http://florian.loitsch.com/publications (bench.tar.gz).

The code is distributed under the MIT license, Copyright (c) 2009 Florian Loitsch.

For a detailed description of the algorithm see:

[1] Loitsch, "Printing Floating-Point Numbers Quickly and Accurately with
    Integers", Proceedings of the ACM SIGPLAN 2010 Conference on Programming
    Language Design and Implementation, PLDI 2010
[2] Burger, Dybvig, "Printing Floating-Point Numbers Quickly and Accurately",
    Proceedings of the ACM SIGPLAN 1996 Conference on Programming Language
    Design and Implementation, PLDI 1996
*/
namespace dtoa_impl
{

template <typename Target, typename Source>
Target reinterpret_bits(const Source source)
{
    static_assert(sizeof(Target) == sizeof(Source), "size mismatch");

    Target target;
    std::memcpy(&target, &source, sizeof(Source));
    return target;
}

struct diyfp // f * 2^e
{
    static constexpr int kPrecision = 64; // = q

    std::uint64_t f = 0;
    int e = 0;

    constexpr diyfp(std::uint64_t f_, int e_) noexcept : f(f_), e(e_) {}

    /*!
    @brief returns x - y
    @pre x.e == y.e and x.f >= y.f
    */
    static diyfp sub(const diyfp& x, const diyfp& y) noexcept
    {
        assert(x.e == y.e);
        assert(x.f >= y.f);

        return {x.f - y.f, x.e};
    }

    /*!
    @brief returns x * y
    @note The result is rounded. (Only the upper q bits are returned.)
    */
    static diyfp mul(const diyfp& x, const diyfp& y) noexcept
    {
        static_assert(kPrecision == 64, "internal error");

        // Computes:
        //  f = round((x.f * y.f) / 2^q)
        //  e = x.e + y.e + q

        // Emulate the 64-bit * 64-bit multiplication:
        //
        // p = u * v
        //   = (u_lo + 2^32 u_hi) (v_lo + 2^32 v_hi)
        //   = (u_lo v_lo         ) + 2^32 ((u_lo v_hi         ) + (u_hi v_lo         )) + 2^64 (u_hi v_hi         )
        //   = (p0                ) + 2^32 ((p1                ) + (p2                )) + 2^64 (p3                )
        //   = (p0_lo + 2^32 p0_hi) + 2^32 ((p1_lo + 2^32 p1_hi) + (p2_lo + 2^32 p2_hi)) + 2^64 (p3                )
        //   = (p0_lo             ) + 2^32 (p0_hi + p1_lo + p2_lo                      ) + 2^64 (p1_hi + p2_hi + p3)
        //   = (p0_lo             ) + 2^32 (Q                                          ) + 2^64 (H                 )
        //   = (p0_lo             ) + 2^32 (Q_lo + 2^32 Q_hi                           ) + 2^64 (H                 )
        //
        // (Since Q might be larger than 2^32 - 1)
        //
        //   = (p0_lo + 2^32 Q_lo) + 2^64 (Q_hi + H)
        //
        // (Q_hi + H does not overflow a 64-bit int)
        //
        //   = p_lo + 2^64 p_hi

        const std::uint64_t u_lo = x.f & 0xFFFFFFFFu;
        const std::uint64_t u_hi = x.f >> 32u;
        const std::uint64_t v_lo = y.f & 0xFFFFFFFFu;
        const std::uint64_t v_hi = y.f >> 32u;

        const std::uint64_t p0 = u_lo * v_lo;
        const std::uint64_t p1 = u_lo * v_hi;
        const std::uint64_t p2 = u_hi * v_lo;
        const std::uint64_t p3 = u_hi * v_hi;

        const std::uint64_t p0_hi = p0 >> 32u;
        const std::uint64_t p1_lo = p1 & 0xFFFFFFFFu;
        const std::uint64_t p1_hi = p1 >> 32u;
        const std::uint64_t p2_lo = p2 & 0xFFFFFFFFu;
        const std::uint64_t p2_hi = p2 >> 32u;

        std::uint64_t Q = p0_hi + p1_lo + p2_lo;

        // The full product might now be computed as
        //
        // p_hi = p3 + p2_hi + p1_hi + (Q >> 32)
        // p_lo = p0_lo + (Q << 32)
        //
        // But in this particular case here, the full p_lo is not required.
        // Effectively we only need to add the highest bit in p_lo to p_hi (and
        // Q_hi + 1 does not overflow).

        Q += std::uint64_t{1} << (64u - 32u - 1u); // round, ties up

        const std::uint64_t h = p3 + p2_hi + p1_hi + (Q >> 32u);

        return {h, x.e + y.e + 64};
    }

    /*!
    @brief normalize x such that the significand is >= 2^(q-1)
    @pre x.f != 0
    */
    static diyfp normalize(diyfp x) noexcept
    {
        assert(x.f != 0);

        while ((x.f >> 63u) == 0)
        {
            x.f <<= 1u;
            x.e--;
        }

        return x;
    }

    /*!
    @brief normalize x such that the result has the exponent E
    @pre e >= x.e and the upper e - x.e bits of x.f must be zero.
    */
    static diyfp normalize_to(const diyfp& x, const int target_exponent) noexcept
    {
        const int delta = x.e - target_exponent;

        assert(delta >= 0);
        assert(((x.f << delta) >> delta) == x.f);

        return {x.f << delta, target_exponent};
    }
};

struct boundaries
{
    diyfp w;
    diyfp minus;
    diyfp plus;
};

/*!
Compute the (normalized) diyfp representing the input number 'value' and its
boundaries.

@pre value must be finite and positive
*/
template <typename FloatType>
boundaries compute_boundaries(FloatType value)
{
    assert(std::isfinite(value));
    assert(value > 0);

    // Convert the IEEE representation into a diyfp.
    //
    // If v is denormal:
    //      value = 0.F * 2^(1 - bias) = (          F) * 2^(1 - bias - (p-1))
    // If v is normalized:
    //      value = 1.F * 2^(E - bias) = (2^(p-1) + F) * 2^(E - bias - (p-1))

    static_assert(std::numeric_limits<FloatType>::is_iec559,
                  "internal error: dtoa_short requires an IEEE-754 floating-point implementation");

    constexpr int      kPrecision = std::numeric_limits<FloatType>::digits; // = p (includes the hidden bit)
    constexpr int      kBias      = std::numeric_limits<FloatType>::max_exponent - 1 + (kPrecision - 1);
    constexpr int      kMinExp    = 1 - kBias;
    constexpr std::uint64_t kHiddenBit = std::uint64_t{1} << (kPrecision - 1); // = 2^(p-1)

    using bits_type = typename std::conditional<kPrecision == 24, std::uint32_t, std::uint64_t >::type;

    const std::uint64_t bits = reinterpret_bits<bits_type>(value);
    const std::uint64_t E = bits >> (kPrecision - 1);
    const std::uint64_t F = bits & (kHiddenBit - 1);

    const bool is_denormal = E == 0;
    const diyfp v = is_denormal
                    ? diyfp(F, kMinExp)
                    : diyfp(F + kHiddenBit, static_cast<int>(E) - kBias);

    // Compute the boundaries m- and m+ of the floating-point value
    // v = f * 2^e.
    //
    // Determine v- and v+, the floating-point predecessor and successor if v,
    // respectively.
    //
    //      v- = v - 2^e        if f != 2^(p-1) or e == e_min                (A)
    //         = v - 2^(e-1)    if f == 2^(p-1) and e > e_min                (B)
    //
    //      v+ = v + 2^e
    //
    // Let m- = (v- + v) / 2 and m+ = (v + v+) / 2. All real numbers _strictly_
    // between m- and m+ round to v, regardless of how the input rounding
    // algorithm breaks ties.
    //
    //      ---+-------------+-------------+-------------+-------------+---  (A)
    //         v-            m-            v             m+            v+
    //
    //      -----------------+------+------+-------------+-------------+---  (B)
    //                       v-     m-     v             m+            v+

    const bool lower_boundary_is_closer = F == 0 and E > 1;
    const diyfp m_plus = diyfp(2 * v.f + 1, v.e - 1);
    const diyfp m_minus = lower_boundary_is_closer
                          ? diyfp(4 * v.f - 1, v.e - 2)  // (B)
                          : diyfp(2 * v.f - 1, v.e - 1); // (A)

    // Determine the normalized w+ = m+.
    const diyfp w_plus = diyfp::normalize(m_plus);

    // Determine w- = m- such that e_(w-) = e_(w+).
    const diyfp w_minus = diyfp::normalize_to(m_minus, w_plus.e);

    return {diyfp::normalize(v), w_minus, w_plus};
}

// Given normalized diyfp w, Grisu needs to find a (normalized) cached
// power-of-ten c, such that the exponent of the product c * w = f * 2^e lies
// within a certain range [alpha, gamma] (Definition 3.2 from [1])
//
//      alpha <= e = e_c + e_w + q <= gamma
//
// or
//
//      f_c * f_w * 2^alpha <= f_c 2^(e_c) * f_w 2^(e_w) * 2^q
//                          <= f_c * f_w * 2^gamma
//
// Since c and w are normalized, i.e. 2^(q-1) <= f < 2^q, this implies
//
//      2^(q-1) * 2^(q-1) * 2^alpha <= c * w * 2^q < 2^q * 2^q * 2^gamma
//
// or
//
//      2^(q - 2 + alpha) <= c * w < 2^(q + gamma)
//
// The choice of (alpha,gamma) determines the size of the table and the form of
// the digit generation procedure. Using (alpha,gamma)=(-60,-32) works out well
// in practice:
//
// The idea is to cut the number c * w = f * 2^e into two parts, which can be
// processed independently: An integral part p1, and a fractional part p2:
//
//      f * 2^e = ( (f div 2^-e) * 2^-e + (f mod 2^-e) ) * 2^e
//              = (f div 2^-e) + (f mod 2^-e) * 2^e
//              = p1 + p2 * 2^e
//
// The conversion of p1 into decimal form requires a series of divisions and
// modulos by (a power of) 10. These operations are faster for 32-bit than for
// 64-bit integers, so p1 should ideally fit into a 32-bit integer. This can be
// achieved by choosing
//
//      -e >= 32   or   e <= -32 := gamma
//
// In order to convert the fractional part
//
//      p2 * 2^e = p2 / 2^-e = d[-1] / 10^1 + d[-2] / 10^2 + ...
//
// into decimal form, the fraction is repeatedly multiplied by 10 and the digits
// d[-i] are extracted in order:
//
//      (10 * p2) div 2^-e = d[-1]
//      (10 * p2) mod 2^-e = d[-2] / 10^1 + ...
//
// The multiplication by 10 must not overflow. It is sufficient to choose
//
//      10 * p2 < 16 * p2 = 2^4 * p2 <= 2^64.
//
// Since p2 = f mod 2^-e < 2^-e,
//
//      -e <= 60   or   e >= -60 := alpha

constexpr int kAlpha = -60;
constexpr int kGamma = -32;

struct cached_power // c = f * 2^e ~= 10^k
{
    std::uint64_t f;
    int e;
    int k;
};

/*!
For a normalized diyfp w = f * 2^e, this function returns a (normalized) cached
power-of-ten c = f_c * 2^e_c, such that the exponent of the product w * c
satisfies (Definition 3.2 from [1])

     alpha <= e_c + e + q <= gamma.
*/
inline cached_power get_cached_power_for_binary_exponent(int e)
{
    // Now
    //
    //      alpha <= e_c + e + q <= gamma                                    (1)
    //      ==> f_c * 2^alpha <= c * 2^e * 2^q
    //
    // and since the c's are normalized, 2^(q-1) <= f_c,
    //
    //      ==> 2^(q - 1 + alpha) <= c * 2^(e + q)
    //      ==> 2^(alpha - e - 1) <= c
    //
    // If c were an exact power of ten, i.e. c = 10^k, one may determine k as
    //
    //      k = ceil( log_10( 2^(alpha - e - 1) ) )
    //        = ceil( (alpha - e - 1) * log_10(2) )
    //
    // From the paper:
    // "In theory the result of the procedure could be wrong since c is rounded,
    //  and the computation itself is approximated [...]. In practice, however,
    //  this simple function is sufficient."
    //
    // For IEEE double precision floating-point numbers converted into
    // normalized diyfp's w = f * 2^e, with q = 64,
    //
    //      e >= -1022      (min IEEE exponent)
    //           -52        (p - 1)
    //           -52        (p - 1, possibly normalize denormal IEEE numbers)
    //           -11        (normalize the diyfp)
    //         = -1137
    //
    // and
    //
    //      e <= +1023      (max IEEE exponent)
    //           -52        (p - 1)
    //           -11        (normalize the diyfp)
    //         = 960
    //
    // This binary exponent range [-1137,960] results in a decimal exponent
    // range [-307,324]. One does not need to store a cached power for each
    // k in this range. For each such k it suffices to find a cached power
    // such that the exponent of the product lies in [alpha,gamma].
    // This implies that the difference of the decimal exponents of adjacent
    // table entries must be less than or equal to
    //
    //      floor( (gamma - alpha) * log_10(2) ) = 8.
    //
    // (A smaller distance gamma-alpha would require a larger table.)

    // NB:
    // Actually this function returns c, such that -60 <= e_c + e + 64 <= -34.

    constexpr int kCachedPowersMinDecExp = -300;
    constexpr int kCachedPowersDecStep = 8;

    static constexpr std::array<cached_power, 79> kCachedPowers =
    {
        {
            { 0xAB70FE17C79AC6CA, -1060, -300 },
            { 0xFF77B1FCBEBCDC4F, -1034, -292 },
            { 0xBE5691EF416BD60C, -1007, -284 },
            { 0x8DD01FAD907FFC3C,  -980, -276 },
            { 0xD3515C2831559A83,  -954, -268 },
            { 0x9D71AC8FADA6C9B5,  -927, -260 },
            { 0xEA9C227723EE8BCB,  -901, -252 },
            { 0xAECC49914078536D,  -874, -244 },
            { 0x823C12795DB6CE57,  -847, -236 },
            { 0xC21094364DFB5637,  -821, -228 },
            { 0x9096EA6F3848984F,  -794, -220 },
            { 0xD77485CB25823AC7,  -768, -212 },
            { 0xA086CFCD97BF97F4,  -741, -204 },
            { 0xEF340A98172AACE5,  -715, -196 },
            { 0xB23867FB2A35B28E,  -688, -188 },
            { 0x84C8D4DFD2C63F3B,  -661, -180 },
            { 0xC5DD44271AD3CDBA,  -635, -172 },
            { 0x936B9FCEBB25C996,  -608, -164 },
            { 0xDBAC6C247D62A584,  -582, -156 },
            { 0xA3AB66580D5FDAF6,  -555, -148 },
            { 0xF3E2F893DEC3F126,  -529, -140 },
            { 0xB5B5ADA8AAFF80B8,  -502, -132 },
            { 0x87625F056C7C4A8B,  -475, -124 },
            { 0xC9BCFF6034C13053,  -449, -116 },
            { 0x964E858C91BA2655,  -422, -108 },
            { 0xDFF9772470297EBD,  -396, -100 },
            { 0xA6DFBD9FB8E5B88F,  -369,  -92 },
            { 0xF8A95FCF88747D94,  -343,  -84 },
            { 0xB94470938FA89BCF,  -316,  -76 },
            { 0x8A08F0F8BF0F156B,  -289,  -68 },
            { 0xCDB02555653131B6,  -263,  -60 },
            { 0x993FE2C6D07B7FAC,  -236,  -52 },
            { 0xE45C10C42A2B3B06,  -210,  -44 },
            { 0xAA242499697392D3,  -183,  -36 },
            { 0xFD87B5F28300CA0E,  -157,  -28 },
            { 0xBCE5086492111AEB,  -130,  -20 },
            { 0x8CBCCC096F5088CC,  -103,  -12 },
            { 0xD1B71758E219652C,   -77,   -4 },
            { 0x9C40000000000000,   -50,    4 },
            { 0xE8D4A51000000000,   -24,   12 },
            { 0xAD78EBC5AC620000,     3,   20 },
            { 0x813F3978F8940984,    30,   28 },
            { 0xC097CE7BC90715B3,    56,   36 },
            { 0x8F7E32CE7BEA5C70,    83,   44 },
            { 0xD5D238A4ABE98068,   109,   52 },
            { 0x9F4F2726179A2245,   136,   60 },
            { 0xED63A231D4C4FB27,   162,   68 },
            { 0xB0DE65388CC8ADA8,   189,   76 },
            { 0x83C7088E1AAB65DB,   216,   84 },
            { 0xC45D1DF942711D9A,   242,   92 },
            { 0x924D692CA61BE758,   269,  100 },
            { 0xDA01EE641A708DEA,   295,  108 },
            { 0xA26DA3999AEF774A,   322,  116 },
            { 0xF209787BB47D6B85,   348,  124 },
            { 0xB454E4A179DD1877,   375,  132 },
            { 0x865B86925B9BC5C2,   402,  140 },
            { 0xC83553C5C8965D3D,   428,  148 },
            { 0x952AB45CFA97A0B3,   455,  156 },
            { 0xDE469FBD99A05FE3,   481,  164 },
            { 0xA59BC234DB398C25,   508,  172 },
            { 0xF6C69A72A3989F5C,   534,  180 },
            { 0xB7DCBF5354E9BECE,   561,  188 },
            { 0x88FCF317F22241E2,   588,  196 },
            { 0xCC20CE9BD35C78A5,   614,  204 },
            { 0x98165AF37B2153DF,   641,  212 },
            { 0xE2A0B5DC971F303A,   667,  220 },
            { 0xA8D9D1535CE3B396,   694,  228 },
            { 0xFB9B7CD9A4A7443C,   720,  236 },
            { 0xBB764C4CA7A44410,   747,  244 },
            { 0x8BAB8EEFB6409C1A,   774,  252 },
            { 0xD01FEF10A657842C,   800,  260 },
            { 0x9B10A4E5E9913129,   827,  268 },
            { 0xE7109BFBA19C0C9D,   853,  276 },
            { 0xAC2820D9623BF429,   880,  284 },
            { 0x80444B5E7AA7CF85,   907,  292 },
            { 0xBF21E44003ACDD2D,   933,  300 },
            { 0x8E679C2F5E44FF8F,   960,  308 },
            { 0xD433179D9C8CB841,   986,  316 },
            { 0x9E19DB92B4E31BA9,  1013,  324 },
        }
    };

    // This computation gives exactly the same results for k as
    //      k = ceil((kAlpha - e - 1) * 0.30102999566398114)
    // for |e| <= 1500, but doesn't require floating-point operations.
    // NB: log_10(2) ~= 78913 / 2^18
    assert(e >= -1500);
    assert(e <=  1500);
    const int f = kAlpha - e - 1;
    const int k = (f * 78913) / (1 << 18) + static_cast<int>(f > 0);

    const int index = (-kCachedPowersMinDecExp + k + (kCachedPowersDecStep - 1)) / kCachedPowersDecStep;
    assert(index >= 0);
    assert(static_cast<std::size_t>(index) < kCachedPowers.size());

    const cached_power cached = kCachedPowers[static_cast<std::size_t>(index)];
    assert(kAlpha <= cached.e + e + 64);
    assert(kGamma >= cached.e + e + 64);

    return cached;
}

/*!
For n != 0, returns k, such that pow10 := 10^(k-1) <= n < 10^k.
For n == 0, returns 1 and sets pow10 := 1.
*/
inline int find_largest_pow10(const std::uint32_t n, std::uint32_t& pow10)
{
    // LCOV_EXCL_START
    if (n >= 1000000000)
    {
        pow10 = 1000000000;
        return 10;
    }
    // LCOV_EXCL_STOP
    else if (n >= 100000000)
    {
        pow10 = 100000000;
        return  9;
    }
    else if (n >= 10000000)
    {
        pow10 = 10000000;
        return  8;
    }
    else if (n >= 1000000)
    {
        pow10 = 1000000;
        return  7;
    }
    else if (n >= 100000)
    {
        pow10 = 100000;
        return  6;
    }
    else if (n >= 10000)
    {
        pow10 = 10000;
        return  5;
    }
    else if (n >= 1000)
    {
        pow10 = 1000;
        return  4;
    }
    else if (n >= 100)
    {
        pow10 = 100;
        return  3;
    }
    else if (n >= 10)
    {
        pow10 = 10;
        return  2;
    }
    else
    {
        pow10 = 1;
        return 1;
    }
}

inline void grisu2_round(char* buf, int len, std::uint64_t dist, std::uint64_t delta,
                         std::uint64_t rest, std::uint64_t ten_k)
{
    assert(len >= 1);
    assert(dist <= delta);
    assert(rest <= delta);
    assert(ten_k > 0);

    //               <--------------------------- delta ---->
    //                                  <---- dist --------->
    // --------------[------------------+-------------------]--------------
    //               M-                 w                   M+
    //
    //                                  ten_k
    //                                <------>
    //                                       <---- rest ---->
    // --------------[------------------+----+--------------]--------------
    //                                  w    V
    //                                       = buf * 10^k
    //
    // ten_k represents a unit-in-the-last-place in the decimal representation
    // stored in buf.
    // Decrement buf by ten_k while this takes buf closer to w.

    // The tests are written in this order to avoid overflow in unsigned
    // integer arithmetic.

    while (rest < dist
            and delta - rest >= ten_k
            and (rest + ten_k < dist or dist - rest > rest + ten_k - dist))
    {
        assert(buf[len - 1] != '0');
        buf[len - 1]--;
        rest += ten_k;
    }
}

/*!
Generates V = buffer * 10^decimal_exponent, such that M- <= V <= M+.
M- and M+ must be normalized and share the same exponent -60 <= e <= -32.
*/
inline void grisu2_digit_gen(char* buffer, int& length, int& decimal_exponent,
                             diyfp M_minus, diyfp w, diyfp M_plus)
{
    static_assert(kAlpha >= -60, "internal error");
    static_assert(kGamma <= -32, "internal error");

    // Generates the digits (and the exponent) of a decimal floating-point
    // number V = buffer * 10^decimal_exponent in the range [M-, M+]. The diyfp's
    // w, M- and M+ share the same exponent e, which satisfies alpha <= e <= gamma.
    //
    //               <--------------------------- delta ---->
    //                                  <---- dist --------->
    // --------------[------------------+-------------------]--------------
    //               M-                 w                   M+
    //
    // Grisu2 generates the digits of M+ from left to right and stops as soon as
    // V is in [M-,M+].

    assert(M_plus.e >= kAlpha);
    assert(M_plus.e <= kGamma);

    std::uint64_t delta = diyfp::sub(M_plus, M_minus).f; // (significand of (M+ - M-), implicit exponent is e)
    std::uint64_t dist  = diyfp::sub(M_plus, w      ).f; // (significand of (M+ - w ), implicit exponent is e)

    // Split M+ = f * 2^e into two parts p1 and p2 (note: e < 0):
    //
    //      M+ = f * 2^e
    //         = ((f div 2^-e) * 2^-e + (f mod 2^-e)) * 2^e
    //         = ((p1        ) * 2^-e + (p2        )) * 2^e
    //         = p1 + p2 * 2^e

    const diyfp one(std::uint64_t{1} << -M_plus.e, M_plus.e);

    auto p1 = static_cast<std::uint32_t>(M_plus.f >> -one.e); // p1 = f div 2^-e (Since -e >= 32, p1 fits into a 32-bit int.)
    std::uint64_t p2 = M_plus.f & (one.f - 1);                    // p2 = f mod 2^-e

    // 1)
    //
    // Generate the digits of the integral part p1 = d[n-1]...d[1]d[0]

    assert(p1 > 0);

    std::uint32_t pow10;
    const int k = find_largest_pow10(p1, pow10);

    //      10^(k-1) <= p1 < 10^k, pow10 = 10^(k-1)
    //
    //      p1 = (p1 div 10^(k-1)) * 10^(k-1) + (p1 mod 10^(k-1))
    //         = (d[k-1]         ) * 10^(k-1) + (p1 mod 10^(k-1))
    //
    //      M+ = p1                                             + p2 * 2^e
    //         = d[k-1] * 10^(k-1) + (p1 mod 10^(k-1))          + p2 * 2^e
    //         = d[k-1] * 10^(k-1) + ((p1 mod 10^(k-1)) * 2^-e + p2) * 2^e
    //         = d[k-1] * 10^(k-1) + (                         rest) * 2^e
    //
    // Now generate the digits d[n] of p1 from left to right (n = k-1,...,0)
    //
    //      p1 = d[k-1]...d[n] * 10^n + d[n-1]...d[0]
    //
    // but stop as soon as
    //
    //      rest * 2^e = (d[n-1]...d[0] * 2^-e + p2) * 2^e <= delta * 2^e

    int n = k;
    while (n > 0)
    {
        // Invariants:
        //      M+ = buffer * 10^n + (p1 + p2 * 2^e)    (buffer = 0 for n = k)
        //      pow10 = 10^(n-1) <= p1 < 10^n
        //
        const std::uint32_t d = p1 / pow10;  // d = p1 div 10^(n-1)
        const std::uint32_t r = p1 % pow10;  // r = p1 mod 10^(n-1)
        //
        //      M+ = buffer * 10^n + (d * 10^(n-1) + r) + p2 * 2^e
        //         = (buffer * 10 + d) * 10^(n-1) + (r + p2 * 2^e)
        //
        assert(d <= 9);
        buffer[length++] = static_cast<char>('0' + d); // buffer := buffer * 10 + d
        //
        //      M+ = buffer * 10^(n-1) + (r + p2 * 2^e)
        //
        p1 = r;
        n--;
        //
        //      M+ = buffer * 10^n + (p1 + p2 * 2^e)
        //      pow10 = 10^n
        //

        // Now check if enough digits have been generated.
        // Compute
        //
        //      p1 + p2 * 2^e = (p1 * 2^-e + p2) * 2^e = rest * 2^e
        //
        // Note:
        // Since rest and delta share the same exponent e, it suffices to
        // compare the significands.
        const std::uint64_t rest = (std::uint64_t{p1} << -one.e) + p2;
        if (rest <= delta)
        {
            // V = buffer * 10^n, with M- <= V <= M+.

            decimal_exponent += n;

            // We may now just stop. But instead look if the buffer could be
            // decremented to bring V closer to w.
            //
            // pow10 = 10^n is now 1 ulp in the decimal representation V.
            // The rounding procedure works with diyfp's with an implicit
            // exponent of e.
            //
            //      10^n = (10^n * 2^-e) * 2^e = ulp * 2^e
            //
            const std::uint64_t ten_n = std::uint64_t{pow10} << -one.e;
            grisu2_round(buffer, length, dist, delta, rest, ten_n);

            return;
        }

        pow10 /= 10;
        //
        //      pow10 = 10^(n-1) <= p1 < 10^n
        // Invariants restored.
    }

    // 2)
    //
    // The digits of the integral part have been generated:
    //
    //      M+ = d[k-1]...d[1]d[0] + p2 * 2^e
    //         = buffer            + p2 * 2^e
    //
    // Now generate the digits of the fractional part p2 * 2^e.
    //
    // Note:
    // No decimal point is generated: the exponent is adjusted instead.
    //
    // p2 actually represents the fraction
    //
    //      p2 * 2^e
    //          = p2 / 2^-e
    //          = d[-1] / 10^1 + d[-2] / 10^2 + ...
    //
    // Now generate the digits d[-m] of p1 from left to right (m = 1,2,...)
    //
    //      p2 * 2^e = d[-1]d[-2]...d[-m] * 10^-m
    //                      + 10^-m * (d[-m-1] / 10^1 + d[-m-2] / 10^2 + ...)
    //
    // using
    //
    //      10^m * p2 = ((10^m * p2) div 2^-e) * 2^-e + ((10^m * p2) mod 2^-e)
    //                = (                   d) * 2^-e + (                   r)
    //
    // or
    //      10^m * p2 * 2^e = d + r * 2^e
    //
    // i.e.
    //
    //      M+ = buffer + p2 * 2^e
    //         = buffer + 10^-m * (d + r * 2^e)
    //         = (buffer * 10^m + d) * 10^-m + 10^-m * r * 2^e
    //
    // and stop as soon as 10^-m * r * 2^e <= delta * 2^e

    assert(p2 > delta);

    int m = 0;
    for (;;)
    {
        // Invariant:
        //      M+ = buffer * 10^-m + 10^-m * (d[-m-1] / 10 + d[-m-2] / 10^2 + ...) * 2^e
        //         = buffer * 10^-m + 10^-m * (p2                                 ) * 2^e
        //         = buffer * 10^-m + 10^-m * (1/10 * (10 * p2)                   ) * 2^e
        //         = buffer * 10^-m + 10^-m * (1/10 * ((10*p2 div 2^-e) * 2^-e + (10*p2 mod 2^-e)) * 2^e
        //
        assert(p2 <= (std::numeric_limits<std::uint64_t>::max)() / 10);
        p2 *= 10;
        const std::uint64_t d = p2 >> -one.e;     // d = (10 * p2) div 2^-e
        const std::uint64_t r = p2 & (one.f - 1); // r = (10 * p2) mod 2^-e
        //
        //      M+ = buffer * 10^-m + 10^-m * (1/10 * (d * 2^-e + r) * 2^e
        //         = buffer * 10^-m + 10^-m * (1/10 * (d + r * 2^e))
        //         = (buffer * 10 + d) * 10^(-m-1) + 10^(-m-1) * r * 2^e
        //
        assert(d <= 9);
        buffer[length++] = static_cast<char>('0' + d); // buffer := buffer * 10 + d
        //
        //      M+ = buffer * 10^(-m-1) + 10^(-m-1) * r * 2^e
        //
        p2 = r;
        m++;
        //
        //      M+ = buffer * 10^-m + 10^-m * p2 * 2^e
        // Invariant restored.

        // Check if enough digits have been generated.
        //
        //      10^-m * p2 * 2^e <= delta * 2^e
        //              p2 * 2^e <= 10^m * delta * 2^e
        //                    p2 <= 10^m * delta
        delta *= 10;
        dist  *= 10;
        if (p2 <= delta)
        {
            break;
        }
    }

    // V = buffer * 10^-m, with M- <= V <= M+.

    decimal_exponent -= m;

    // 1 ulp in the decimal representation is now 10^-m.
    // Since delta and dist are now scaled by 10^m, we need to do the
    // same with ulp in order to keep the units in sync.
    //
    //      10^m * 10^-m = 1 = 2^-e * 2^e = ten_m * 2^e
    //
    const std::uint64_t ten_m = one.f;
    grisu2_round(buffer, length, dist, delta, p2, ten_m);

    // By construction this algorithm generates the shortest possible decimal
    // number (Loitsch, Theorem 6.2) which rounds back to w.
    // For an input number of precision p, at least
    //
    //      N = 1 + ceil(p * log_10(2))
    //
    // decimal digits are sufficient to identify all binary floating-point
    // numbers (Matula, "In-and-Out conversions").
    // This implies that the algorithm does not produce more than N decimal
    // digits.
    //
    //      N = 17 for p = 53 (IEEE double precision)
    //      N = 9  for p = 24 (IEEE single precision)
}

/*!
v = buf * 10^decimal_exponent
len is the length of the buffer (number of decimal digits)
The buffer must be large enough, i.e. >= max_digits10.
*/
JSON_HEDLEY_NON_NULL(1)
inline void grisu2(char* buf, int& len, int& decimal_exponent,
                   diyfp m_minus, diyfp v, diyfp m_plus)
{
    assert(m_plus.e == m_minus.e);
    assert(m_plus.e == v.e);

    //  --------(-----------------------+-----------------------)--------    (A)
    //          m-                      v                       m+
    //
    //  --------------------(-----------+-----------------------)--------    (B)
    //                      m-          v                       m+
    //
    // First scale v (and m- and m+) such that the exponent is in the range
    // [alpha, gamma].

    const cached_power cached = get_cached_power_for_binary_exponent(m_plus.e);

    const diyfp c_minus_k(cached.f, cached.e); // = c ~= 10^-k

    // The exponent of the products is = v.e + c_minus_k.e + q and is in the range [alpha,gamma]
    const diyfp w       = diyfp::mul(v,       c_minus_k);
    const diyfp w_minus = diyfp::mul(m_minus, c_minus_k);
    const diyfp w_plus  = diyfp::mul(m_plus,  c_minus_k);

    //  ----(---+---)---------------(---+---)---------------(---+---)----
    //          w-                      w                       w+
    //          = c*m-                  = c*v                   = c*m+
    //
    // diyfp::mul rounds its result and c_minus_k is approximated too. w, w- and
    // w+ are now off by a small amount.
    // In fact:
    //
    //      w - v * 10^k < 1 ulp
    //
    // To account for this inaccuracy, add resp. subtract 1 ulp.
    //
    //  --------+---[---------------(---+---)---------------]---+--------
    //          w-  M-                  w                   M+  w+
    //
    // Now any number in [M-, M+] (bounds included) will round to w when input,
    // regardless of how the input rounding algorithm breaks ties.
    //
    // And digit_gen generates the shortest possible such number in [M-, M+].
    // Note that this does not mean that Grisu2 always generates the shortest
    // possible number in the interval (m-, m+).
    const diyfp M_minus(w_minus.f + 1, w_minus.e);
    const diyfp M_plus (w_plus.f  - 1, w_plus.e );

    decimal_exponent = -cached.k; // = -(-k) = k

    grisu2_digit_gen(buf, len, decimal_exponent, M_minus, w, M_plus);
}

/*!
v = buf * 10^decimal_exponent
len is the length of the buffer (number of decimal digits)
The buffer must be large enough, i.e. >= max_digits10.
*/
template <typename FloatType>
JSON_HEDLEY_NON_NULL(1)
void grisu2(char* buf, int& len, int& decimal_exponent, FloatType value)
{
    static_assert(diyfp::kPrecision >= std::numeric_limits<FloatType>::digits + 3,
                  "internal error: not enough precision");

    assert(std::isfinite(value));
    assert(value > 0);

    // If the neighbors (and boundaries) of 'value' are always computed for double-precision
    // numbers, all float's can be recovered using strtod (and strtof). However, the resulting
    // decimal representations are not exactly "short".
    //
    // The documentation for 'std::to_chars' (https://en.cppreference.com/w/cpp/utility/to_chars)
    // says "value is converted to a string as if by std::sprintf in the default ("C") locale"
    // and since sprintf promotes float's to double's, I think this is exactly what 'std::to_chars'
    // does.
    // On the other hand, the documentation for 'std::to_chars' requires that "parsing the
    // representation using the corresponding std::from_chars function recovers value exactly". That
    // indicates that single precision floating-point numbers should be recovered using
    // 'std::strtof'.
    //
    // NB: If the neighbors are computed for single-precision numbers, there is a single float
    //     (7.0385307e-26f) which can't be recovered using strtod. The resulting double precision
    //     value is off by 1 ulp.
#if 0
    const boundaries w = compute_boundaries(static_cast<double>(value));
#else
    const boundaries w = compute_boundaries(value);
#endif

    grisu2(buf, len, decimal_exponent, w.minus, w.w, w.plus);
}

/*!
@brief appends a decimal representation of e to buf
@return a pointer to the element following the exponent.
@pre -1000 < e < 1000
*/
JSON_HEDLEY_NON_NULL(1)
JSON_HEDLEY_RETURNS_NON_NULL
inline char* append_exponent(char* buf, int e)
{
    assert(e > -1000);
    assert(e <  1000);

    if (e < 0)
    {
        e = -e;
        *buf++ = '-';
    }
    else
    {
        *buf++ = '+';
    }

    auto k = static_cast<std::uint32_t>(e);
    if (k < 10)
    {
        // Always print at least two digits in the exponent.
        // This is for compatibility with printf("%g").
        *buf++ = '0';
        *buf++ = static_cast<char>('0' + k);
    }
    else if (k < 100)
    {
        *buf++ = static_cast<char>('0' + k / 10);
        k %= 10;
        *buf++ = static_cast<char>('0' + k);
    }
    else
    {
        *buf++ = static_cast<char>('0' + k / 100);
        k %= 100;
        *buf++ = static_cast<char>('0' + k / 10);
        k %= 10;
        *buf++ = static_cast<char>('0' + k);
    }

    return buf;
}

/*!
@brief prettify v = buf * 10^decimal_exponent

If v is in the range [10^min_exp, 10^max_exp) it will be printed in fixed-point
notation. Otherwise it will be printed in exponential notation.

@pre min_exp < 0
@pre max_exp > 0
*/
JSON_HEDLEY_NON_NULL(1)
JSON_HEDLEY_RETURNS_NON_NULL
inline char* format_buffer(char* buf, int len, int decimal_exponent,
                           int min_exp, int max_exp)
{
    assert(min_exp < 0);
    assert(max_exp > 0);

    const int k = len;
    const int n = len + decimal_exponent;

    // v = buf * 10^(n-k)
    // k is the length of the buffer (number of decimal digits)
    // n is the position of the decimal point relative to the start of the buffer.

    if (k <= n and n <= max_exp)
    {
        // digits[000]
        // len <= max_exp + 2

        std::memset(buf + k, '0', static_cast<size_t>(n - k));
        // Make it look like a floating-point number (#362, #378)
        buf[n + 0] = '.';
        buf[n + 1] = '0';
        return buf + (n + 2);
    }

    if (0 < n and n <= max_exp)
    {
        // dig.its
        // len <= max_digits10 + 1

        assert(k > n);

        std::memmove(buf + (n + 1), buf + n, static_cast<size_t>(k - n));
        buf[n] = '.';
        return buf + (k + 1);
    }

    if (min_exp < n and n <= 0)
    {
        // 0.[000]digits
        // len <= 2 + (-min_exp - 1) + max_digits10

        std::memmove(buf + (2 + -n), buf, static_cast<size_t>(k));
        buf[0] = '0';
        buf[1] = '.';
        std::memset(buf + 2, '0', static_cast<size_t>(-n));
        return buf + (2 + (-n) + k);
    }

    if (k == 1)
    {
        // dE+123
        // len <= 1 + 5

        buf += 1;
    }
    else
    {
        // d.igitsE+123
        // len <= max_digits10 + 1 + 5

        std::memmove(buf + 2, buf + 1, static_cast<size_t>(k - 1));
        buf[1] = '.';
        buf += 1 + k;
    }

    *buf++ = 'e';
    return append_exponent(buf, n - 1);
}

} // namespace dtoa_impl

/*!
@brief generates a decimal representation of the floating-point number value in [first, last).

The format of the resulting decimal representation is similar to printf's %g
format. Returns an iterator pointing past-the-end of the decimal representation.

@note The input number must be finite, i.e. NaN's and Inf's are not supported.
@note The buffer must be large enough.
@note The result is NOT null-terminated.
*/
template <typename FloatType>
JSON_HEDLEY_NON_NULL(1, 2)
JSON_HEDLEY_RETURNS_NON_NULL
char* to_chars(char* first, const char* last, FloatType value)
{
    static_cast<void>(last); // maybe unused - fix warning
    assert(std::isfinite(value));

    // Use signbit(value) instead of (value < 0) since signbit works for -0.
    if (std::signbit(value))
    {
        value = -value;
        *first++ = '-';
    }

    if (value == 0) // +-0
    {
        *first++ = '0';
        // Make it look like a floating-point number (#362, #378)
        *first++ = '.';
        *first++ = '0';
        return first;
    }

    assert(last - first >= std::numeric_limits<FloatType>::max_digits10);

    // Compute v = buffer * 10^decimal_exponent.
    // The decimal digits are stored in the buffer, which needs to be interpreted
    // as an unsigned decimal integer.
    // len is the length of the buffer, i.e. the number of decimal digits.
    int len = 0;
    int decimal_exponent = 0;
    dtoa_impl::grisu2(first, len, decimal_exponent, value);

    assert(len <= std::numeric_limits<FloatType>::max_digits10);

    // Format the buffer like printf("%.*g", prec, value)
    constexpr int kMinExp = -4;
    // Use digits10 here to increase compatibility with version 2.
    constexpr int kMaxExp = std::numeric_limits<FloatType>::digits10;

    assert(last - first >= kMaxExp + 2);
    assert(last - first >= 2 + (-kMinExp - 1) + std::numeric_limits<FloatType>::max_digits10);
    assert(last - first >= std::numeric_limits<FloatType>::max_digits10 + 6);

    return dtoa_impl::format_buffer(first, len, decimal_exponent, kMinExp, kMaxExp);
}

} // namespace detail
} // namespace nlohmann