grisu2.h

#pragma once

#include <cstring>
#include <cstdint>
#include <array>
#include <cmath>

namespace grisu2 {
/*!
implements the Grisu2 algorithm for binary to decimal floating-point
conversion.
Adapted from JSON for Modern C++

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
*/

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 {

    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 {

    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;

    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) {

  // 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 && 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
  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;

  const cached_power cached = kCachedPowers[static_cast<std::size_t>(index)];

  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) {

  //               <--------------------------- 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 && delta - rest >= ten_k &&
         (rest + ten_k < dist || dist - rest > rest + ten_k - dist)) {
    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+].

  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]

  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)
    //
    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

  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
    //
    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
    //
    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.
*/
inline void grisu2_core(char *buf, int &len, int &decimal_exponent, diyfp m_minus,
                   diyfp v, diyfp m_plus) {

  //  --------(-----------------------+-----------------------)--------    (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>
void grisu2_wrap(char *buf, int &len, int &decimal_exponent, FloatType value) {
  static_assert(diyfp::kPrecision >= std::numeric_limits<FloatType>::digits + 3,
                "internal error: not enough precision");

  // 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_core(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
*/
inline char *append_exponent(char *buf, int e) {

  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
*/
inline char *format_buffer(char *buf, int len, int decimal_exponent,
                           int min_exp, int max_exp) {

  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 && n <= max_exp) {
    // digits[000]
    // len <= max_exp + 2

    std::memset(buf + k, '0', static_cast<size_t>(n) - static_cast<size_t>(k));
    return buf + (static_cast<size_t>(n));
  }

  if (0 < n && n <= max_exp) {
    // dig.its
    // len <= max_digits10 + 1
    std::memmove(buf + (static_cast<size_t>(n) + 1), buf + n,
                 static_cast<size_t>(k) - static_cast<size_t>(n));
    buf[n] = '.';
    return buf + (static_cast<size_t>(k) + 1U);
  }

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

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

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

/*!
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>
char *to_chars(char *first, FloatType value) {
  bool negative = std::signbit(value);
  if (negative) {
    value = -value;
    *first++ = '-';
  }
  if (value == 0) // +-0
  {
    *first++ = '0';
    return first;
  }
  // 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;
  grisu2_wrap(first, len, decimal_exponent, value);
  // Format the buffer like printf("%.*g", prec, value)
  constexpr int kMinExp = -4;
  constexpr int kMaxExp = std::numeric_limits<double>::digits10;

  return format_buffer(first, len, decimal_exponent, kMinExp, kMaxExp);
}
} // namespace grisu2