ToChars.cpp

/*!********************************************************************

 Audacity: A Digital Audio Editor

 @file ToChars.h
 @brief Define functions to convert numeric types to string representation.

 Dmitry Vedenko
 **********************************************************************/

#include "ToChars.h"

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

#if defined(_MSC_VER) && defined(_M_X64)

#include <intrin.h>

#endif

namespace internal
{
/*

Adaptation of itoa implementation by James Edward Anhalt III: https://github.com/jeaiii/itoa/blob/main/itoa/itoa_jeaiii.cpp

MIT License
Copyright (c) 2017 James Edward Anhalt III - https://github.com/jeaiii/itoa
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
*/
namespace itoa_impl
{
struct pair final
{
   char t, o;
};

#define P(T) T, '0', T, '1', T, '2', T, '3', T, '4', T, '5', T, '6', T, '7', T, '8', T, '9'

static const pair s_pairs[] = { P('0'), P('1'), P('2'), P('3'), P('4'), P('5'), P('6'), P('7'), P('8'), P('9') };

#define W(N, I) *(pair*)&b[N] = s_pairs[I]

#define A(N) t = (uint64_t(1) << (32 + N / 5 * N * 53 / 16)) / uint32_t(1e##N) + 1 + N / 6 - N / 8, t *= u, t >>= N / 5 * N * 53 / 16, t += N / 6 * 4, W(0, t >> 32)

#define S(N) b[N] = char(uint64_t(10) * uint32_t(t) >> 32) + '0'

#define D(N) t = uint64_t(100) * uint32_t(t), W(N, t >> 32)

#define L0 b[0] = char(u) + '0'
#define L1 W(0, u)
#define L2 A(1), S(2)
#define L3 A(2), D(2)
#define L4 A(3), D(2), S(4)
#define L5 A(4), D(2), D(4)
#define L6 A(5), D(2), D(4), S(6)
#define L7 A(6), D(2), D(4), D(6)
#define L8 A(7), D(2), D(4), D(6), S(8)
#define L9 A(8), D(2), D(4), D(6), D(8)

#define LN(N) (L##N, b += N + 1)
#define LZ LN
// if you want to '\0' terminate
//#define LZ(N) &(L##N, b[N + 1] = '\0')

#define LG(F)                                              \
   (u < 100        ? u < 10 ? F(0) : F(1) :                \
    u < 1000000    ? u < 10000  ? u < 1000 ? F(2) : F(3) : \
                     u < 100000 ? F(4) :                   \
                                  F(5) :                   \
    u < 100000000  ? u < 10000000 ? F(6) : F(7) :          \
    u < 1000000000 ? F(8) :                                \
                     F(9))

char* u64toa_jeaiii(uint64_t n, char* b)
{
   uint32_t u;
   uint64_t t;

   if (uint32_t(n >> 32) == 0)
   {
      u = uint32_t(n);
      return LG(LZ);
   }

   uint64_t a = n / 100000000;

   if (uint32_t(a >> 32) == 0)
   {
      u = uint32_t(a);
      LG(LN);
   }
   else
   {
      u = uint32_t(a / 100000000);
      LG(LN);
      u = a % 100000000;
      LN(7);
   }

   u = n % 100000000;
   return LZ(7);
}

} // namespace itoa_impl
/*
Implements the Grisu2 algorithm for binary to decimal floating-point
conversion.

Adapted from simdjson: https://github.com/simdjson/simdjson/blob/master/src/to_chars.cpp

Changes:

* SIMD is returned where possible.
* Buffer size is obeyed
* Some Audacity specific cases covered

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
*/
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
   {

      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

#if defined(_MSC_VER) && defined(_M_X64)

      uint64_t h = 0;
      uint64_t l = _umul128(x.f, y.f, &h);
      h += l >> 63; // round, ties up: [h, l] += 2^q / 2

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

#elif defined(__GNUC__) && defined(__SIZEOF_INT128__)

    __extension__ using Uint128 = unsigned __int128;

    Uint128 const p = Uint128{x.f} * Uint128{y.f};

    uint64_t h = static_cast<uint64_t>(p >> 64);
    uint64_t l = static_cast<uint64_t>(p);
    h += l >> 63; // round, ties up: [h, l] += 2^q / 2

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

#else

      // 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 };
#endif
   }

   /*!
   @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 bool grisu2_digit_gen(
   char* buffer, char* last, 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");

   const int max_length = static_cast<int>(last - buffer);

   // 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)
   {
      // Check that we are able to write the next symbol into the buffer
      if (length >= max_length)
         return false;

      // 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 true;
      }

      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 (;;)
   {
      // Check that we are able to write the next symbol into the buffer
      if (length >= max_length)
         return false;
      // 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)

   return true;
}

/*!
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 bool grisu2(
   char* buf, char* last, 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

   return grisu2_digit_gen(buf, last, 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>
bool grisu2(char* buf, char* last, 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

   return grisu2(buf, last, 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 ToCharsResult append_exponent(char* buf, char* last, int e)
{
   if (buf >= last)
      return { last, std::errc::value_too_large };

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

   auto k = static_cast<std::uint32_t>(e);

   const int requiredSymbolsCount = k < 100 ? 2 : 3;
   char* requiredLast = buf + requiredSymbolsCount + 1;

   if (requiredLast > last)
      return { last, std::errc::value_too_large };

   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, std::errc() };
}

/*!
@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 ToCharsResult format_buffer(
   char* buf, char* last, 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)
   {
      char* requiredLast = buf + (static_cast<size_t>(n));
      if (requiredLast > last)
         return { last, std::errc::value_too_large };
      // digits[000]
      // len <= max_exp + 2

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

   if (0 < n && n <= max_exp)
   {
      char* requiredLast = buf + (static_cast<size_t>(k) + 1U);

      if (requiredLast > last)
         return { last, std::errc::value_too_large };
      // 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 { requiredLast, std::errc() };
   }

   if (min_exp < n && n <= 0)
   {
      char* requiredLast =
         buf + (2U + static_cast<size_t>(-n) + static_cast<size_t>(k));

      if (requiredLast > last)
         return { last, std::errc::value_too_large };
      // 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 { requiredLast, std::errc() };
   }

   if (k == 1)
   {
      char* requiredLast = buf + 1;

      if (requiredLast > last)
         return { last, std::errc::value_too_large };
      // dE+123
      // len <= 1 + 5

      buf += 1;
   }
   else
   {
      char* requiredLast = buf + 1 + static_cast<size_t>(k);

      if (requiredLast > last)
         return { last, std::errc::value_too_large };
      // 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, last, n - 1);
}

} // namespace dtoa_impl

/*!
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 T>
ToCharsResult float_to_chars(
   char* first, char* last, T value, int digitsAfterDecimalPoint)
{
   if (first >= last || first == nullptr)
      return { last, std::errc::value_too_large };

   if (value == 0)
   {
      *first++ = '0';

      return { first, std::errc() };
   }

   if (std::signbit(value))
   {
      value = -value;
      *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;
   if (!dtoa_impl::grisu2(first, last, len, decimal_exponent, value))
      return { last, std::errc::value_too_large };
   // Format the buffer like printf("%.*g", prec, value)
   const int kMinExp = digitsAfterDecimalPoint < 0 ? -4 : -digitsAfterDecimalPoint;
   constexpr int kMaxExp = std::numeric_limits<double>::digits10;

   // Audacity specific extension imitating Internat::ToDisplayString
   // for a consistent behavior
   if (digitsAfterDecimalPoint >= 0)
   {
      if (len + decimal_exponent > 0 && -decimal_exponent > digitsAfterDecimalPoint)
      {
         const int difference = digitsAfterDecimalPoint + decimal_exponent;
         decimal_exponent = -digitsAfterDecimalPoint;
         len += difference;
      }
   }

   return dtoa_impl::format_buffer(
      first, last, len, decimal_exponent, kMinExp, kMaxExp);
}
} // namespace internal

STRING_UTILS_API ToCharsResult ToChars(
   char* buffer, char* last, float value,
   int digitsAfterDecimalPoint) noexcept
{
   return internal::float_to_chars(
      buffer, last, value, digitsAfterDecimalPoint);
}

STRING_UTILS_API ToCharsResult ToChars(
   char* buffer, char* last, double value,
   int digitsAfterDecimalPoint) noexcept
{
   return internal::float_to_chars(
      buffer, last, value, digitsAfterDecimalPoint);
}

ToCharsResult
ToChars(char* buffer, char* last, long long value) noexcept
{
   if (buffer >= last || buffer == nullptr)
      return { last, std::errc::value_too_large };

   if (value < 0)
   {
      *buffer++ = '-';
      value = -value;
   }

   return ToChars(buffer, last, static_cast<unsigned long long>(value));
}

ToCharsResult ToChars(char* buffer, char* last, unsigned long long value) noexcept
{
   if (buffer >= last || buffer == nullptr)
      return { last, std::errc::value_too_large };

   if (value == 0)
   {
      *buffer++ = '0';
      return { buffer, std::errc() };
   }

   constexpr size_t safeSize =
      std::numeric_limits<unsigned long long>::digits10 + 2;

   const size_t bufferSize = static_cast<size_t>(last - buffer);

   if (bufferSize >= safeSize)
      return { internal::itoa_impl::u64toa_jeaiii(value, buffer), std::errc() };

   char tempBuffer[safeSize];
   char* tempLast = internal::itoa_impl::u64toa_jeaiii(value, tempBuffer);

   const size_t resultSize = static_cast<size_t>(tempLast - tempBuffer);

   if (resultSize > bufferSize)
      return { last, std::errc::value_too_large };

   std::copy(tempBuffer, tempLast, buffer);

   return { buffer + resultSize, std::errc() };
}