text: // Formatting library for C++ - implementation
  //
  // Copyright (c) 2012 - 2016, Victor Zverovich
  // All rights reserved.
  //
  // For the license information refer to format.h.
  
  #ifndef FMT_FORMAT_INL_H_
  #define FMT_FORMAT_INL_H_
  
  #include <algorithm>
  #include <cctype>
  #include <cerrno>  // errno
  #include <climits>
  #include <cmath>
  #include <cstdarg>
  #include <cstring>  // std::memmove
  #include <cwchar>
  #include <exception>
  
  #ifndef FMT_STATIC_THOUSANDS_SEPARATOR
  #  include <locale>
  #endif
  
  #ifdef _WIN32
  #  include <io.h>  // _isatty
  #endif
  
  #include "format.h"
  
  FMT_BEGIN_NAMESPACE
  namespace detail {
  
  FMT_FUNC void assert_fail(const char* file, int line, const char* message) {
    // Use unchecked std::fprintf to avoid triggering another assertion when
    // writing to stderr fails
    std::fprintf(stderr, "%s:%d: assertion failed: %s", file, line, message);
    // Chosen instead of std::abort to satisfy Clang in CUDA mode during device
    // code pass.
    std::terminate();
  }
  
  FMT_FUNC void throw_format_error(const char* message) {
    FMT_THROW(format_error(message));
  }
  
  #ifndef _MSC_VER
  #  define FMT_SNPRINTF snprintf
  #else  // _MSC_VER
  inline int fmt_snprintf(char* buffer, size_t size, const char* format, ...) {
    va_list args;
    va_start(args, format);
    int result = vsnprintf_s(buffer, size, _TRUNCATE, format, args);
    va_end(args);
    return result;
  }
  #  define FMT_SNPRINTF fmt_snprintf
  #endif  // _MSC_VER
  
  FMT_FUNC void format_error_code(detail::buffer<char>& out, int error_code,
                                  string_view message) FMT_NOEXCEPT {
    // Report error code making sure that the output fits into
    // inline_buffer_size to avoid dynamic memory allocation and potential
    // bad_alloc.
    out.try_resize(0);
    static const char SEP[] = ": ";
    static const char ERROR_STR[] = "error ";
    // Subtract 2 to account for terminating null characters in SEP and ERROR_STR.
    size_t error_code_size = sizeof(SEP) + sizeof(ERROR_STR) - 2;
    auto abs_value = static_cast<uint32_or_64_or_128_t<int>>(error_code);
    if (detail::is_negative(error_code)) {
      abs_value = 0 - abs_value;
      ++error_code_size;
    }
    error_code_size += detail::to_unsigned(detail::count_digits(abs_value));
    auto it = buffer_appender<char>(out);
    if (message.size() <= inline_buffer_size - error_code_size)
      format_to(it, FMT_STRING("{}{}"), message, SEP);
    format_to(it, FMT_STRING("{}{}"), ERROR_STR, error_code);
    FMT_ASSERT(out.size() <= inline_buffer_size, "");
  }
  
  FMT_FUNC void report_error(format_func func, int error_code,
                             const char* message) FMT_NOEXCEPT {
    memory_buffer full_message;
    func(full_message, error_code, message);
    // Don't use fwrite_fully because the latter may throw.
    if (std::fwrite(full_message.data(), full_message.size(), 1, stderr) > 0)
      std::fputc('\n', stderr);
  }
  
  // A wrapper around fwrite that throws on error.
  inline void fwrite_fully(const void* ptr, size_t size, size_t count,
                           FILE* stream) {
    size_t written = std::fwrite(ptr, size, count, stream);
    if (written < count) FMT_THROW(system_error(errno, "cannot write to file"));
  }
  
  #ifndef FMT_STATIC_THOUSANDS_SEPARATOR
  template <typename Locale>
  locale_ref::locale_ref(const Locale& loc) : locale_(&loc) {
    static_assert(std::is_same<Locale, std::locale>::value, "");
  }
  
  template <typename Locale> Locale locale_ref::get() const {
    static_assert(std::is_same<Locale, std::locale>::value, "");
    return locale_ ? *static_cast<const std::locale*>(locale_) : std::locale();
  }
  
  template <typename Char>
  FMT_FUNC auto thousands_sep_impl(locale_ref loc) -> thousands_sep_result<Char> {
    auto& facet = std::use_facet<std::numpunct<Char>>(loc.get<std::locale>());
    auto grouping = facet.grouping();
    auto thousands_sep = grouping.empty() ? Char() : facet.thousands_sep();
    return {std::move(grouping), thousands_sep};
  }
  template <typename Char> FMT_FUNC Char decimal_point_impl(locale_ref loc) {
    return std::use_facet<std::numpunct<Char>>(loc.get<std::locale>())
        .decimal_point();
  }
  #else
  template <typename Char>
  FMT_FUNC auto thousands_sep_impl(locale_ref) -> thousands_sep_result<Char> {
    return {"\03", FMT_STATIC_THOUSANDS_SEPARATOR};
  }
  template <typename Char> FMT_FUNC Char decimal_point_impl(locale_ref) {
    return '.';
  }
  #endif
  }  // namespace detail
  
  #if !FMT_MSC_VER
  FMT_API FMT_FUNC format_error::~format_error() FMT_NOEXCEPT = default;
  #endif
  
  FMT_FUNC std::system_error vsystem_error(int error_code, string_view format_str,
                                           format_args args) {
    auto ec = std::error_code(error_code, std::generic_category());
    return std::system_error(ec, vformat(format_str, args));
  }
  
  namespace detail {
  
  template <> FMT_FUNC int count_digits<4>(detail::fallback_uintptr n) {
    // fallback_uintptr is always stored in little endian.
    int i = static_cast<int>(sizeof(void*)) - 1;
    while (i > 0 && n.value[i] == 0) --i;
    auto char_digits = std::numeric_limits<unsigned char>::digits / 4;
    return i >= 0 ? i * char_digits + count_digits<4, unsigned>(n.value[i]) : 1;
  }
  
  // log10(2) = 0x0.4d104d427de7fbcc...
  static constexpr uint64_t log10_2_significand = 0x4d104d427de7fbcc;
  
  template <typename T = void> struct basic_impl_data {
    // Normalized 64-bit significands of pow(10, k), for k = -348, -340, ..., 340.
    // These are generated by support/compute-powers.py.
    static constexpr uint64_t pow10_significands[87] = {
        0xfa8fd5a0081c0288, 0xbaaee17fa23ebf76, 0x8b16fb203055ac76,
        0xcf42894a5dce35ea, 0x9a6bb0aa55653b2d, 0xe61acf033d1a45df,
        0xab70fe17c79ac6ca, 0xff77b1fcbebcdc4f, 0xbe5691ef416bd60c,
        0x8dd01fad907ffc3c, 0xd3515c2831559a83, 0x9d71ac8fada6c9b5,
        0xea9c227723ee8bcb, 0xaecc49914078536d, 0x823c12795db6ce57,
        0xc21094364dfb5637, 0x9096ea6f3848984f, 0xd77485cb25823ac7,
        0xa086cfcd97bf97f4, 0xef340a98172aace5, 0xb23867fb2a35b28e,
        0x84c8d4dfd2c63f3b, 0xc5dd44271ad3cdba, 0x936b9fcebb25c996,
        0xdbac6c247d62a584, 0xa3ab66580d5fdaf6, 0xf3e2f893dec3f126,
        0xb5b5ada8aaff80b8, 0x87625f056c7c4a8b, 0xc9bcff6034c13053,
        0x964e858c91ba2655, 0xdff9772470297ebd, 0xa6dfbd9fb8e5b88f,
        0xf8a95fcf88747d94, 0xb94470938fa89bcf, 0x8a08f0f8bf0f156b,
        0xcdb02555653131b6, 0x993fe2c6d07b7fac, 0xe45c10c42a2b3b06,
        0xaa242499697392d3, 0xfd87b5f28300ca0e, 0xbce5086492111aeb,
        0x8cbccc096f5088cc, 0xd1b71758e219652c, 0x9c40000000000000,
        0xe8d4a51000000000, 0xad78ebc5ac620000, 0x813f3978f8940984,
        0xc097ce7bc90715b3, 0x8f7e32ce7bea5c70, 0xd5d238a4abe98068,
        0x9f4f2726179a2245, 0xed63a231d4c4fb27, 0xb0de65388cc8ada8,
        0x83c7088e1aab65db, 0xc45d1df942711d9a, 0x924d692ca61be758,
        0xda01ee641a708dea, 0xa26da3999aef774a, 0xf209787bb47d6b85,
        0xb454e4a179dd1877, 0x865b86925b9bc5c2, 0xc83553c5c8965d3d,
        0x952ab45cfa97a0b3, 0xde469fbd99a05fe3, 0xa59bc234db398c25,
        0xf6c69a72a3989f5c, 0xb7dcbf5354e9bece, 0x88fcf317f22241e2,
        0xcc20ce9bd35c78a5, 0x98165af37b2153df, 0xe2a0b5dc971f303a,
        0xa8d9d1535ce3b396, 0xfb9b7cd9a4a7443c, 0xbb764c4ca7a44410,
        0x8bab8eefb6409c1a, 0xd01fef10a657842c, 0x9b10a4e5e9913129,
        0xe7109bfba19c0c9d, 0xac2820d9623bf429, 0x80444b5e7aa7cf85,
        0xbf21e44003acdd2d, 0x8e679c2f5e44ff8f, 0xd433179d9c8cb841,
        0x9e19db92b4e31ba9, 0xeb96bf6ebadf77d9, 0xaf87023b9bf0ee6b,
    };
  
  #if FMT_GCC_VERSION && FMT_GCC_VERSION < 409
    FMT_GCC_PRAGMA("GCC diagnostic push")
    FMT_GCC_PRAGMA("GCC diagnostic ignored \"-Wnarrowing\"")
  #endif
    // Binary exponents of pow(10, k), for k = -348, -340, ..., 340, corresponding
    // to significands above.
    static constexpr int16_t pow10_exponents[87] = {
        -1220, -1193, -1166, -1140, -1113, -1087, -1060, -1034, -1007, -980, -954,
        -927,  -901,  -874,  -847,  -821,  -794,  -768,  -741,  -715,  -688, -661,
        -635,  -608,  -582,  -555,  -529,  -502,  -475,  -449,  -422,  -396, -369,
        -343,  -316,  -289,  -263,  -236,  -210,  -183,  -157,  -130,  -103, -77,
        -50,   -24,   3,     30,    56,    83,    109,   136,   162,   189,  216,
        242,   269,   295,   322,   348,   375,   402,   428,   455,   481,  508,
        534,   561,   588,   614,   641,   667,   694,   720,   747,   774,  800,
        827,   853,   880,   907,   933,   960,   986,   1013,  1039,  1066};
  #if FMT_GCC_VERSION && FMT_GCC_VERSION < 409
    FMT_GCC_PRAGMA("GCC diagnostic pop")
  #endif
  
    static constexpr uint64_t power_of_10_64[20] = {
        1, FMT_POWERS_OF_10(1ULL), FMT_POWERS_OF_10(1000000000ULL),
        10000000000000000000ULL};
  };
  
  // This is a struct rather than an alias to avoid shadowing warnings in gcc.
  struct impl_data : basic_impl_data<> {};
  
  #if __cplusplus < 201703L
  template <typename T>
  constexpr uint64_t basic_impl_data<T>::pow10_significands[];
  template <typename T> constexpr int16_t basic_impl_data<T>::pow10_exponents[];
  template <typename T> constexpr uint64_t basic_impl_data<T>::power_of_10_64[];
  #endif
  
  template <typename T> struct bits {
    static FMT_CONSTEXPR_DECL const int value =
        static_cast<int>(sizeof(T) * std::numeric_limits<unsigned char>::digits);
  };
  
  class fp;
  template <int SHIFT = 0> FMT_CONSTEXPR fp normalize(fp value);
  
  // Lower (upper) boundary is a value half way between a floating-point value
  // and its predecessor (successor). Boundaries have the same exponent as the
  // value so only significands are stored.
  struct boundaries {
    uint64_t lower;
    uint64_t upper;
  };
  
  // A handmade floating-point number f * pow(2, e).
  class fp {
   private:
    using significand_type = uint64_t;
  
    template <typename Float>
    using is_supported_float = bool_constant<sizeof(Float) == sizeof(uint64_t) ||
                                             sizeof(Float) == sizeof(uint32_t)>;
  
   public:
    significand_type f;
    int e;
  
    // All sizes are in bits.
    // Subtract 1 to account for an implicit most significant bit in the
    // normalized form.
    static FMT_CONSTEXPR_DECL const int double_significand_size =
        std::numeric_limits<double>::digits - 1;
    static FMT_CONSTEXPR_DECL const uint64_t implicit_bit =
        1ULL << double_significand_size;
    static FMT_CONSTEXPR_DECL const int significand_size =
        bits<significand_type>::value;
  
    constexpr fp() : f(0), e(0) {}
    constexpr fp(uint64_t f_val, int e_val) : f(f_val), e(e_val) {}
  
    // Constructs fp from an IEEE754 double. It is a template to prevent compile
    // errors on platforms where double is not IEEE754.
    template <typename Double> explicit FMT_CONSTEXPR fp(Double d) { assign(d); }
  
    // Assigns d to this and return true iff predecessor is closer than successor.
    template <typename Float, FMT_ENABLE_IF(is_supported_float<Float>::value)>
    FMT_CONSTEXPR bool assign(Float d) {
      // Assume float is in the format [sign][exponent][significand].
      using limits = std::numeric_limits<Float>;
      const int float_significand_size = limits::digits - 1;
      const int exponent_size =
          bits<Float>::value - float_significand_size - 1;  // -1 for sign
      const uint64_t float_implicit_bit = 1ULL << float_significand_size;
      const uint64_t significand_mask = float_implicit_bit - 1;
      const uint64_t exponent_mask = (~0ULL >> 1) & ~significand_mask;
      const int exponent_bias = (1 << exponent_size) - limits::max_exponent - 1;
      constexpr bool is_double = sizeof(Float) == sizeof(uint64_t);
      auto u = bit_cast<conditional_t<is_double, uint64_t, uint32_t>>(d);
      f = u & significand_mask;
      int biased_e =
          static_cast<int>((u & exponent_mask) >> float_significand_size);
      // Predecessor is closer if d is a normalized power of 2 (f == 0) other than
      // the smallest normalized number (biased_e > 1).
      bool is_predecessor_closer = f == 0 && biased_e > 1;
      if (biased_e != 0)
        f += float_implicit_bit;
      else
        biased_e = 1;  // Subnormals use biased exponent 1 (min exponent).
      e = biased_e - exponent_bias - float_significand_size;
      return is_predecessor_closer;
    }
  
    template <typename Float, FMT_ENABLE_IF(!is_supported_float<Float>::value)>
    bool assign(Float) {
      *this = fp();
      return false;
    }
  };
  
  // Normalizes the value converted from double and multiplied by (1 << SHIFT).
  template <int SHIFT> FMT_CONSTEXPR fp normalize(fp value) {
    // Handle subnormals.
    const auto shifted_implicit_bit = fp::implicit_bit << SHIFT;
    while ((value.f & shifted_implicit_bit) == 0) {
      value.f <<= 1;
      --value.e;
    }
    // Subtract 1 to account for hidden bit.
    const auto offset =
        fp::significand_size - fp::double_significand_size - SHIFT - 1;
    value.f <<= offset;
    value.e -= offset;
    return value;
  }
  
  inline bool operator==(fp x, fp y) { return x.f == y.f && x.e == y.e; }
  
  // Computes lhs * rhs / pow(2, 64) rounded to nearest with half-up tie breaking.
  FMT_CONSTEXPR inline uint64_t multiply(uint64_t lhs, uint64_t rhs) {
  #if FMT_USE_INT128
    auto product = static_cast<__uint128_t>(lhs) * rhs;
    auto f = static_cast<uint64_t>(product >> 64);
    return (static_cast<uint64_t>(product) & (1ULL << 63)) != 0 ? f + 1 : f;
  #else
    // Multiply 32-bit parts of significands.
    uint64_t mask = (1ULL << 32) - 1;
    uint64_t a = lhs >> 32, b = lhs & mask;
    uint64_t c = rhs >> 32, d = rhs & mask;
    uint64_t ac = a * c, bc = b * c, ad = a * d, bd = b * d;
    // Compute mid 64-bit of result and round.
    uint64_t mid = (bd >> 32) + (ad & mask) + (bc & mask) + (1U << 31);
    return ac + (ad >> 32) + (bc >> 32) + (mid >> 32);
  #endif
  }
  
  FMT_CONSTEXPR inline fp operator*(fp x, fp y) {
    return {multiply(x.f, y.f), x.e + y.e + 64};
  }
  
  // Returns a cached power of 10 `c_k = c_k.f * pow(2, c_k.e)` such that its
  // (binary) exponent satisfies `min_exponent <= c_k.e <= min_exponent + 28`.
  FMT_CONSTEXPR inline fp get_cached_power(int min_exponent,
                                           int& pow10_exponent) {
    const int shift = 32;
    const auto significand = static_cast<int64_t>(log10_2_significand);
    int index = static_cast<int>(
        ((min_exponent + fp::significand_size - 1) * (significand >> shift) +
         ((int64_t(1) << shift) - 1))  // ceil
        >> 32                          // arithmetic shift
    );
    // Decimal exponent of the first (smallest) cached power of 10.
    const int first_dec_exp = -348;
    // Difference between 2 consecutive decimal exponents in cached powers of 10.
    const int dec_exp_step = 8;
    index = (index - first_dec_exp - 1) / dec_exp_step + 1;
    pow10_exponent = first_dec_exp + index * dec_exp_step;
    return {impl_data::pow10_significands[index],
            impl_data::pow10_exponents[index]};
  }
  
  // A simple accumulator to hold the sums of terms in bigint::square if uint128_t
  // is not available.
  struct accumulator {
    uint64_t lower;
    uint64_t upper;
  
    constexpr accumulator() : lower(0), upper(0) {}
    constexpr explicit operator uint32_t() const {
      return static_cast<uint32_t>(lower);
    }
  
    FMT_CONSTEXPR void operator+=(uint64_t n) {
      lower += n;
      if (lower < n) ++upper;
    }
    FMT_CONSTEXPR void operator>>=(int shift) {
      FMT_ASSERT(shift == 32, "");
      (void)shift;
      lower = (upper << 32) | (lower >> 32);
      upper >>= 32;
    }
  };
  
  class bigint {
   private:
    // A bigint is stored as an array of bigits (big digits), with bigit at index
    // 0 being the least significant one.
    using bigit = uint32_t;
    using double_bigit = uint64_t;
    enum { bigits_capacity = 32 };
    basic_memory_buffer<bigit, bigits_capacity> bigits_;
    int exp_;
  
    FMT_CONSTEXPR20 bigit operator[](int index) const {
      return bigits_[to_unsigned(index)];
    }
    FMT_CONSTEXPR20 bigit& operator[](int index) {
      return bigits_[to_unsigned(index)];
    }
  
    static FMT_CONSTEXPR_DECL const int bigit_bits = bits<bigit>::value;
  
    friend struct formatter<bigint>;
  
    FMT_CONSTEXPR20 void subtract_bigits(int index, bigit other, bigit& borrow) {
      auto result = static_cast<double_bigit>((*this)[index]) - other - borrow;
      (*this)[index] = static_cast<bigit>(result);
      borrow = static_cast<bigit>(result >> (bigit_bits * 2 - 1));
    }
  
    FMT_CONSTEXPR20 void remove_leading_zeros() {
      int num_bigits = static_cast<int>(bigits_.size()) - 1;
      while (num_bigits > 0 && (*this)[num_bigits] == 0) --num_bigits;
      bigits_.resize(to_unsigned(num_bigits + 1));
    }
  
    // Computes *this -= other assuming aligned bigints and *this >= other.
    FMT_CONSTEXPR20 void subtract_aligned(const bigint& other) {
      FMT_ASSERT(other.exp_ >= exp_, "unaligned bigints");
      FMT_ASSERT(compare(*this, other) >= 0, "");
      bigit borrow = 0;
      int i = other.exp_ - exp_;
      for (size_t j = 0, n = other.bigits_.size(); j != n; ++i, ++j)
        subtract_bigits(i, other.bigits_[j], borrow);
      while (borrow > 0) subtract_bigits(i, 0, borrow);
      remove_leading_zeros();
    }
  
    FMT_CONSTEXPR20 void multiply(uint32_t value) {
      const double_bigit wide_value = value;
      bigit carry = 0;
      for (size_t i = 0, n = bigits_.size(); i < n; ++i) {
        double_bigit result = bigits_[i] * wide_value + carry;
        bigits_[i] = static_cast<bigit>(result);
        carry = static_cast<bigit>(result >> bigit_bits);
      }
      if (carry != 0) bigits_.push_back(carry);
    }
  
    FMT_CONSTEXPR20 void multiply(uint64_t value) {
      const bigit mask = ~bigit(0);
      const double_bigit lower = value & mask;
      const double_bigit upper = value >> bigit_bits;
      double_bigit carry = 0;
      for (size_t i = 0, n = bigits_.size(); i < n; ++i) {
        double_bigit result = bigits_[i] * lower + (carry & mask);
        carry =
            bigits_[i] * upper + (result >> bigit_bits) + (carry >> bigit_bits);
        bigits_[i] = static_cast<bigit>(result);
      }
      while (carry != 0) {
        bigits_.push_back(carry & mask);
        carry >>= bigit_bits;
      }
    }
  
   public:
    FMT_CONSTEXPR20 bigint() : exp_(0) {}
    explicit bigint(uint64_t n) { assign(n); }
    FMT_CONSTEXPR20 ~bigint() {
      FMT_ASSERT(bigits_.capacity() <= bigits_capacity, "");
    }
  
    bigint(const bigint&) = delete;
    void operator=(const bigint&) = delete;
  
    FMT_CONSTEXPR20 void assign(const bigint& other) {
      auto size = other.bigits_.size();
      bigits_.resize(size);
      auto data = other.bigits_.data();
      std::copy(data, data + size, make_checked(bigits_.data(), size));
      exp_ = other.exp_;
    }
  
    FMT_CONSTEXPR20 void assign(uint64_t n) {
      size_t num_bigits = 0;
      do {
        bigits_[num_bigits++] = n & ~bigit(0);
        n >>= bigit_bits;
      } while (n != 0);
      bigits_.resize(num_bigits);
      exp_ = 0;
    }
  
    FMT_CONSTEXPR20 int num_bigits() const {
      return static_cast<int>(bigits_.size()) + exp_;
    }
  
    FMT_NOINLINE FMT_CONSTEXPR20 bigint& operator<<=(int shift) {
      FMT_ASSERT(shift >= 0, "");
      exp_ += shift / bigit_bits;
      shift %= bigit_bits;
      if (shift == 0) return *this;
      bigit carry = 0;
      for (size_t i = 0, n = bigits_.size(); i < n; ++i) {
        bigit c = bigits_[i] >> (bigit_bits - shift);
        bigits_[i] = (bigits_[i] << shift) + carry;
        carry = c;
      }
      if (carry != 0) bigits_.push_back(carry);
      return *this;
    }
  
    template <typename Int> FMT_CONSTEXPR20 bigint& operator*=(Int value) {
      FMT_ASSERT(value > 0, "");
      multiply(uint32_or_64_or_128_t<Int>(value));
      return *this;
    }
  
    friend FMT_CONSTEXPR20 int compare(const bigint& lhs, const bigint& rhs) {
      int num_lhs_bigits = lhs.num_bigits(), num_rhs_bigits = rhs.num_bigits();
      if (num_lhs_bigits != num_rhs_bigits)
        return num_lhs_bigits > num_rhs_bigits ? 1 : -1;
      int i = static_cast<int>(lhs.bigits_.size()) - 1;
      int j = static_cast<int>(rhs.bigits_.size()) - 1;
      int end = i - j;
      if (end < 0) end = 0;
      for (; i >= end; --i, --j) {
        bigit lhs_bigit = lhs[i], rhs_bigit = rhs[j];
        if (lhs_bigit != rhs_bigit) return lhs_bigit > rhs_bigit ? 1 : -1;
      }
      if (i != j) return i > j ? 1 : -1;
      return 0;
    }
  
    // Returns compare(lhs1 + lhs2, rhs).
    friend FMT_CONSTEXPR20 int add_compare(const bigint& lhs1, const bigint& lhs2,
                                           const bigint& rhs) {
      int max_lhs_bigits = (std::max)(lhs1.num_bigits(), lhs2.num_bigits());
      int num_rhs_bigits = rhs.num_bigits();
      if (max_lhs_bigits + 1 < num_rhs_bigits) return -1;
      if (max_lhs_bigits > num_rhs_bigits) return 1;
      auto get_bigit = [](const bigint& n, int i) -> bigit {
        return i >= n.exp_ && i < n.num_bigits() ? n[i - n.exp_] : 0;
      };
      double_bigit borrow = 0;
      int min_exp = (std::min)((std::min)(lhs1.exp_, lhs2.exp_), rhs.exp_);
      for (int i = num_rhs_bigits - 1; i >= min_exp; --i) {
        double_bigit sum =
            static_cast<double_bigit>(get_bigit(lhs1, i)) + get_bigit(lhs2, i);
        bigit rhs_bigit = get_bigit(rhs, i);
        if (sum > rhs_bigit + borrow) return 1;
        borrow = rhs_bigit + borrow - sum;
        if (borrow > 1) return -1;
        borrow <<= bigit_bits;
      }
      return borrow != 0 ? -1 : 0;
    }
  
    // Assigns pow(10, exp) to this bigint.
    FMT_CONSTEXPR20 void assign_pow10(int exp) {
      FMT_ASSERT(exp >= 0, "");
      if (exp == 0) return assign(1);
      // Find the top bit.
      int bitmask = 1;
      while (exp >= bitmask) bitmask <<= 1;
      bitmask >>= 1;
      // pow(10, exp) = pow(5, exp) * pow(2, exp). First compute pow(5, exp) by
      // repeated squaring and multiplication.
      assign(5);
      bitmask >>= 1;
      while (bitmask != 0) {
        square();
        if ((exp & bitmask) != 0) *this *= 5;
        bitmask >>= 1;
      }
      *this <<= exp;  // Multiply by pow(2, exp) by shifting.
    }
  
    FMT_CONSTEXPR20 void square() {
      int num_bigits = static_cast<int>(bigits_.size());
      int num_result_bigits = 2 * num_bigits;
      basic_memory_buffer<bigit, bigits_capacity> n(std::move(bigits_));
      bigits_.resize(to_unsigned(num_result_bigits));
      using accumulator_t = conditional_t<FMT_USE_INT128, uint128_t, accumulator>;
      auto sum = accumulator_t();
      for (int bigit_index = 0; bigit_index < num_bigits; ++bigit_index) {
        // Compute bigit at position bigit_index of the result by adding
        // cross-product terms n[i] * n[j] such that i + j == bigit_index.
        for (int i = 0, j = bigit_index; j >= 0; ++i, --j) {
          // Most terms are multiplied twice which can be optimized in the future.
          sum += static_cast<double_bigit>(n[i]) * n[j];
        }
        (*this)[bigit_index] = static_cast<bigit>(sum);
        sum >>= bits<bigit>::value;  // Compute the carry.
      }
      // Do the same for the top half.
      for (int bigit_index = num_bigits; bigit_index < num_result_bigits;
           ++bigit_index) {
        for (int j = num_bigits - 1, i = bigit_index - j; i < num_bigits;)
          sum += static_cast<double_bigit>(n[i++]) * n[j--];
        (*this)[bigit_index] = static_cast<bigit>(sum);
        sum >>= bits<bigit>::value;
      }
      remove_leading_zeros();
      exp_ *= 2;
    }
  
    // If this bigint has a bigger exponent than other, adds trailing zero to make
    // exponents equal. This simplifies some operations such as subtraction.
    FMT_CONSTEXPR20 void align(const bigint& other) {
      int exp_difference = exp_ - other.exp_;
      if (exp_difference <= 0) return;
      int num_bigits = static_cast<int>(bigits_.size());
      bigits_.resize(to_unsigned(num_bigits + exp_difference));
      for (int i = num_bigits - 1, j = i + exp_difference; i >= 0; --i, --j)
        bigits_[j] = bigits_[i];
      std::uninitialized_fill_n(bigits_.data(), exp_difference, 0);
      exp_ -= exp_difference;
    }
  
    // Divides this bignum by divisor, assigning the remainder to this and
    // returning the quotient.
    FMT_CONSTEXPR20 int divmod_assign(const bigint& divisor) {
      FMT_ASSERT(this != &divisor, "");
      if (compare(*this, divisor) < 0) return 0;
      FMT_ASSERT(divisor.bigits_[divisor.bigits_.size() - 1u] != 0, "");
      align(divisor);
      int quotient = 0;
      do {
        subtract_aligned(divisor);
        ++quotient;
      } while (compare(*this, divisor) >= 0);
      return quotient;
    }
  };
  
  enum class round_direction { unknown, up, down };
  
  // Given the divisor (normally a power of 10), the remainder = v % divisor for
  // some number v and the error, returns whether v should be rounded up, down, or
  // whether the rounding direction can't be determined due to error.
  // error should be less than divisor / 2.
  FMT_CONSTEXPR inline round_direction get_round_direction(uint64_t divisor,
                                                           uint64_t remainder,
                                                           uint64_t error) {
    FMT_ASSERT(remainder < divisor, "");  // divisor - remainder won't overflow.
    FMT_ASSERT(error < divisor, "");      // divisor - error won't overflow.
    FMT_ASSERT(error < divisor - error, "");  // error * 2 won't overflow.
    // Round down if (remainder + error) * 2 <= divisor.
    if (remainder <= divisor - remainder && error * 2 <= divisor - remainder * 2)
      return round_direction::down;
    // Round up if (remainder - error) * 2 >= divisor.
    if (remainder >= error &&
        remainder - error >= divisor - (remainder - error)) {
      return round_direction::up;
    }
    return round_direction::unknown;
  }
  
  namespace digits {
  enum result {
    more,  // Generate more digits.
    done,  // Done generating digits.
    error  // Digit generation cancelled due to an error.
  };
  }
  
  // Generates output using the Grisu digit-gen algorithm.
  // error: the size of the region (lower, upper) outside of which numbers
  // definitely do not round to value (Delta in Grisu3).
  template <typename Handler>
  FMT_INLINE FMT_CONSTEXPR digits::result grisu_gen_digits(fp value,
                                                           uint64_t error,
                                                           int& exp,
                                                           Handler& handler) {
    const fp one(1ULL << -value.e, value.e);
    // The integral part of scaled value (p1 in Grisu) = value / one. It cannot be
    // zero because it contains a product of two 64-bit numbers with MSB set (due
    // to normalization) - 1, shifted right by at most 60 bits.
    auto integral = static_cast<uint32_t>(value.f >> -one.e);
    FMT_ASSERT(integral != 0, "");
    FMT_ASSERT(integral == value.f >> -one.e, "");
    // The fractional part of scaled value (p2 in Grisu) c = value % one.
    uint64_t fractional = value.f & (one.f - 1);
    exp = count_digits(integral);  // kappa in Grisu.
    // Divide by 10 to prevent overflow.
    auto result = handler.on_start(impl_data::power_of_10_64[exp - 1] << -one.e,
                                   value.f / 10, error * 10, exp);
    if (result != digits::more) return result;
    // Generate digits for the integral part. This can produce up to 10 digits.
    do {
      uint32_t digit = 0;
      auto divmod_integral = [&](uint32_t divisor) {
        digit = integral / divisor;
        integral %= divisor;
      };
      // This optimization by Milo Yip reduces the number of integer divisions by
      // one per iteration.
      switch (exp) {
      case 10:
        divmod_integral(1000000000);
        break;
      case 9:
        divmod_integral(100000000);
        break;
      case 8:
        divmod_integral(10000000);
        break;
      case 7:
        divmod_integral(1000000);
        break;
      case 6:
        divmod_integral(100000);
        break;
      case 5:
        divmod_integral(10000);
        break;
      case 4:
        divmod_integral(1000);
        break;
      case 3:
        divmod_integral(100);
        break;
      case 2:
        divmod_integral(10);
        break;
      case 1:
        digit = integral;
        integral = 0;
        break;
      default:
        FMT_ASSERT(false, "invalid number of digits");
      }
      --exp;
      auto remainder = (static_cast<uint64_t>(integral) << -one.e) + fractional;
      result = handler.on_digit(static_cast<char>('0' + digit),
                                impl_data::power_of_10_64[exp] << -one.e,
                                remainder, error, exp, true);
      if (result != digits::more) return result;
    } while (exp > 0);
    // Generate digits for the fractional part.
    for (;;) {
      fractional *= 10;
      error *= 10;
      char digit = static_cast<char>('0' + (fractional >> -one.e));
      fractional &= one.f - 1;
      --exp;
      result = handler.on_digit(digit, one.f, fractional, error, exp, false);
      if (result != digits::more) return result;
    }
  }
  
  // The fixed precision digit handler.
  struct fixed_handler {
    char* buf;
    int size;
    int precision;
    int exp10;
    bool fixed;
  
    FMT_CONSTEXPR digits::result on_start(uint64_t divisor, uint64_t remainder,
                                          uint64_t error, int& exp) {
      // Non-fixed formats require at least one digit and no precision adjustment.
      if (!fixed) return digits::more;
      // Adjust fixed precision by exponent because it is relative to decimal
      // point.
      precision += exp + exp10;
      // Check if precision is satisfied just by leading zeros, e.g.
      // format("{:.2f}", 0.001) gives "0.00" without generating any digits.
      if (precision > 0) return digits::more;
      if (precision < 0) return digits::done;
      auto dir = get_round_direction(divisor, remainder, error);
      if (dir == round_direction::unknown) return digits::error;
      buf[size++] = dir == round_direction::up ? '1' : '0';
      return digits::done;
    }
  
    FMT_CONSTEXPR digits::result on_digit(char digit, uint64_t divisor,
                                          uint64_t remainder, uint64_t error, int,
                                          bool integral) {
      FMT_ASSERT(remainder < divisor, "");
      buf[size++] = digit;
      if (!integral && error >= remainder) return digits::error;
      if (size < precision) return digits::more;
      if (!integral) {
        // Check if error * 2 < divisor with overflow prevention.
        // The check is not needed for the integral part because error = 1
        // and divisor > (1 << 32) there.
        if (error >= divisor || error >= divisor - error) return digits::error;
      } else {
        FMT_ASSERT(error == 1 && divisor > 2, "");
      }
      auto dir = get_round_direction(divisor, remainder, error);
      if (dir != round_direction::up)
        return dir == round_direction::down ? digits::done : digits::error;
      ++buf[size - 1];
      for (int i = size - 1; i > 0 && buf[i] > '9'; --i) {
        buf[i] = '0';
        ++buf[i - 1];
      }
      if (buf[0] > '9') {
        buf[0] = '1';
        if (fixed)
          buf[size++] = '0';
        else
          ++exp10;
      }
      return digits::done;
    }
  };
  
  // A 128-bit integer type used internally,
  struct uint128_wrapper {
    uint128_wrapper() = default;
  
  #if FMT_USE_INT128
    uint128_t internal_;
  
    constexpr uint128_wrapper(uint64_t high, uint64_t low) FMT_NOEXCEPT
        : internal_{static_cast<uint128_t>(low) |
                    (static_cast<uint128_t>(high) << 64)} {}
  
    constexpr uint128_wrapper(uint128_t u) : internal_{u} {}
  
    constexpr uint64_t high() const FMT_NOEXCEPT {
      return uint64_t(internal_ >> 64);
    }
    constexpr uint64_t low() const FMT_NOEXCEPT { return uint64_t(internal_); }
  
    uint128_wrapper& operator+=(uint64_t n) FMT_NOEXCEPT {
      internal_ += n;
      return *this;
    }
  #else
    uint64_t high_;
    uint64_t low_;
  
    constexpr uint128_wrapper(uint64_t high, uint64_t low) FMT_NOEXCEPT
        : high_{high},
          low_{low} {}
  
    constexpr uint64_t high() const FMT_NOEXCEPT { return high_; }
    constexpr uint64_t low() const FMT_NOEXCEPT { return low_; }
  
    uint128_wrapper& operator+=(uint64_t n) FMT_NOEXCEPT {
  #  if defined(_MSC_VER) && defined(_M_X64)
      unsigned char carry = _addcarry_u64(0, low_, n, &low_);
      _addcarry_u64(carry, high_, 0, &high_);
      return *this;
  #  else
      uint64_t sum = low_ + n;
      high_ += (sum < low_ ? 1 : 0);
      low_ = sum;
      return *this;
  #  endif
    }
  #endif
  };
  
  // Implementation of Dragonbox algorithm: https://github.com/jk-jeon/dragonbox.
  namespace dragonbox {
  // Computes 128-bit result of multiplication of two 64-bit unsigned integers.
  inline uint128_wrapper umul128(uint64_t x, uint64_t y) FMT_NOEXCEPT {
  #if FMT_USE_INT128
    return static_cast<uint128_t>(x) * static_cast<uint128_t>(y);
  #elif defined(_MSC_VER) && defined(_M_X64)
    uint128_wrapper result;
    result.low_ = _umul128(x, y, &result.high_);
    return result;
  #else
    const uint64_t mask = (uint64_t(1) << 32) - uint64_t(1);
  
    uint64_t a = x >> 32;
    uint64_t b = x & mask;
    uint64_t c = y >> 32;
    uint64_t d = y & mask;
  
    uint64_t ac = a * c;
    uint64_t bc = b * c;
    uint64_t ad = a * d;
    uint64_t bd = b * d;
  
    uint64_t intermediate = (bd >> 32) + (ad & mask) + (bc & mask);
  
    return {ac + (intermediate >> 32) + (ad >> 32) + (bc >> 32),
            (intermediate << 32) + (bd & mask)};
  #endif
  }
  
  // Computes upper 64 bits of multiplication of two 64-bit unsigned integers.
  inline uint64_t umul128_upper64(uint64_t x, uint64_t y) FMT_NOEXCEPT {
  #if FMT_USE_INT128
    auto p = static_cast<uint128_t>(x) * static_cast<uint128_t>(y);
    return static_cast<uint64_t>(p >> 64);
  #elif defined(_MSC_VER) && defined(_M_X64)
    return __umulh(x, y);
  #else
    return umul128(x, y).high();
  #endif
  }
  
  // Computes upper 64 bits of multiplication of a 64-bit unsigned integer and a
  // 128-bit unsigned integer.
  inline uint64_t umul192_upper64(uint64_t x, uint128_wrapper y) FMT_NOEXCEPT {
    uint128_wrapper g0 = umul128(x, y.high());
    g0 += umul128_upper64(x, y.low());
    return g0.high();
  }
  
  // Computes upper 32 bits of multiplication of a 32-bit unsigned integer and a
  // 64-bit unsigned integer.
  inline uint32_t umul96_upper32(uint32_t x, uint64_t y) FMT_NOEXCEPT {
    return static_cast<uint32_t>(umul128_upper64(x, y));
  }
  
  // Computes middle 64 bits of multiplication of a 64-bit unsigned integer and a
  // 128-bit unsigned integer.
  inline uint64_t umul192_middle64(uint64_t x, uint128_wrapper y) FMT_NOEXCEPT {
    uint64_t g01 = x * y.high();
    uint64_t g10 = umul128_upper64(x, y.low());
    return g01 + g10;
  }
  
  // Computes lower 64 bits of multiplication of a 32-bit unsigned integer and a
  // 64-bit unsigned integer.
  inline uint64_t umul96_lower64(uint32_t x, uint64_t y) FMT_NOEXCEPT {
    return x * y;
  }
  
  // Computes floor(log10(pow(2, e))) for e in [-1700, 1700] using the method from
  // https://fmt.dev/papers/Grisu-Exact.pdf#page=5, section 3.4.
  inline int floor_log10_pow2(int e) FMT_NOEXCEPT {
    FMT_ASSERT(e <= 1700 && e >= -1700, "too large exponent");
    const int shift = 22;
    return (e * static_cast<int>(log10_2_significand >> (64 - shift))) >> shift;
  }
  
  // Various fast log computations.
  inline int floor_log2_pow10(int e) FMT_NOEXCEPT {
    FMT_ASSERT(e <= 1233 && e >= -1233, "too large exponent");
    const uint64_t log2_10_integer_part = 3;
    const uint64_t log2_10_fractional_digits = 0x5269e12f346e2bf9;
    const int shift_amount = 19;
    return (e * static_cast<int>(
                    (log2_10_integer_part << shift_amount) |
                    (log2_10_fractional_digits >> (64 - shift_amount)))) >>
           shift_amount;
  }
  inline int floor_log10_pow2_minus_log10_4_over_3(int e) FMT_NOEXCEPT {
    FMT_ASSERT(e <= 1700 && e >= -1700, "too large exponent");
    const uint64_t log10_4_over_3_fractional_digits = 0x1ffbfc2bbc780375;
    const int shift_amount = 22;
    return (e * static_cast<int>(log10_2_significand >> (64 - shift_amount)) -
            static_cast<int>(log10_4_over_3_fractional_digits >>
                             (64 - shift_amount))) >>
           shift_amount;
  }
  
  // Returns true iff x is divisible by pow(2, exp).
  inline bool divisible_by_power_of_2(uint32_t x, int exp) FMT_NOEXCEPT {
    FMT_ASSERT(exp >= 1, "");
    FMT_ASSERT(x != 0, "");
  #ifdef FMT_BUILTIN_CTZ
    return FMT_BUILTIN_CTZ(x) >= exp;
  #else
    return exp < num_bits<uint32_t>() && x == ((x >> exp) << exp);
  #endif
  }
  inline bool divisible_by_power_of_2(uint64_t x, int exp) FMT_NOEXCEPT {
    FMT_ASSERT(exp >= 1, "");
    FMT_ASSERT(x != 0, "");
  #ifdef FMT_BUILTIN_CTZLL
    return FMT_BUILTIN_CTZLL(x) >= exp;
  #else
    return exp < num_bits<uint64_t>() && x == ((x >> exp) << exp);
  #endif
  }
  
  // Table entry type for divisibility test.
  template <typename T> struct divtest_table_entry {
    T mod_inv;
    T max_quotient;
  };
  
  // Returns true iff x is divisible by pow(5, exp).
  inline bool divisible_by_power_of_5(uint32_t x, int exp) FMT_NOEXCEPT {
    FMT_ASSERT(exp <= 10, "too large exponent");
    static constexpr const divtest_table_entry<uint32_t> divtest_table[] = {
        {0x00000001, 0xffffffff}, {0xcccccccd, 0x33333333},
        {0xc28f5c29, 0x0a3d70a3}, {0x26e978d5, 0x020c49ba},
        {0x3afb7e91, 0x0068db8b}, {0x0bcbe61d, 0x0014f8b5},
        {0x68c26139, 0x000431bd}, {0xae8d46a5, 0x0000d6bf},
        {0x22e90e21, 0x00002af3}, {0x3a2e9c6d, 0x00000897},
        {0x3ed61f49, 0x000001b7}};
    return x * divtest_table[exp].mod_inv <= divtest_table[exp].max_quotient;
  }
  inline bool divisible_by_power_of_5(uint64_t x, int exp) FMT_NOEXCEPT {
    FMT_ASSERT(exp <= 23, "too large exponent");
    static constexpr const divtest_table_entry<uint64_t> divtest_table[] = {
        {0x0000000000000001, 0xffffffffffffffff},
        {0xcccccccccccccccd, 0x3333333333333333},
        {0x8f5c28f5c28f5c29, 0x0a3d70a3d70a3d70},
        {0x1cac083126e978d5, 0x020c49ba5e353f7c},
        {0xd288ce703afb7e91, 0x0068db8bac710cb2},
        {0x5d4e8fb00bcbe61d, 0x0014f8b588e368f0},
        {0x790fb65668c26139, 0x000431bde82d7b63},
        {0xe5032477ae8d46a5, 0x0000d6bf94d5e57a},
        {0xc767074b22e90e21, 0x00002af31dc46118},
        {0x8e47ce423a2e9c6d, 0x0000089705f4136b},
        {0x4fa7f60d3ed61f49, 0x000001b7cdfd9d7b},
        {0x0fee64690c913975, 0x00000057f5ff85e5},
        {0x3662e0e1cf503eb1, 0x000000119799812d},
        {0xa47a2cf9f6433fbd, 0x0000000384b84d09},
        {0x54186f653140a659, 0x00000000b424dc35},
        {0x7738164770402145, 0x0000000024075f3d},
        {0xe4a4d1417cd9a041, 0x000000000734aca5},
        {0xc75429d9e5c5200d, 0x000000000170ef54},
        {0xc1773b91fac10669, 0x000000000049c977},
        {0x26b172506559ce15, 0x00000000000ec1e4},
        {0xd489e3a9addec2d1, 0x000000000002f394},
        {0x90e860bb892c8d5d, 0x000000000000971d},
        {0x502e79bf1b6f4f79, 0x0000000000001e39},
        {0xdcd618596be30fe5, 0x000000000000060b}};
    return x * divtest_table[exp].mod_inv <= divtest_table[exp].max_quotient;
  }
  
  // Replaces n by floor(n / pow(5, N)) returning true if and only if n is
  // divisible by pow(5, N).
  // Precondition: n <= 2 * pow(5, N + 1).
  template <int N>
  bool check_divisibility_and_divide_by_pow5(uint32_t& n) FMT_NOEXCEPT {
    static constexpr struct {
      uint32_t magic_number;
      int bits_for_comparison;
      uint32_t threshold;
      int shift_amount;
    } infos[] = {{0xcccd, 16, 0x3333, 18}, {0xa429, 8, 0x0a, 20}};
    constexpr auto info = infos[N - 1];
    n *= info.magic_number;
    const uint32_t comparison_mask = (1u << info.bits_for_comparison) - 1;
    bool result = (n & comparison_mask) <= info.threshold;
    n >>= info.shift_amount;
    return result;
  }
  
  // Computes floor(n / pow(10, N)) for small n and N.
  // Precondition: n <= pow(10, N + 1).
  template <int N> uint32_t small_division_by_pow10(uint32_t n) FMT_NOEXCEPT {
    static constexpr struct {
      uint32_t magic_number;
      int shift_amount;
      uint32_t divisor_times_10;
    } infos[] = {{0xcccd, 19, 100}, {0xa3d8, 22, 1000}};
    constexpr auto info = infos[N - 1];
    FMT_ASSERT(n <= info.divisor_times_10, "n is too large");
    return n * info.magic_number >> info.shift_amount;
  }
  
  // Computes floor(n / 10^(kappa + 1)) (float)
  inline uint32_t divide_by_10_to_kappa_plus_1(uint32_t n) FMT_NOEXCEPT {
    return n / float_info<float>::big_divisor;
  }
  // Computes floor(n / 10^(kappa + 1)) (double)
  inline uint64_t divide_by_10_to_kappa_plus_1(uint64_t n) FMT_NOEXCEPT {
    return umul128_upper64(n, 0x83126e978d4fdf3c) >> 9;
  }
  
  // Various subroutines using pow10 cache
  template <class T> struct cache_accessor;
  
  template <> struct cache_accessor<float> {
    using carrier_uint = float_info<float>::carrier_uint;
    using cache_entry_type = uint64_t;
  
    static uint64_t get_cached_power(int k) FMT_NOEXCEPT {
      FMT_ASSERT(k >= float_info<float>::min_k && k <= float_info<float>::max_k,
                 "k is out of range");
      static constexpr const uint64_t pow10_significands[] = {
          0x81ceb32c4b43fcf5, 0xa2425ff75e14fc32, 0xcad2f7f5359a3b3f,
          0xfd87b5f28300ca0e, 0x9e74d1b791e07e49, 0xc612062576589ddb,
          0xf79687aed3eec552, 0x9abe14cd44753b53, 0xc16d9a0095928a28,
          0xf1c90080baf72cb2, 0x971da05074da7bef, 0xbce5086492111aeb,
          0xec1e4a7db69561a6, 0x9392ee8e921d5d08, 0xb877aa3236a4b44a,
          0xe69594bec44de15c, 0x901d7cf73ab0acda, 0xb424dc35095cd810,
          0xe12e13424bb40e14, 0x8cbccc096f5088cc, 0xafebff0bcb24aaff,
          0xdbe6fecebdedd5bf, 0x89705f4136b4a598, 0xabcc77118461cefd,
          0xd6bf94d5e57a42bd, 0x8637bd05af6c69b6, 0xa7c5ac471b478424,
          0xd1b71758e219652c, 0x83126e978d4fdf3c, 0xa3d70a3d70a3d70b,
          0xcccccccccccccccd, 0x8000000000000000, 0xa000000000000000,
          0xc800000000000000, 0xfa00000000000000, 0x9c40000000000000,
          0xc350000000000000, 0xf424000000000000, 0x9896800000000000,
          0xbebc200000000000, 0xee6b280000000000, 0x9502f90000000000,
          0xba43b74000000000, 0xe8d4a51000000000, 0x9184e72a00000000,
          0xb5e620f480000000, 0xe35fa931a0000000, 0x8e1bc9bf04000000,
          0xb1a2bc2ec5000000, 0xde0b6b3a76400000, 0x8ac7230489e80000,
          0xad78ebc5ac620000, 0xd8d726b7177a8000, 0x878678326eac9000,
          0xa968163f0a57b400, 0xd3c21bcecceda100, 0x84595161401484a0,
          0xa56fa5b99019a5c8, 0xcecb8f27f4200f3a, 0x813f3978f8940984,
          0xa18f07d736b90be5, 0xc9f2c9cd04674ede, 0xfc6f7c4045812296,
          0x9dc5ada82b70b59d, 0xc5371912364ce305, 0xf684df56c3e01bc6,
          0x9a130b963a6c115c, 0xc097ce7bc90715b3, 0xf0bdc21abb48db20,
          0x96769950b50d88f4, 0xbc143fa4e250eb31, 0xeb194f8e1ae525fd,
          0x92efd1b8d0cf37be, 0xb7abc627050305ad, 0xe596b7b0c643c719,
          0x8f7e32ce7bea5c6f, 0xb35dbf821ae4f38b, 0xe0352f62a19e306e};
      return pow10_significands[k - float_info<float>::min_k];
    }
  
    static carrier_uint compute_mul(carrier_uint u,
                                    const cache_entry_type& cache) FMT_NOEXCEPT {
      return umul96_upper32(u, cache);
    }
  
    static uint32_t compute_delta(const cache_entry_type& cache,
                                  int beta_minus_1) FMT_NOEXCEPT {
      return static_cast<uint32_t>(cache >> (64 - 1 - beta_minus_1));
    }
  
    static bool compute_mul_parity(carrier_uint two_f,
                                   const cache_entry_type& cache,
                                   int beta_minus_1) FMT_NOEXCEPT {
      FMT_ASSERT(beta_minus_1 >= 1, "");
      FMT_ASSERT(beta_minus_1 < 64, "");
  
      return ((umul96_lower64(two_f, cache) >> (64 - beta_minus_1)) & 1) != 0;
    }
  
    static carrier_uint compute_left_endpoint_for_shorter_interval_case(
        const cache_entry_type& cache, int beta_minus_1) FMT_NOEXCEPT {
      return static_cast<carrier_uint>(
          (cache - (cache >> (float_info<float>::significand_bits + 2))) >>
          (64 - float_info<float>::significand_bits - 1 - beta_minus_1));
    }
  
    static carrier_uint compute_right_endpoint_for_shorter_interval_case(
        const cache_entry_type& cache, int beta_minus_1) FMT_NOEXCEPT {
      return static_cast<carrier_uint>(
          (cache + (cache >> (float_info<float>::significand_bits + 1))) >>
          (64 - float_info<float>::significand_bits - 1 - beta_minus_1));
    }
  
    static carrier_uint compute_round_up_for_shorter_interval_case(
        const cache_entry_type& cache, int beta_minus_1) FMT_NOEXCEPT {
      return (static_cast<carrier_uint>(
                  cache >>
                  (64 - float_info<float>::significand_bits - 2 - beta_minus_1)) +
              1) /
             2;
    }
  };
  
  template <> struct cache_accessor<double> {
    using carrier_uint = float_info<double>::carrier_uint;
    using cache_entry_type = uint128_wrapper;
  
    static uint128_wrapper get_cached_power(int k) FMT_NOEXCEPT {
      FMT_ASSERT(k >= float_info<double>::min_k && k <= float_info<double>::max_k,
                 "k is out of range");
  
      static constexpr const uint128_wrapper pow10_significands[] = {
  #if FMT_USE_FULL_CACHE_DRAGONBOX
        {0xff77b1fcbebcdc4f, 0x25e8e89c13bb0f7b},
        {0x9faacf3df73609b1, 0x77b191618c54e9ad},
        {0xc795830d75038c1d, 0xd59df5b9ef6a2418},
        {0xf97ae3d0d2446f25, 0x4b0573286b44ad1e},
        {0x9becce62836ac577, 0x4ee367f9430aec33},
        {0xc2e801fb244576d5, 0x229c41f793cda740},
        {0xf3a20279ed56d48a, 0x6b43527578c11110},
        {0x9845418c345644d6, 0x830a13896b78aaaa},
        {0xbe5691ef416bd60c, 0x23cc986bc656d554},
        {0xedec366b11c6cb8f, 0x2cbfbe86b7ec8aa9},
        {0x94b3a202eb1c3f39, 0x7bf7d71432f3d6aa},
        {0xb9e08a83a5e34f07, 0xdaf5ccd93fb0cc54},
        {0xe858ad248f5c22c9, 0xd1b3400f8f9cff69},
        {0x91376c36d99995be, 0x23100809b9c21fa2},
        {0xb58547448ffffb2d, 0xabd40a0c2832a78b},
        {0xe2e69915b3fff9f9, 0x16c90c8f323f516d},
        {0x8dd01fad907ffc3b, 0xae3da7d97f6792e4},
        {0xb1442798f49ffb4a, 0x99cd11cfdf41779d},
        {0xdd95317f31c7fa1d, 0x40405643d711d584},
        {0x8a7d3eef7f1cfc52, 0x482835ea666b2573},
        {0xad1c8eab5ee43b66, 0xda3243650005eed0},
        {0xd863b256369d4a40, 0x90bed43e40076a83},
        {0x873e4f75e2224e68, 0x5a7744a6e804a292},
        {0xa90de3535aaae202, 0x711515d0a205cb37},
        {0xd3515c2831559a83, 0x0d5a5b44ca873e04},
        {0x8412d9991ed58091, 0xe858790afe9486c3},
        {0xa5178fff668ae0b6, 0x626e974dbe39a873},
        {0xce5d73ff402d98e3, 0xfb0a3d212dc81290},
        {0x80fa687f881c7f8e, 0x7ce66634bc9d0b9a},
        {0xa139029f6a239f72, 0x1c1fffc1ebc44e81},
        {0xc987434744ac874e, 0xa327ffb266b56221},
        {0xfbe9141915d7a922, 0x4bf1ff9f0062baa9},
        {0x9d71ac8fada6c9b5, 0x6f773fc3603db4aa},
        {0xc4ce17b399107c22, 0xcb550fb4384d21d4},
        {0xf6019da07f549b2b, 0x7e2a53a146606a49},
        {0x99c102844f94e0fb, 0x2eda7444cbfc426e},
        {0xc0314325637a1939, 0xfa911155fefb5309},
        {0xf03d93eebc589f88, 0x793555ab7eba27cb},
        {0x96267c7535b763b5, 0x4bc1558b2f3458df},
        {0xbbb01b9283253ca2, 0x9eb1aaedfb016f17},
        {0xea9c227723ee8bcb, 0x465e15a979c1cadd},
        {0x92a1958a7675175f, 0x0bfacd89ec191eca},
        {0xb749faed14125d36, 0xcef980ec671f667c},
        {0xe51c79a85916f484, 0x82b7e12780e7401b},
        {0x8f31cc0937ae58d2, 0xd1b2ecb8b0908811},
        {0xb2fe3f0b8599ef07, 0x861fa7e6dcb4aa16},
        {0xdfbdcece67006ac9, 0x67a791e093e1d49b},
        {0x8bd6a141006042bd, 0xe0c8bb2c5c6d24e1},
        {0xaecc49914078536d, 0x58fae9f773886e19},
        {0xda7f5bf590966848, 0xaf39a475506a899f},
        {0x888f99797a5e012d, 0x6d8406c952429604},
        {0xaab37fd7d8f58178, 0xc8e5087ba6d33b84},
        {0xd5605fcdcf32e1d6, 0xfb1e4a9a90880a65},
        {0x855c3be0a17fcd26, 0x5cf2eea09a550680},
        {0xa6b34ad8c9dfc06f, 0xf42faa48c0ea481f},
        {0xd0601d8efc57b08b, 0xf13b94daf124da27},
        {0x823c12795db6ce57, 0x76c53d08d6b70859},
        {0xa2cb1717b52481ed, 0x54768c4b0c64ca6f},
        {0xcb7ddcdda26da268, 0xa9942f5dcf7dfd0a},
        {0xfe5d54150b090b02, 0xd3f93b35435d7c4d},
        {0x9efa548d26e5a6e1, 0xc47bc5014a1a6db0},
        {0xc6b8e9b0709f109a, 0x359ab6419ca1091c},
        {0xf867241c8cc6d4c0, 0xc30163d203c94b63},
        {0x9b407691d7fc44f8, 0x79e0de63425dcf1e},
        {0xc21094364dfb5636, 0x985915fc12f542e5},
        {0xf294b943e17a2bc4, 0x3e6f5b7b17b2939e},
        {0x979cf3ca6cec5b5a, 0xa705992ceecf9c43},
        {0xbd8430bd08277231, 0x50c6ff782a838354},
        {0xece53cec4a314ebd, 0xa4f8bf5635246429},
        {0x940f4613ae5ed136, 0x871b7795e136be9a},
        {0xb913179899f68584, 0x28e2557b59846e40},
        {0xe757dd7ec07426e5, 0x331aeada2fe589d0},
        {0x9096ea6f3848984f, 0x3ff0d2c85def7622},
        {0xb4bca50b065abe63, 0x0fed077a756b53aa},
        {0xe1ebce4dc7f16dfb, 0xd3e8495912c62895},
        {0x8d3360f09cf6e4bd, 0x64712dd7abbbd95d},
        {0xb080392cc4349dec, 0xbd8d794d96aacfb4},
        {0xdca04777f541c567, 0xecf0d7a0fc5583a1},
        {0x89e42caaf9491b60, 0xf41686c49db57245},
        {0xac5d37d5b79b6239, 0x311c2875c522ced6},
        {0xd77485cb25823ac7, 0x7d633293366b828c},
        {0x86a8d39ef77164bc, 0xae5dff9c02033198},
        {0xa8530886b54dbdeb, 0xd9f57f830283fdfd},
        {0xd267caa862a12d66, 0xd072df63c324fd7c},
        {0x8380dea93da4bc60, 0x4247cb9e59f71e6e},
        {0xa46116538d0deb78, 0x52d9be85f074e609},
        {0xcd795be870516656, 0x67902e276c921f8c},
        {0x806bd9714632dff6, 0x00ba1cd8a3db53b7},
        {0xa086cfcd97bf97f3, 0x80e8a40eccd228a5},
        {0xc8a883c0fdaf7df0, 0x6122cd128006b2ce},
        {0xfad2a4b13d1b5d6c, 0x796b805720085f82},
        {0x9cc3a6eec6311a63, 0xcbe3303674053bb1},
        {0xc3f490aa77bd60fc, 0xbedbfc4411068a9d},
        {0xf4f1b4d515acb93b, 0xee92fb5515482d45},
        {0x991711052d8bf3c5, 0x751bdd152d4d1c4b},
        {0xbf5cd54678eef0b6, 0xd262d45a78a0635e},
        {0xef340a98172aace4, 0x86fb897116c87c35},
        {0x9580869f0e7aac0e, 0xd45d35e6ae3d4da1},
        {0xbae0a846d2195712, 0x8974836059cca10a},
        {0xe998d258869facd7, 0x2bd1a438703fc94c},
        {0x91ff83775423cc06, 0x7b6306a34627ddd0},
        {0xb67f6455292cbf08, 0x1a3bc84c17b1d543},
        {0xe41f3d6a7377eeca, 0x20caba5f1d9e4a94},
        {0x8e938662882af53e, 0x547eb47b7282ee9d},
        {0xb23867fb2a35b28d, 0xe99e619a4f23aa44},
        {0xdec681f9f4c31f31, 0x6405fa00e2ec94d5},
        {0x8b3c113c38f9f37e, 0xde83bc408dd3dd05},
        {0xae0b158b4738705e, 0x9624ab50b148d446},
        {0xd98ddaee19068c76, 0x3badd624dd9b0958},
        {0x87f8a8d4cfa417c9, 0xe54ca5d70a80e5d7},
        {0xa9f6d30a038d1dbc, 0x5e9fcf4ccd211f4d},
        {0xd47487cc8470652b, 0x7647c32000696720},
        {0x84c8d4dfd2c63f3b, 0x29ecd9f40041e074},
        {0xa5fb0a17c777cf09, 0xf468107100525891},
        {0xcf79cc9db955c2cc, 0x7182148d4066eeb5},
        {0x81ac1fe293d599bf, 0xc6f14cd848405531},
        {0xa21727db38cb002f, 0xb8ada00e5a506a7d},
        {0xca9cf1d206fdc03b, 0xa6d90811f0e4851d},
        {0xfd442e4688bd304a, 0x908f4a166d1da664},
        {0x9e4a9cec15763e2e, 0x9a598e4e043287ff},
        {0xc5dd44271ad3cdba, 0x40eff1e1853f29fe},
        {0xf7549530e188c128, 0xd12bee59e68ef47d},
        {0x9a94dd3e8cf578b9, 0x82bb74f8301958cf},
        {0xc13a148e3032d6e7, 0xe36a52363c1faf02},
        {0xf18899b1bc3f8ca1, 0xdc44e6c3cb279ac2},
        {0x96f5600f15a7b7e5, 0x29ab103a5ef8c0ba},
        {0xbcb2b812db11a5de, 0x7415d448f6b6f0e8},
        {0xebdf661791d60f56, 0x111b495b3464ad22},
        {0x936b9fcebb25c995, 0xcab10dd900beec35},
        {0xb84687c269ef3bfb, 0x3d5d514f40eea743},
        {0xe65829b3046b0afa, 0x0cb4a5a3112a5113},
        {0x8ff71a0fe2c2e6dc, 0x47f0e785eaba72ac},
        {0xb3f4e093db73a093, 0x59ed216765690f57},
        {0xe0f218b8d25088b8, 0x306869c13ec3532d},
        {0x8c974f7383725573, 0x1e414218c73a13fc},
        {0xafbd2350644eeacf, 0xe5d1929ef90898fb},
        {0xdbac6c247d62a583, 0xdf45f746b74abf3a},
        {0x894bc396ce5da772, 0x6b8bba8c328eb784},
        {0xab9eb47c81f5114f, 0x066ea92f3f326565},
        {0xd686619ba27255a2, 0xc80a537b0efefebe},
        {0x8613fd0145877585, 0xbd06742ce95f5f37},
        {0xa798fc4196e952e7, 0x2c48113823b73705},
        {0xd17f3b51fca3a7a0, 0xf75a15862ca504c6},
        {0x82ef85133de648c4, 0x9a984d73dbe722fc},
        {0xa3ab66580d5fdaf5, 0xc13e60d0d2e0ebbb},
        {0xcc963fee10b7d1b3, 0x318df905079926a9},
        {0xffbbcfe994e5c61f, 0xfdf17746497f7053},
        {0x9fd561f1fd0f9bd3, 0xfeb6ea8bedefa634},
        {0xc7caba6e7c5382c8, 0xfe64a52ee96b8fc1},
        {0xf9bd690a1b68637b, 0x3dfdce7aa3c673b1},
        {0x9c1661a651213e2d, 0x06bea10ca65c084f},
        {0xc31bfa0fe5698db8, 0x486e494fcff30a63},
        {0xf3e2f893dec3f126, 0x5a89dba3c3efccfb},
        {0x986ddb5c6b3a76b7, 0xf89629465a75e01d},
        {0xbe89523386091465, 0xf6bbb397f1135824},
        {0xee2ba6c0678b597f, 0x746aa07ded582e2d},
        {0x94db483840b717ef, 0xa8c2a44eb4571cdd},
        {0xba121a4650e4ddeb, 0x92f34d62616ce414},
        {0xe896a0d7e51e1566, 0x77b020baf9c81d18},
        {0x915e2486ef32cd60, 0x0ace1474dc1d122f},
        {0xb5b5ada8aaff80b8, 0x0d819992132456bb},
        {0xe3231912d5bf60e6, 0x10e1fff697ed6c6a},
        {0x8df5efabc5979c8f, 0xca8d3ffa1ef463c2},
        {0xb1736b96b6fd83b3, 0xbd308ff8a6b17cb3},
        {0xddd0467c64bce4a0, 0xac7cb3f6d05ddbdf},
        {0x8aa22c0dbef60ee4, 0x6bcdf07a423aa96c},
        {0xad4ab7112eb3929d, 0x86c16c98d2c953c7},
        {0xd89d64d57a607744, 0xe871c7bf077ba8b8},
        {0x87625f056c7c4a8b, 0x11471cd764ad4973},
        {0xa93af6c6c79b5d2d, 0xd598e40d3dd89bd0},
        {0xd389b47879823479, 0x4aff1d108d4ec2c4},
        {0x843610cb4bf160cb, 0xcedf722a585139bb},
        {0xa54394fe1eedb8fe, 0xc2974eb4ee658829},
        {0xce947a3da6a9273e, 0x733d226229feea33},
        {0x811ccc668829b887, 0x0806357d5a3f5260},
        {0xa163ff802a3426a8, 0xca07c2dcb0cf26f8},
        {0xc9bcff6034c13052, 0xfc89b393dd02f0b6},
        {0xfc2c3f3841f17c67, 0xbbac2078d443ace3},
        {0x9d9ba7832936edc0, 0xd54b944b84aa4c0e},
        {0xc5029163f384a931, 0x0a9e795e65d4df12},
        {0xf64335bcf065d37d, 0x4d4617b5ff4a16d6},
        {0x99ea0196163fa42e, 0x504bced1bf8e4e46},
        {0xc06481fb9bcf8d39, 0xe45ec2862f71e1d7},
        {0xf07da27a82c37088, 0x5d767327bb4e5a4d},
        {0x964e858c91ba2655, 0x3a6a07f8d510f870},
        {0xbbe226efb628afea, 0x890489f70a55368c},
        {0xeadab0aba3b2dbe5, 0x2b45ac74ccea842f},
        {0x92c8ae6b464fc96f, 0x3b0b8bc90012929e},
        {0xb77ada0617e3bbcb, 0x09ce6ebb40173745},
        {0xe55990879ddcaabd, 0xcc420a6a101d0516},
        {0x8f57fa54c2a9eab6, 0x9fa946824a12232e},
        {0xb32df8e9f3546564, 0x47939822dc96abfa},
        {0xdff9772470297ebd, 0x59787e2b93bc56f8},
        {0x8bfbea76c619ef36, 0x57eb4edb3c55b65b},
        {0xaefae51477a06b03, 0xede622920b6b23f2},
        {0xdab99e59958885c4, 0xe95fab368e45ecee},
        {0x88b402f7fd75539b, 0x11dbcb0218ebb415},
        {0xaae103b5fcd2a881, 0xd652bdc29f26a11a},
        {0xd59944a37c0752a2, 0x4be76d3346f04960},
        {0x857fcae62d8493a5, 0x6f70a4400c562ddc},
        {0xa6dfbd9fb8e5b88e, 0xcb4ccd500f6bb953},
        {0xd097ad07a71f26b2, 0x7e2000a41346a7a8},
        {0x825ecc24c873782f, 0x8ed400668c0c28c9},
        {0xa2f67f2dfa90563b, 0x728900802f0f32fb},
        {0xcbb41ef979346bca, 0x4f2b40a03ad2ffba},
        {0xfea126b7d78186bc, 0xe2f610c84987bfa9},
        {0x9f24b832e6b0f436, 0x0dd9ca7d2df4d7ca},
        {0xc6ede63fa05d3143, 0x91503d1c79720dbc},
        {0xf8a95fcf88747d94, 0x75a44c6397ce912b},
        {0x9b69dbe1b548ce7c, 0xc986afbe3ee11abb},
        {0xc24452da229b021b, 0xfbe85badce996169},
        {0xf2d56790ab41c2a2, 0xfae27299423fb9c4},
        {0x97c560ba6b0919a5, 0xdccd879fc967d41b},
        {0xbdb6b8e905cb600f, 0x5400e987bbc1c921},
        {0xed246723473e3813, 0x290123e9aab23b69},
        {0x9436c0760c86e30b, 0xf9a0b6720aaf6522},
        {0xb94470938fa89bce, 0xf808e40e8d5b3e6a},
        {0xe7958cb87392c2c2, 0xb60b1d1230b20e05},
        {0x90bd77f3483bb9b9, 0xb1c6f22b5e6f48c3},
        {0xb4ecd5f01a4aa828, 0x1e38aeb6360b1af4},
        {0xe2280b6c20dd5232, 0x25c6da63c38de1b1},
        {0x8d590723948a535f, 0x579c487e5a38ad0f},
        {0xb0af48ec79ace837, 0x2d835a9df0c6d852},
        {0xdcdb1b2798182244, 0xf8e431456cf88e66},
        {0x8a08f0f8bf0f156b, 0x1b8e9ecb641b5900},
        {0xac8b2d36eed2dac5, 0xe272467e3d222f40},
        {0xd7adf884aa879177, 0x5b0ed81dcc6abb10},
        {0x86ccbb52ea94baea, 0x98e947129fc2b4ea},
        {0xa87fea27a539e9a5, 0x3f2398d747b36225},
        {0xd29fe4b18e88640e, 0x8eec7f0d19a03aae},
        {0x83a3eeeef9153e89, 0x1953cf68300424ad},
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        {0xa1075a24e4421730, 0xb24cf65b8612f81f},
        {0xc94930ae1d529cfc, 0xdee033f26797b627},
        {0xfb9b7cd9a4a7443c, 0x169840ef017da3b1},
        {0x9d412e0806e88aa5, 0x8e1f289560ee864e},
        {0xc491798a08a2ad4e, 0xf1a6f2bab92a27e2},
        {0xf5b5d7ec8acb58a2, 0xae10af696774b1db},
        {0x9991a6f3d6bf1765, 0xacca6da1e0a8ef29},
        {0xbff610b0cc6edd3f, 0x17fd090a58d32af3},
        {0xeff394dcff8a948e, 0xddfc4b4cef07f5b0},
        {0x95f83d0a1fb69cd9, 0x4abdaf101564f98e},
        {0xbb764c4ca7a4440f, 0x9d6d1ad41abe37f1},
        {0xea53df5fd18d5513, 0x84c86189216dc5ed},
        {0x92746b9be2f8552c, 0x32fd3cf5b4e49bb4},
        {0xb7118682dbb66a77, 0x3fbc8c33221dc2a1},
        {0xe4d5e82392a40515, 0x0fabaf3feaa5334a},
        {0x8f05b1163ba6832d, 0x29cb4d87f2a7400e},
        {0xb2c71d5bca9023f8, 0x743e20e9ef511012},
        {0xdf78e4b2bd342cf6, 0x914da9246b255416},
        {0x8bab8eefb6409c1a, 0x1ad089b6c2f7548e},
        {0xae9672aba3d0c320, 0xa184ac2473b529b1},
        {0xda3c0f568cc4f3e8, 0xc9e5d72d90a2741e},
        {0x8865899617fb1871, 0x7e2fa67c7a658892},
        {0xaa7eebfb9df9de8d, 0xddbb901b98feeab7},
        {0xd51ea6fa85785631, 0x552a74227f3ea565},
        {0x8533285c936b35de, 0xd53a88958f87275f},
        {0xa67ff273b8460356, 0x8a892abaf368f137},
        {0xd01fef10a657842c, 0x2d2b7569b0432d85},
        {0x8213f56a67f6b29b, 0x9c3b29620e29fc73},
        {0xa298f2c501f45f42, 0x8349f3ba91b47b8f},
        {0xcb3f2f7642717713, 0x241c70a936219a73},
        {0xfe0efb53d30dd4d7, 0xed238cd383aa0110},
        {0x9ec95d1463e8a506, 0xf4363804324a40aa},
        {0xc67bb4597ce2ce48, 0xb143c6053edcd0d5},
        {0xf81aa16fdc1b81da, 0xdd94b7868e94050a},
        {0x9b10a4e5e9913128, 0xca7cf2b4191c8326},
        {0xc1d4ce1f63f57d72, 0xfd1c2f611f63a3f0},
        {0xf24a01a73cf2dccf, 0xbc633b39673c8cec},
        {0x976e41088617ca01, 0xd5be0503e085d813},
        {0xbd49d14aa79dbc82, 0x4b2d8644d8a74e18},
        {0xec9c459d51852ba2, 0xddf8e7d60ed1219e},
        {0x93e1ab8252f33b45, 0xcabb90e5c942b503},
        {0xb8da1662e7b00a17, 0x3d6a751f3b936243},
        {0xe7109bfba19c0c9d, 0x0cc512670a783ad4},
        {0x906a617d450187e2, 0x27fb2b80668b24c5},
        {0xb484f9dc9641e9da, 0xb1f9f660802dedf6},
        {0xe1a63853bbd26451, 0x5e7873f8a0396973},
        {0x8d07e33455637eb2, 0xdb0b487b6423e1e8},
        {0xb049dc016abc5e5f, 0x91ce1a9a3d2cda62},
        {0xdc5c5301c56b75f7, 0x7641a140cc7810fb},
        {0x89b9b3e11b6329ba, 0xa9e904c87fcb0a9d},
        {0xac2820d9623bf429, 0x546345fa9fbdcd44},
        {0xd732290fbacaf133, 0xa97c177947ad4095},
        {0x867f59a9d4bed6c0, 0x49ed8eabcccc485d},
        {0xa81f301449ee8c70, 0x5c68f256bfff5a74},
        {0xd226fc195c6a2f8c, 0x73832eec6fff3111},
        {0x83585d8fd9c25db7, 0xc831fd53c5ff7eab},
        {0xa42e74f3d032f525, 0xba3e7ca8b77f5e55},
        {0xcd3a1230c43fb26f, 0x28ce1bd2e55f35eb},
        {0x80444b5e7aa7cf85, 0x7980d163cf5b81b3},
        {0xa0555e361951c366, 0xd7e105bcc332621f},
        {0xc86ab5c39fa63440, 0x8dd9472bf3fefaa7},
        {0xfa856334878fc150, 0xb14f98f6f0feb951},
        {0x9c935e00d4b9d8d2, 0x6ed1bf9a569f33d3},
        {0xc3b8358109e84f07, 0x0a862f80ec4700c8},
        {0xf4a642e14c6262c8, 0xcd27bb612758c0fa},
        {0x98e7e9cccfbd7dbd, 0x8038d51cb897789c},
        {0xbf21e44003acdd2c, 0xe0470a63e6bd56c3},
        {0xeeea5d5004981478, 0x1858ccfce06cac74},
        {0x95527a5202df0ccb, 0x0f37801e0c43ebc8},
        {0xbaa718e68396cffd, 0xd30560258f54e6ba},
        {0xe950df20247c83fd, 0x47c6b82ef32a2069},
        {0x91d28b7416cdd27e, 0x4cdc331d57fa5441},
        {0xb6472e511c81471d, 0xe0133fe4adf8e952},
        {0xe3d8f9e563a198e5, 0x58180fddd97723a6},
        {0x8e679c2f5e44ff8f, 0x570f09eaa7ea7648},
        {0xb201833b35d63f73, 0x2cd2cc6551e513da},
        {0xde81e40a034bcf4f, 0xf8077f7ea65e58d1},
        {0x8b112e86420f6191, 0xfb04afaf27faf782},
        {0xadd57a27d29339f6, 0x79c5db9af1f9b563},
        {0xd94ad8b1c7380874, 0x18375281ae7822bc},
        {0x87cec76f1c830548, 0x8f2293910d0b15b5},
        {0xa9c2794ae3a3c69a, 0xb2eb3875504ddb22},
        {0xd433179d9c8cb841, 0x5fa60692a46151eb},
        {0x849feec281d7f328, 0xdbc7c41ba6bcd333},
        {0xa5c7ea73224deff3, 0x12b9b522906c0800},
        {0xcf39e50feae16bef, 0xd768226b34870a00},
        {0x81842f29f2cce375, 0xe6a1158300d46640},
        {0xa1e53af46f801c53, 0x60495ae3c1097fd0},
        {0xca5e89b18b602368, 0x385bb19cb14bdfc4},
        {0xfcf62c1dee382c42, 0x46729e03dd9ed7b5},
        {0x9e19db92b4e31ba9, 0x6c07a2c26a8346d1},
        {0xc5a05277621be293, 0xc7098b7305241885},
        { 0xf70867153aa2db38,
          0xb8cbee4fc66d1ea7 }
  #else
        {0xff77b1fcbebcdc4f, 0x25e8e89c13bb0f7b},
        {0xce5d73ff402d98e3, 0xfb0a3d212dc81290},
        {0xa6b34ad8c9dfc06f, 0xf42faa48c0ea481f},
        {0x86a8d39ef77164bc, 0xae5dff9c02033198},
        {0xd98ddaee19068c76, 0x3badd624dd9b0958},
        {0xafbd2350644eeacf, 0xe5d1929ef90898fb},
        {0x8df5efabc5979c8f, 0xca8d3ffa1ef463c2},
        {0xe55990879ddcaabd, 0xcc420a6a101d0516},
        {0xb94470938fa89bce, 0xf808e40e8d5b3e6a},
        {0x95a8637627989aad, 0xdde7001379a44aa9},
        {0xf1c90080baf72cb1, 0x5324c68b12dd6339},
        {0xc350000000000000, 0x0000000000000000},
        {0x9dc5ada82b70b59d, 0xf020000000000000},
        {0xfee50b7025c36a08, 0x02f236d04753d5b4},
        {0xcde6fd5e09abcf26, 0xed4c0226b55e6f86},
        {0xa6539930bf6bff45, 0x84db8346b786151c},
        {0x865b86925b9bc5c2, 0x0b8a2392ba45a9b2},
        {0xd910f7ff28069da4, 0x1b2ba1518094da04},
        {0xaf58416654a6babb, 0x387ac8d1970027b2},
        {0x8da471a9de737e24, 0x5ceaecfed289e5d2},
        {0xe4d5e82392a40515, 0x0fabaf3feaa5334a},
        {0xb8da1662e7b00a17, 0x3d6a751f3b936243},
        { 0x95527a5202df0ccb,
          0x0f37801e0c43ebc8 }
  #endif
      };
  
  #if FMT_USE_FULL_CACHE_DRAGONBOX
      return pow10_significands[k - float_info<double>::min_k];
  #else
      static constexpr const uint64_t powers_of_5_64[] = {
          0x0000000000000001, 0x0000000000000005, 0x0000000000000019,
          0x000000000000007d, 0x0000000000000271, 0x0000000000000c35,
          0x0000000000003d09, 0x000000000001312d, 0x000000000005f5e1,
          0x00000000001dcd65, 0x00000000009502f9, 0x0000000002e90edd,
          0x000000000e8d4a51, 0x0000000048c27395, 0x000000016bcc41e9,
          0x000000071afd498d, 0x0000002386f26fc1, 0x000000b1a2bc2ec5,
          0x000003782dace9d9, 0x00001158e460913d, 0x000056bc75e2d631,
          0x0001b1ae4d6e2ef5, 0x000878678326eac9, 0x002a5a058fc295ed,
          0x00d3c21bcecceda1, 0x0422ca8b0a00a425, 0x14adf4b7320334b9};
  
      static constexpr const uint32_t pow10_recovery_errors[] = {
          0x50001400, 0x54044100, 0x54014555, 0x55954415, 0x54115555, 0x00000001,
          0x50000000, 0x00104000, 0x54010004, 0x05004001, 0x55555544, 0x41545555,
          0x54040551, 0x15445545, 0x51555514, 0x10000015, 0x00101100, 0x01100015,
          0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x04450514, 0x45414110,
          0x55555145, 0x50544050, 0x15040155, 0x11054140, 0x50111514, 0x11451454,
          0x00400541, 0x00000000, 0x55555450, 0x10056551, 0x10054011, 0x55551014,
          0x69514555, 0x05151109, 0x00155555};
  
      static const int compression_ratio = 27;
  
      // Compute base index.
      int cache_index = (k - float_info<double>::min_k) / compression_ratio;
      int kb = cache_index * compression_ratio + float_info<double>::min_k;
      int offset = k - kb;
  
      // Get base cache.
      uint128_wrapper base_cache = pow10_significands[cache_index];
      if (offset == 0) return base_cache;
  
      // Compute the required amount of bit-shift.
      int alpha = floor_log2_pow10(kb + offset) - floor_log2_pow10(kb) - offset;
      FMT_ASSERT(alpha > 0 && alpha < 64, "shifting error detected");
  
      // Try to recover the real cache.
      uint64_t pow5 = powers_of_5_64[offset];
      uint128_wrapper recovered_cache = umul128(base_cache.high(), pow5);
      uint128_wrapper middle_low =
          umul128(base_cache.low() - (kb < 0 ? 1u : 0u), pow5);
  
      recovered_cache += middle_low.high();
  
      uint64_t high_to_middle = recovered_cache.high() << (64 - alpha);
      uint64_t middle_to_low = recovered_cache.low() << (64 - alpha);
  
      recovered_cache =
          uint128_wrapper{(recovered_cache.low() >> alpha) | high_to_middle,
                          ((middle_low.low() >> alpha) | middle_to_low)};
  
      if (kb < 0) recovered_cache += 1;
  
      // Get error.
      int error_idx = (k - float_info<double>::min_k) / 16;
      uint32_t error = (pow10_recovery_errors[error_idx] >>
                        ((k - float_info<double>::min_k) % 16) * 2) &
                       0x3;
  
      // Add the error back.
      FMT_ASSERT(recovered_cache.low() + error >= recovered_cache.low(), "");
      return {recovered_cache.high(), recovered_cache.low() + error};
  #endif
    }
  
    static carrier_uint compute_mul(carrier_uint u,
                                    const cache_entry_type& cache) FMT_NOEXCEPT {
      return umul192_upper64(u, cache);
    }
  
    static uint32_t compute_delta(cache_entry_type const& cache,
                                  int beta_minus_1) FMT_NOEXCEPT {
      return static_cast<uint32_t>(cache.high() >> (64 - 1 - beta_minus_1));
    }
  
    static bool compute_mul_parity(carrier_uint two_f,
                                   const cache_entry_type& cache,
                                   int beta_minus_1) FMT_NOEXCEPT {
      FMT_ASSERT(beta_minus_1 >= 1, "");
      FMT_ASSERT(beta_minus_1 < 64, "");
  
      return ((umul192_middle64(two_f, cache) >> (64 - beta_minus_1)) & 1) != 0;
    }
  
    static carrier_uint compute_left_endpoint_for_shorter_interval_case(
        const cache_entry_type& cache, int beta_minus_1) FMT_NOEXCEPT {
      return (cache.high() -
              (cache.high() >> (float_info<double>::significand_bits + 2))) >>
             (64 - float_info<double>::significand_bits - 1 - beta_minus_1);
    }
  
    static carrier_uint compute_right_endpoint_for_shorter_interval_case(
        const cache_entry_type& cache, int beta_minus_1) FMT_NOEXCEPT {
      return (cache.high() +
              (cache.high() >> (float_info<double>::significand_bits + 1))) >>
             (64 - float_info<double>::significand_bits - 1 - beta_minus_1);
    }
  
    static carrier_uint compute_round_up_for_shorter_interval_case(
        const cache_entry_type& cache, int beta_minus_1) FMT_NOEXCEPT {
      return ((cache.high() >>
               (64 - float_info<double>::significand_bits - 2 - beta_minus_1)) +
              1) /
             2;
    }
  };
  
  // Various integer checks
  template <class T>
  bool is_left_endpoint_integer_shorter_interval(int exponent) FMT_NOEXCEPT {
    return exponent >=
               float_info<
                   T>::case_shorter_interval_left_endpoint_lower_threshold &&
           exponent <=
               float_info<T>::case_shorter_interval_left_endpoint_upper_threshold;
  }
  template <class T>
  bool is_endpoint_integer(typename float_info<T>::carrier_uint two_f,
                           int exponent, int minus_k) FMT_NOEXCEPT {
    if (exponent < float_info<T>::case_fc_pm_half_lower_threshold) return false;
    // For k >= 0.
    if (exponent <= float_info<T>::case_fc_pm_half_upper_threshold) return true;
    // For k < 0.
    if (exponent > float_info<T>::divisibility_check_by_5_threshold) return false;
    return divisible_by_power_of_5(two_f, minus_k);
  }
  
  template <class T>
  bool is_center_integer(typename float_info<T>::carrier_uint two_f, int exponent,
                         int minus_k) FMT_NOEXCEPT {
    // Exponent for 5 is negative.
    if (exponent > float_info<T>::divisibility_check_by_5_threshold) return false;
    if (exponent > float_info<T>::case_fc_upper_threshold)
      return divisible_by_power_of_5(two_f, minus_k);
    // Both exponents are nonnegative.
    if (exponent >= float_info<T>::case_fc_lower_threshold) return true;
    // Exponent for 2 is negative.
    return divisible_by_power_of_2(two_f, minus_k - exponent + 1);
  }
  
  // Remove trailing zeros from n and return the number of zeros removed (float)
  FMT_INLINE int remove_trailing_zeros(uint32_t& n) FMT_NOEXCEPT {
  #ifdef FMT_BUILTIN_CTZ
    int t = FMT_BUILTIN_CTZ(n);
  #else
    int t = ctz(n);
  #endif
    if (t > float_info<float>::max_trailing_zeros)
      t = float_info<float>::max_trailing_zeros;
  
    const uint32_t mod_inv1 = 0xcccccccd;
    const uint32_t max_quotient1 = 0x33333333;
    const uint32_t mod_inv2 = 0xc28f5c29;
    const uint32_t max_quotient2 = 0x0a3d70a3;
  
    int s = 0;
    for (; s < t - 1; s += 2) {
      if (n * mod_inv2 > max_quotient2) break;
      n *= mod_inv2;
    }
    if (s < t && n * mod_inv1 <= max_quotient1) {
      n *= mod_inv1;
      ++s;
    }
    n >>= s;
    return s;
  }
  
  // Removes trailing zeros and returns the number of zeros removed (double)
  FMT_INLINE int remove_trailing_zeros(uint64_t& n) FMT_NOEXCEPT {
  #ifdef FMT_BUILTIN_CTZLL
    int t = FMT_BUILTIN_CTZLL(n);
  #else
    int t = ctzll(n);
  #endif
    if (t > float_info<double>::max_trailing_zeros)
      t = float_info<double>::max_trailing_zeros;
    // Divide by 10^8 and reduce to 32-bits
    // Since ret_value.significand <= (2^64 - 1) / 1000 < 10^17,
    // both of the quotient and the r should fit in 32-bits
  
    const uint32_t mod_inv1 = 0xcccccccd;
    const uint32_t max_quotient1 = 0x33333333;
    const uint64_t mod_inv8 = 0xc767074b22e90e21;
    const uint64_t max_quotient8 = 0x00002af31dc46118;
  
    // If the number is divisible by 1'0000'0000, work with the quotient
    if (t >= 8) {
      auto quotient_candidate = n * mod_inv8;
  
      if (quotient_candidate <= max_quotient8) {
        auto quotient = static_cast<uint32_t>(quotient_candidate >> 8);
  
        int s = 8;
        for (; s < t; ++s) {
          if (quotient * mod_inv1 > max_quotient1) break;
          quotient *= mod_inv1;
        }
        quotient >>= (s - 8);
        n = quotient;
        return s;
      }
    }
  
    // Otherwise, work with the remainder
    auto quotient = static_cast<uint32_t>(n / 100000000);
    auto remainder = static_cast<uint32_t>(n - 100000000 * quotient);
  
    if (t == 0 || remainder * mod_inv1 > max_quotient1) {
      return 0;
    }
    remainder *= mod_inv1;
  
    if (t == 1 || remainder * mod_inv1 > max_quotient1) {
      n = (remainder >> 1) + quotient * 10000000ull;
      return 1;
    }
    remainder *= mod_inv1;
  
    if (t == 2 || remainder * mod_inv1 > max_quotient1) {
      n = (remainder >> 2) + quotient * 1000000ull;
      return 2;
    }
    remainder *= mod_inv1;
  
    if (t == 3 || remainder * mod_inv1 > max_quotient1) {
      n = (remainder >> 3) + quotient * 100000ull;
      return 3;
    }
    remainder *= mod_inv1;
  
    if (t == 4 || remainder * mod_inv1 > max_quotient1) {
      n = (remainder >> 4) + quotient * 10000ull;
      return 4;
    }
    remainder *= mod_inv1;
  
    if (t == 5 || remainder * mod_inv1 > max_quotient1) {
      n = (remainder >> 5) + quotient * 1000ull;
      return 5;
    }
    remainder *= mod_inv1;
  
    if (t == 6 || remainder * mod_inv1 > max_quotient1) {
      n = (remainder >> 6) + quotient * 100ull;
      return 6;
    }
    remainder *= mod_inv1;
  
    n = (remainder >> 7) + quotient * 10ull;
    return 7;
  }
  
  // The main algorithm for shorter interval case
  template <class T>
  FMT_INLINE decimal_fp<T> shorter_interval_case(int exponent) FMT_NOEXCEPT {
    decimal_fp<T> ret_value;
    // Compute k and beta
    const int minus_k = floor_log10_pow2_minus_log10_4_over_3(exponent);
    const int beta_minus_1 = exponent + floor_log2_pow10(-minus_k);
  
    // Compute xi and zi
    using cache_entry_type = typename cache_accessor<T>::cache_entry_type;
    const cache_entry_type cache = cache_accessor<T>::get_cached_power(-minus_k);
  
    auto xi = cache_accessor<T>::compute_left_endpoint_for_shorter_interval_case(
        cache, beta_minus_1);
    auto zi = cache_accessor<T>::compute_right_endpoint_for_shorter_interval_case(
        cache, beta_minus_1);
  
    // If the left endpoint is not an integer, increase it
    if (!is_left_endpoint_integer_shorter_interval<T>(exponent)) ++xi;
  
    // Try bigger divisor
    ret_value.significand = zi / 10;
  
    // If succeed, remove trailing zeros if necessary and return
    if (ret_value.significand * 10 >= xi) {
      ret_value.exponent = minus_k + 1;
      ret_value.exponent += remove_trailing_zeros(ret_value.significand);
      return ret_value;
    }
  
    // Otherwise, compute the round-up of y
    ret_value.significand =
        cache_accessor<T>::compute_round_up_for_shorter_interval_case(
            cache, beta_minus_1);
    ret_value.exponent = minus_k;
  
    // When tie occurs, choose one of them according to the rule
    if (exponent >= float_info<T>::shorter_interval_tie_lower_threshold &&
        exponent <= float_info<T>::shorter_interval_tie_upper_threshold) {
      ret_value.significand = ret_value.significand % 2 == 0
                                  ? ret_value.significand
                                  : ret_value.significand - 1;
    } else if (ret_value.significand < xi) {
      ++ret_value.significand;
    }
    return ret_value;
  }
  
  template <typename T> decimal_fp<T> to_decimal(T x) FMT_NOEXCEPT {
    // Step 1: integer promotion & Schubfach multiplier calculation.
  
    using carrier_uint = typename float_info<T>::carrier_uint;
    using cache_entry_type = typename cache_accessor<T>::cache_entry_type;
    auto br = bit_cast<carrier_uint>(x);
  
    // Extract significand bits and exponent bits.
    const carrier_uint significand_mask =
        (static_cast<carrier_uint>(1) << float_info<T>::significand_bits) - 1;
    carrier_uint significand = (br & significand_mask);
    int exponent = static_cast<int>((br & exponent_mask<T>()) >>
                                    float_info<T>::significand_bits);
  
    if (exponent != 0) {  // Check if normal.
      exponent += float_info<T>::exponent_bias - float_info<T>::significand_bits;
  
      // Shorter interval case; proceed like Schubfach.
      if (significand == 0) return shorter_interval_case<T>(exponent);
  
      significand |=
          (static_cast<carrier_uint>(1) << float_info<T>::significand_bits);
    } else {
      // Subnormal case; the interval is always regular.
      if (significand == 0) return {0, 0};
      exponent = float_info<T>::min_exponent - float_info<T>::significand_bits;
    }
  
    const bool include_left_endpoint = (significand % 2 == 0);
    const bool include_right_endpoint = include_left_endpoint;
  
    // Compute k and beta.
    const int minus_k = floor_log10_pow2(exponent) - float_info<T>::kappa;
    const cache_entry_type cache = cache_accessor<T>::get_cached_power(-minus_k);
    const int beta_minus_1 = exponent + floor_log2_pow10(-minus_k);
  
    // Compute zi and deltai
    // 10^kappa <= deltai < 10^(kappa + 1)
    const uint32_t deltai = cache_accessor<T>::compute_delta(cache, beta_minus_1);
    const carrier_uint two_fc = significand << 1;
    const carrier_uint two_fr = two_fc | 1;
    const carrier_uint zi =
        cache_accessor<T>::compute_mul(two_fr << beta_minus_1, cache);
  
    // Step 2: Try larger divisor; remove trailing zeros if necessary
  
    // Using an upper bound on zi, we might be able to optimize the division
    // better than the compiler; we are computing zi / big_divisor here
    decimal_fp<T> ret_value;
    ret_value.significand = divide_by_10_to_kappa_plus_1(zi);
    uint32_t r = static_cast<uint32_t>(zi - float_info<T>::big_divisor *
                                                ret_value.significand);
  
    if (r > deltai) {
      goto small_divisor_case_label;
    } else if (r < deltai) {
      // Exclude the right endpoint if necessary
      if (r == 0 && !include_right_endpoint &&
          is_endpoint_integer<T>(two_fr, exponent, minus_k)) {
        --ret_value.significand;
        r = float_info<T>::big_divisor;
        goto small_divisor_case_label;
      }
    } else {
      // r == deltai; compare fractional parts
      // Check conditions in the order different from the paper
      // to take advantage of short-circuiting
      const carrier_uint two_fl = two_fc - 1;
      if ((!include_left_endpoint ||
           !is_endpoint_integer<T>(two_fl, exponent, minus_k)) &&
          !cache_accessor<T>::compute_mul_parity(two_fl, cache, beta_minus_1)) {
        goto small_divisor_case_label;
      }
    }
    ret_value.exponent = minus_k + float_info<T>::kappa + 1;
  
    // We may need to remove trailing zeros
    ret_value.exponent += remove_trailing_zeros(ret_value.significand);
    return ret_value;
  
    // Step 3: Find the significand with the smaller divisor
  
  small_divisor_case_label:
    ret_value.significand *= 10;
    ret_value.exponent = minus_k + float_info<T>::kappa;
  
    const uint32_t mask = (1u << float_info<T>::kappa) - 1;
    auto dist = r - (deltai / 2) + (float_info<T>::small_divisor / 2);
  
    // Is dist divisible by 2^kappa?
    if ((dist & mask) == 0) {
      const bool approx_y_parity =
          ((dist ^ (float_info<T>::small_divisor / 2)) & 1) != 0;
      dist >>= float_info<T>::kappa;
  
      // Is dist divisible by 5^kappa?
      if (check_divisibility_and_divide_by_pow5<float_info<T>::kappa>(dist)) {
        ret_value.significand += dist;
  
        // Check z^(f) >= epsilon^(f)
        // We have either yi == zi - epsiloni or yi == (zi - epsiloni) - 1,
        // where yi == zi - epsiloni if and only if z^(f) >= epsilon^(f)
        // Since there are only 2 possibilities, we only need to care about the
        // parity. Also, zi and r should have the same parity since the divisor
        // is an even number
        if (cache_accessor<T>::compute_mul_parity(two_fc, cache, beta_minus_1) !=
            approx_y_parity) {
          --ret_value.significand;
        } else {
          // If z^(f) >= epsilon^(f), we might have a tie
          // when z^(f) == epsilon^(f), or equivalently, when y is an integer
          if (is_center_integer<T>(two_fc, exponent, minus_k)) {
            ret_value.significand = ret_value.significand % 2 == 0
                                        ? ret_value.significand
                                        : ret_value.significand - 1;
          }
        }
      }
      // Is dist not divisible by 5^kappa?
      else {
        ret_value.significand += dist;
      }
    }
    // Is dist not divisible by 2^kappa?
    else {
      // Since we know dist is small, we might be able to optimize the division
      // better than the compiler; we are computing dist / small_divisor here
      ret_value.significand +=
          small_division_by_pow10<float_info<T>::kappa>(dist);
    }
    return ret_value;
  }
  }  // namespace dragonbox
  
  // Formats value using a variation of the Fixed-Precision Positive
  // Floating-Point Printout ((FPP)^2) algorithm by Steele & White:
  // https://fmt.dev/papers/p372-steele.pdf.
  template <typename Double>
  FMT_CONSTEXPR20 void fallback_format(Double d, int num_digits, bool binary32,
                                       buffer<char>& buf, int& exp10) {
    bigint numerator;    // 2 * R in (FPP)^2.
    bigint denominator;  // 2 * S in (FPP)^2.
    // lower and upper are differences between value and corresponding boundaries.
    bigint lower;             // (M^- in (FPP)^2).
    bigint upper_store;       // upper's value if different from lower.
    bigint* upper = nullptr;  // (M^+ in (FPP)^2).
    fp value;
    // Shift numerator and denominator by an extra bit or two (if lower boundary
    // is closer) to make lower and upper integers. This eliminates multiplication
    // by 2 during later computations.
    const bool is_predecessor_closer =
        binary32 ? value.assign(static_cast<float>(d)) : value.assign(d);
    int shift = is_predecessor_closer ? 2 : 1;
    uint64_t significand = value.f << shift;
    if (value.e >= 0) {
      numerator.assign(significand);
      numerator <<= value.e;
      lower.assign(1);
      lower <<= value.e;
      if (shift != 1) {
        upper_store.assign(1);
        upper_store <<= value.e + 1;
        upper = &upper_store;
      }
      denominator.assign_pow10(exp10);
      denominator <<= shift;
    } else if (exp10 < 0) {
      numerator.assign_pow10(-exp10);
      lower.assign(numerator);
      if (shift != 1) {
        upper_store.assign(numerator);
        upper_store <<= 1;
        upper = &upper_store;
      }
      numerator *= significand;
      denominator.assign(1);
      denominator <<= shift - value.e;
    } else {
      numerator.assign(significand);
      denominator.assign_pow10(exp10);
      denominator <<= shift - value.e;
      lower.assign(1);
      if (shift != 1) {
        upper_store.assign(1ULL << 1);
        upper = &upper_store;
      }
    }
    // Invariant: value == (numerator / denominator) * pow(10, exp10).
    if (num_digits < 0) {
      // Generate the shortest representation.
      if (!upper) upper = &lower;
      bool even = (value.f & 1) == 0;
      num_digits = 0;
      char* data = buf.data();
      for (;;) {
        int digit = numerator.divmod_assign(denominator);
        bool low = compare(numerator, lower) - even < 0;  // numerator <[=] lower.
        // numerator + upper >[=] pow10:
        bool high = add_compare(numerator, *upper, denominator) + even > 0;
        data[num_digits++] = static_cast<char>('0' + digit);
        if (low || high) {
          if (!low) {
            ++data[num_digits - 1];
          } else if (high) {
            int result = add_compare(numerator, numerator, denominator);
            // Round half to even.
            if (result > 0 || (result == 0 && (digit % 2) != 0))
              ++data[num_digits - 1];
          }
          buf.try_resize(to_unsigned(num_digits));
          exp10 -= num_digits - 1;
          return;
        }
        numerator *= 10;
        lower *= 10;
        if (upper != &lower) *upper *= 10;
      }
    }
    // Generate the given number of digits.
    exp10 -= num_digits - 1;
    if (num_digits == 0) {
      buf.try_resize(1);
      denominator *= 10;
      buf[0] = add_compare(numerator, numerator, denominator) > 0 ? '1' : '0';
      return;
    }
    buf.try_resize(to_unsigned(num_digits));
    for (int i = 0; i < num_digits - 1; ++i) {
      int digit = numerator.divmod_assign(denominator);
      buf[i] = static_cast<char>('0' + digit);
      numerator *= 10;
    }
    int digit = numerator.divmod_assign(denominator);
    auto result = add_compare(numerator, numerator, denominator);
    if (result > 0 || (result == 0 && (digit % 2) != 0)) {
      if (digit == 9) {
        const auto overflow = '0' + 10;
        buf[num_digits - 1] = overflow;
        // Propagate the carry.
        for (int i = num_digits - 1; i > 0 && buf[i] == overflow; --i) {
          buf[i] = '0';
          ++buf[i - 1];
        }
        if (buf[0] == overflow) {
          buf[0] = '1';
          ++exp10;
        }
        return;
      }
      ++digit;
    }
    buf[num_digits - 1] = static_cast<char>('0' + digit);
  }
  
  template <typename T>
  FMT_HEADER_ONLY_CONSTEXPR20 int format_float(T value, int precision,
                                               float_specs specs,
                                               buffer<char>& buf) {
    static_assert(!std::is_same<T, float>::value, "");
    FMT_ASSERT(value >= 0, "value is negative");
  
    const bool fixed = specs.format == float_format::fixed;
    if (value <= 0) {  // <= instead of == to silence a warning.
      if (precision <= 0 || !fixed) {
        buf.push_back('0');
        return 0;
      }
      buf.try_resize(to_unsigned(precision));
      fill_n(buf.data(), precision, '0');
      return -precision;
    }
  
    if (!specs.use_grisu) return snprintf_float(value, precision, specs, buf);
  
    if (!is_constant_evaluated() && precision < 0) {
      // Use Dragonbox for the shortest format.
      if (specs.binary32) {
        auto dec = dragonbox::to_decimal(static_cast<float>(value));
        write<char>(buffer_appender<char>(buf), dec.significand);
        return dec.exponent;
      }
      auto dec = dragonbox::to_decimal(static_cast<double>(value));
      write<char>(buffer_appender<char>(buf), dec.significand);
      return dec.exponent;
    }
  
    // Use Grisu + Dragon4 for the given precision:
    // https://www.cs.tufts.edu/~nr/cs257/archive/florian-loitsch/printf.pdf.
    int exp = 0;
    const int min_exp = -60;  // alpha in Grisu.
    int cached_exp10 = 0;     // K in Grisu.
    fp normalized = normalize(fp(value));
    const auto cached_pow = get_cached_power(
        min_exp - (normalized.e + fp::significand_size), cached_exp10);
    normalized = normalized * cached_pow;
    // Limit precision to the maximum possible number of significant digits in an
    // IEEE754 double because we don't need to generate zeros.
    const int max_double_digits = 767;
    if (precision > max_double_digits) precision = max_double_digits;
    fixed_handler handler{buf.data(), 0, precision, -cached_exp10, fixed};
    if (grisu_gen_digits(normalized, 1, exp, handler) == digits::error ||
        is_constant_evaluated()) {
      exp += handler.size - cached_exp10 - 1;
      fallback_format(value, handler.precision, specs.binary32, buf, exp);
    } else {
      exp += handler.exp10;
      buf.try_resize(to_unsigned(handler.size));
    }
    if (!fixed && !specs.showpoint) {
      // Remove trailing zeros.
      auto num_digits = buf.size();
      while (num_digits > 0 && buf[num_digits - 1] == '0') {
        --num_digits;
        ++exp;
      }
      buf.try_resize(num_digits);
    }
    return exp;
  }  // namespace detail
  
  template <typename T>
  int snprintf_float(T value, int precision, float_specs specs,
                     buffer<char>& buf) {
    // Buffer capacity must be non-zero, otherwise MSVC's vsnprintf_s will fail.
    FMT_ASSERT(buf.capacity() > buf.size(), "empty buffer");
    static_assert(!std::is_same<T, float>::value, "");
  
    // Subtract 1 to account for the difference in precision since we use %e for
    // both general and exponent format.
    if (specs.format == float_format::general ||
        specs.format == float_format::exp)
      precision = (precision >= 0 ? precision : 6) - 1;
  
    // Build the format string.
    enum { max_format_size = 7 };  // The longest format is "%#.*Le".
    char format[max_format_size];
    char* format_ptr = format;
    *format_ptr++ = '%';
    if (specs.showpoint && specs.format == float_format::hex) *format_ptr++ = '#';
    if (precision >= 0) {
      *format_ptr++ = '.';
      *format_ptr++ = '*';
    }
    if (std::is_same<T, long double>()) *format_ptr++ = 'L';
    *format_ptr++ = specs.format != float_format::hex
                        ? (specs.format == float_format::fixed ? 'f' : 'e')
                        : (specs.upper ? 'A' : 'a');
    *format_ptr = '\0';
  
    // Format using snprintf.
    auto offset = buf.size();
    for (;;) {
      auto begin = buf.data() + offset;
      auto capacity = buf.capacity() - offset;
  #ifdef FMT_FUZZ
      if (precision > 100000)
        throw std::runtime_error(
            "fuzz mode - avoid large allocation inside snprintf");
  #endif
      // Suppress the warning about a nonliteral format string.
      // Cannot use auto because of a bug in MinGW (#1532).
      int (*snprintf_ptr)(char*, size_t, const char*, ...) = FMT_SNPRINTF;
      int result = precision >= 0
                       ? snprintf_ptr(begin, capacity, format, precision, value)
                       : snprintf_ptr(begin, capacity, format, value);
      if (result < 0) {
        // The buffer will grow exponentially.
        buf.try_reserve(buf.capacity() + 1);
        continue;
      }
      auto size = to_unsigned(result);
      // Size equal to capacity means that the last character was truncated.
      if (size >= capacity) {
        buf.try_reserve(size + offset + 1);  // Add 1 for the terminating '\0'.
        continue;
      }
      auto is_digit = [](char c) { return c >= '0' && c <= '9'; };
      if (specs.format == float_format::fixed) {
        if (precision == 0) {
          buf.try_resize(size);
          return 0;
        }
        // Find and remove the decimal point.
        auto end = begin + size, p = end;
        do {
          --p;
        } while (is_digit(*p));
        int fraction_size = static_cast<int>(end - p - 1);
        std::memmove(p, p + 1, to_unsigned(fraction_size));
        buf.try_resize(size - 1);
        return -fraction_size;
      }
      if (specs.format == float_format::hex) {
        buf.try_resize(size + offset);
        return 0;
      }
      // Find and parse the exponent.
      auto end = begin + size, exp_pos = end;
      do {
        --exp_pos;
      } while (*exp_pos != 'e');
      char sign = exp_pos[1];
      FMT_ASSERT(sign == '+' || sign == '-', "");
      int exp = 0;
      auto p = exp_pos + 2;  // Skip 'e' and sign.
      do {
        FMT_ASSERT(is_digit(*p), "");
        exp = exp * 10 + (*p++ - '0');
      } while (p != end);
      if (sign == '-') exp = -exp;
      int fraction_size = 0;
      if (exp_pos != begin + 1) {
        // Remove trailing zeros.
        auto fraction_end = exp_pos - 1;
        while (*fraction_end == '0') --fraction_end;
        // Move the fractional part left to get rid of the decimal point.
        fraction_size = static_cast<int>(fraction_end - begin - 1);
        std::memmove(begin + 1, begin + 2, to_unsigned(fraction_size));
      }
      buf.try_resize(to_unsigned(fraction_size) + offset + 1);
      return exp - fraction_size;
    }
  }
  }  // namespace detail
  
  template <> struct formatter<detail::bigint> {
    FMT_CONSTEXPR format_parse_context::iterator parse(
        format_parse_context& ctx) {
      return ctx.begin();
    }
  
    format_context::iterator format(const detail::bigint& n,
                                    format_context& ctx) {
      auto out = ctx.out();
      bool first = true;
      for (auto i = n.bigits_.size(); i > 0; --i) {
        auto value = n.bigits_[i - 1u];
        if (first) {
          out = format_to(out, FMT_STRING("{:x}"), value);
          first = false;
          continue;
        }
        out = format_to(out, FMT_STRING("{:08x}"), value);
      }
      if (n.exp_ > 0)
        out = format_to(out, FMT_STRING("p{}"),
                        n.exp_ * detail::bigint::bigit_bits);
      return out;
    }
  };
  
  FMT_FUNC detail::utf8_to_utf16::utf8_to_utf16(string_view s) {
    for_each_codepoint(s, [this](uint32_t cp, string_view) {
      if (cp == invalid_code_point) FMT_THROW(std::runtime_error("invalid utf8"));
      if (cp <= 0xFFFF) {
        buffer_.push_back(static_cast<wchar_t>(cp));
      } else {
        cp -= 0x10000;
        buffer_.push_back(static_cast<wchar_t>(0xD800 + (cp >> 10)));
        buffer_.push_back(static_cast<wchar_t>(0xDC00 + (cp & 0x3FF)));
      }
      return true;
    });
    buffer_.push_back(0);
  }
  
  FMT_FUNC void format_system_error(detail::buffer<char>& out, int error_code,
                                    const char* message) FMT_NOEXCEPT {
    FMT_TRY {
      auto ec = std::error_code(error_code, std::generic_category());
      write(std::back_inserter(out), std::system_error(ec, message).what());
      return;
    }
    FMT_CATCH(...) {}
    format_error_code(out, error_code, message);
  }
  
  FMT_FUNC void report_system_error(int error_code,
                                    const char* message) FMT_NOEXCEPT {
    report_error(format_system_error, error_code, message);
  }
  
  FMT_FUNC std::string vformat(string_view fmt, format_args args) {
    // Don't optimize the "{}" case to keep the binary size small and because it
    // can be better optimized in fmt::format anyway.
    auto buffer = memory_buffer();
    detail::vformat_to(buffer, fmt, args);
    return to_string(buffer);
  }
  
  #ifdef _WIN32
  namespace detail {
  using dword = conditional_t<sizeof(long) == 4, unsigned long, unsigned>;
  extern "C" __declspec(dllimport) int __stdcall WriteConsoleW(  //
      void*, const void*, dword, dword*, void*);
  }  // namespace detail
  #endif
  
  namespace detail {
  FMT_FUNC void print(std::FILE* f, string_view text) {
  #ifdef _WIN32
    auto fd = _fileno(f);
    if (_isatty(fd)) {
      detail::utf8_to_utf16 u16(string_view(text.data(), text.size()));
      auto written = detail::dword();
      if (detail::WriteConsoleW(reinterpret_cast<void*>(_get_osfhandle(fd)),
                                u16.c_str(), static_cast<uint32_t>(u16.size()),
                                &written, nullptr)) {
        return;
      }
      // Fallback to fwrite on failure. It can happen if the output has been
      // redirected to NUL.
    }
  #endif
    detail::fwrite_fully(text.data(), 1, text.size(), f);
  }
  }  // namespace detail
  
  FMT_FUNC void vprint(std::FILE* f, string_view format_str, format_args args) {
    memory_buffer buffer;
    detail::vformat_to(buffer, format_str, args);
    detail::print(f, {buffer.data(), buffer.size()});
  }
  
  #ifdef _WIN32
  // Print assuming legacy (non-Unicode) encoding.
  FMT_FUNC void detail::vprint_mojibake(std::FILE* f, string_view format_str,
                                        format_args args) {
    memory_buffer buffer;
    detail::vformat_to(buffer, format_str,
                       basic_format_args<buffer_context<char>>(args));
    fwrite_fully(buffer.data(), 1, buffer.size(), f);
  }
  #endif
  
  FMT_FUNC void vprint(string_view format_str, format_args args) {
    vprint(stdout, format_str, args);
  }
  
  FMT_END_NAMESPACE
  
  #endif  // FMT_FORMAT_INL_H_
  