internal::dtoa_impl Namespace Reference

Classes
struct	boundaries
struct	cached_power
struct	diyfp

Functions
template<typename Target , typename Source >
Target	reinterpret_bits (const Source source)
template<typename FloatType >
boundaries	compute_boundaries (FloatType value)
cached_power	get_cached_power_for_binary_exponent (int e)
int	find_largest_pow10 (const std::uint32_t n, std::uint32_t &pow10)
void	grisu2_round (char *buf, int len, std::uint64_t dist, std::uint64_t delta, std::uint64_t rest, std::uint64_t ten_k)
bool	grisu2_digit_gen (char *buffer, char *last, int &length, int &decimal_exponent, diyfp M_minus, diyfp w, diyfp M_plus)
bool	grisu2 (char *buf, char *last, int &len, int &decimal_exponent, diyfp m_minus, diyfp v, diyfp m_plus)
template<typename FloatType >
bool	grisu2 (char *buf, char *last, int &len, int &decimal_exponent, FloatType value)
ToCharsResult	append_exponent (char *buf, char *last, int e)
	appends a decimal representation of e to buf More...
ToCharsResult	format_buffer (char *buf, char *last, int len, int decimal_exponent, int min_exp, int max_exp)
	prettify v = buf * 10^decimal_exponent If v is in the range [10^min_exp, 10^max_exp) it will be printed in fixed-point notation. Otherwise it will be printed in exponential notation. More...

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

Function Documentation

append_exponent()

ToCharsResult internal::dtoa_impl::append_exponent ( char *  buf, char *  last, int  e  )

inline

appends a decimal representation of e to buf

Returns

a pointer to the element following the exponent.

Precondition

-1000 < e < 1000

Definition at line 1021 of file ToChars.cpp.

{
   if (buf >= last)
      return { last, std::errc::value_too_large };
 
   if (e < 0)
   {
      e = -e;
      *buf++ = '-';
   }
   else
   {
      *buf++ = '+';
   }
 
   auto k = static_cast<std::uint32_t>(e);
 
   const int requiredSymbolsCount = k < 100 ? 2 : 3;
   char* requiredLast = buf + requiredSymbolsCount + 1;
 
   if (requiredLast > last)
      return { last, std::errc::value_too_large };
 
   if (k < 10)
   {
      // Always print at least two digits in the exponent.
      // This is for compatibility with printf("%g").
      *buf++ = '0';
      *buf++ = static_cast<char>('0' + k);
   }
   else if (k < 100)
   {
      *buf++ = static_cast<char>('0' + k / 10);
      k %= 10;
      *buf++ = static_cast<char>('0' + k);
   }
   else
   {
      *buf++ = static_cast<char>('0' + k / 100);
      k %= 100;
      *buf++ = static_cast<char>('0' + k / 10);
      k %= 10;
      *buf++ = static_cast<char>('0' + k);
   }
 
   return { buf, std::errc() };
}

Referenced by format_buffer().

Here is the caller graph for this function:

compute_boundaries()

template<typename FloatType >

boundaries internal::dtoa_impl::compute_boundaries

(

FloatType

value

)

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

Precondition

value must be finite and positive

Definition at line 316 of file ToChars.cpp.

{
 
   // Convert the IEEE representation into a diyfp.
   //
   // If v is denormal:
   //      value = 0.F * 2^(1 - bias) = (          F) * 2^(1 - bias - (p-1))
   // If v is normalized:
   //      value = 1.F * 2^(E - bias) = (2^(p-1) + F) * 2^(E - bias - (p-1))
 
   static_assert(
      std::numeric_limits<FloatType>::is_iec559,
      "internal error: dtoa_short requires an IEEE-754 "
      "floating-point implementation");
 
   constexpr int kPrecision =
      std::numeric_limits<FloatType>::digits; // = p (includes the hidden bit)
   constexpr int kBias =
      std::numeric_limits<FloatType>::max_exponent - 1 + (kPrecision - 1);
   constexpr int kMinExp = 1 - kBias;
   constexpr std::uint64_t kHiddenBit = std::uint64_t { 1 }
                                        << (kPrecision - 1); // = 2^(p-1)
 
   using bits_type = std::conditional_t<
      kPrecision == 24, std::uint32_t, std::uint64_t>;
 
   const std::uint64_t bits = reinterpret_bits<bits_type>(value);
   const std::uint64_t E = bits >> (kPrecision - 1);
   const std::uint64_t F = bits & (kHiddenBit - 1);
 
   const bool is_denormal = E == 0;
   const diyfp v = is_denormal ?
                      diyfp(F, kMinExp) :
                      diyfp(F + kHiddenBit, static_cast<int>(E) - kBias);
 
   // Compute the boundaries m- and m+ of the floating-point value
   // v = f * 2^e.
   //
   // Determine v- and v+, the floating-point predecessor and successor if v,
   // respectively.
   //
   //      v- = v - 2^e        if f != 2^(p-1) or e == e_min                (A)
   //         = v - 2^(e-1)    if f == 2^(p-1) and e > e_min                (B)
   //
   //      v+ = v + 2^e
   //
   // Let m- = (v- + v) / 2 and m+ = (v + v+) / 2. All real numbers _strictly_
   // between m- and m+ round to v, regardless of how the input rounding
   // algorithm breaks ties.
   //
   //      ---+-------------+-------------+-------------+-------------+---  (A)
   //         v-            m-            v             m+            v+
   //
   //      -----------------+------+------+-------------+-------------+---  (B)
   //                       v-     m-     v             m+            v+
 
   const bool lower_boundary_is_closer = F == 0 && E > 1;
   const diyfp m_plus = diyfp(2 * v.f + 1, v.e - 1);
   const diyfp m_minus = lower_boundary_is_closer ?
                            diyfp(4 * v.f - 1, v.e - 2) // (B)
                            :
                            diyfp(2 * v.f - 1, v.e - 1); // (A)
 
   // Determine the normalized w+ = m+.
   const diyfp w_plus = diyfp::normalize(m_plus);
 
   // Determine w- = m- such that e_(w-) = e_(w+).
   const diyfp w_minus = diyfp::normalize_to(m_minus, w_plus.e);
 
   return { diyfp::normalize(v), w_minus, w_plus };
}

References internal::dtoa_impl::diyfp::e, internal::dtoa_impl::diyfp::f, internal::dtoa_impl::diyfp::normalize(), and internal::dtoa_impl::diyfp::normalize_to().

Referenced by grisu2().

Here is the call graph for this function:

Here is the caller graph for this function:

find_largest_pow10()

int internal::dtoa_impl::find_largest_pow10 ( const std::uint32_t  n, std::uint32_t &  pow10  )

inline

For n != 0, returns k, such that pow10 := 10^(k-1) <= n < 10^k. For n == 0, returns 1 and sets pow10 := 1.

Definition at line 577 of file ToChars.cpp.

{
   // LCOV_EXCL_START
   if (n >= 1000000000)
   {
      pow10 = 1000000000;
      return 10;
   }
   // LCOV_EXCL_STOP
   else if (n >= 100000000)
   {
      pow10 = 100000000;
      return 9;
   }
   else if (n >= 10000000)
   {
      pow10 = 10000000;
      return 8;
   }
   else if (n >= 1000000)
   {
      pow10 = 1000000;
      return 7;
   }
   else if (n >= 100000)
   {
      pow10 = 100000;
      return 6;
   }
   else if (n >= 10000)
   {
      pow10 = 10000;
      return 5;
   }
   else if (n >= 1000)
   {
      pow10 = 1000;
      return 4;
   }
   else if (n >= 100)
   {
      pow10 = 100;
      return 3;
   }
   else if (n >= 10)
   {
      pow10 = 10;
      return 2;
   }
   else
   {
      pow10 = 1;
      return 1;
   }
}

Referenced by grisu2_digit_gen().

Here is the caller graph for this function:

format_buffer()

ToCharsResult internal::dtoa_impl::format_buffer ( char *  buf, char *  last, int  len, int  decimal_exponent, int  min_exp, int  max_exp  )

inline

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

Precondition

min_exp < 0

max_exp > 0

Definition at line 1076 of file ToChars.cpp.

{
   const int k = len;
   const int n = len + decimal_exponent;
 
   // v = buf * 10^(n-k)
   // k is the length of the buffer (number of decimal digits)
   // n is the position of the decimal point relative to the start of the
   // buffer.
 
   if (k <= n && n <= max_exp)
   {
      char* requiredLast = buf + (static_cast<size_t>(n));
      if (requiredLast > last)
         return { last, std::errc::value_too_large };
      // digits[000]
      // len <= max_exp + 2
 
      std::memset(
         buf + k, '0', static_cast<size_t>(n) - static_cast<size_t>(k));
      // Make it look like a floating-point number (#362, #378)
      // buf[n + 0] = '.';
      // buf[n + 1] = '0';
      return { requiredLast, std::errc() };
   }
 
   if (0 < n && n <= max_exp)
   {
      char* requiredLast = buf + (static_cast<size_t>(k) + 1U);
 
      if (requiredLast > last)
         return { last, std::errc::value_too_large };
      // dig.its
      // len <= max_digits10 + 1
      std::memmove(
         buf + (static_cast<size_t>(n) + 1), buf + n,
         static_cast<size_t>(k) - static_cast<size_t>(n));
      buf[n] = '.';
 
      return { requiredLast, std::errc() };
   }
 
   if (min_exp < n && n <= 0)
   {
      char* requiredLast =
         buf + (2U + static_cast<size_t>(-n) + static_cast<size_t>(k));
 
      if (requiredLast > last)
         return { last, std::errc::value_too_large };
      // 0.[000]digits
      // len <= 2 + (-min_exp - 1) + max_digits10
 
      std::memmove(
         buf + (2 + static_cast<size_t>(-n)), buf, static_cast<size_t>(k));
      buf[0] = '0';
      buf[1] = '.';
      std::memset(buf + 2, '0', static_cast<size_t>(-n));
 
      return { requiredLast, std::errc() };
   }
 
   if (k == 1)
   {
      char* requiredLast = buf + 1;
 
      if (requiredLast > last)
         return { last, std::errc::value_too_large };
      // dE+123
      // len <= 1 + 5
 
      buf += 1;
   }
   else
   {
      char* requiredLast = buf + 1 + static_cast<size_t>(k);
 
      if (requiredLast > last)
         return { last, std::errc::value_too_large };
      // d.igitsE+123
      // len <= max_digits10 + 1 + 5
 
      std::memmove(buf + 2, buf + 1, static_cast<size_t>(k) - 1);
 
      buf[1] = '.';
      buf += 1 + static_cast<size_t>(k);
   }
 
   *buf++ = 'e';
   return append_exponent(buf, last, n - 1);
}

References append_exponent().

Referenced by internal::float_to_chars().

Here is the call graph for this function:

Here is the caller graph for this function:

get_cached_power_for_binary_exponent()

cached_power internal::dtoa_impl::get_cached_power_for_binary_exponent ( int  e )

inline

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

Definition at line 459 of file ToChars.cpp.

{
   // Now
   //
   //      alpha <= e_c + e + q <= gamma                                    (1)
   //      ==> f_c * 2^alpha <= c * 2^e * 2^q
   //
   // and since the c's are normalized, 2^(q-1) <= f_c,
   //
   //      ==> 2^(q - 1 + alpha) <= c * 2^(e + q)
   //      ==> 2^(alpha - e - 1) <= c
   //
   // If c were an exact power of ten, i.e. c = 10^k, one may determine k as
   //
   //      k = ceil( log_10( 2^(alpha - e - 1) ) )
   //        = ceil( (alpha - e - 1) * log_10(2) )
   //
   // From the paper:
   // "In theory the result of the procedure could be wrong since c is rounded,
   //  and the computation itself is approximated [...]. In practice, however,
   //  this simple function is sufficient."
   //
   // For IEEE double precision floating-point numbers converted into
   // normalized diyfp's w = f * 2^e, with q = 64,
   //
   //      e >= -1022      (min IEEE exponent)
   //           -52        (p - 1)
   //           -52        (p - 1, possibly normalize denormal IEEE numbers)
   //           -11        (normalize the diyfp)
   //         = -1137
   //
   // and
   //
   //      e <= +1023      (max IEEE exponent)
   //           -52        (p - 1)
   //           -11        (normalize the diyfp)
   //         = 960
   //
   // This binary exponent range [-1137,960] results in a decimal exponent
   // range [-307,324]. One does not need to store a cached power for each
   // k in this range. For each such k it suffices to find a cached power
   // such that the exponent of the product lies in [alpha,gamma].
   // This implies that the difference of the decimal exponents of adjacent
   // table entries must be less than or equal to
   //
   //      floor( (gamma - alpha) * log_10(2) ) = 8.
   //
   // (A smaller distance gamma-alpha would require a larger table.)
 
   // NB:
   // Actually this function returns c, such that -60 <= e_c + e + 64 <= -34.
 
   constexpr int kCachedPowersMinDecExp = -300;
   constexpr int kCachedPowersDecStep = 8;
 
   static constexpr std::array<cached_power, 79> kCachedPowers = { {
      { 0xAB70FE17C79AC6CA, -1060, -300 }, { 0xFF77B1FCBEBCDC4F, -1034, -292 },
      { 0xBE5691EF416BD60C, -1007, -284 }, { 0x8DD01FAD907FFC3C, -980, -276 },
      { 0xD3515C2831559A83, -954, -268 },  { 0x9D71AC8FADA6C9B5, -927, -260 },
      { 0xEA9C227723EE8BCB, -901, -252 },  { 0xAECC49914078536D, -874, -244 },
      { 0x823C12795DB6CE57, -847, -236 },  { 0xC21094364DFB5637, -821, -228 },
      { 0x9096EA6F3848984F, -794, -220 },  { 0xD77485CB25823AC7, -768, -212 },
      { 0xA086CFCD97BF97F4, -741, -204 },  { 0xEF340A98172AACE5, -715, -196 },
      { 0xB23867FB2A35B28E, -688, -188 },  { 0x84C8D4DFD2C63F3B, -661, -180 },
      { 0xC5DD44271AD3CDBA, -635, -172 },  { 0x936B9FCEBB25C996, -608, -164 },
      { 0xDBAC6C247D62A584, -582, -156 },  { 0xA3AB66580D5FDAF6, -555, -148 },
      { 0xF3E2F893DEC3F126, -529, -140 },  { 0xB5B5ADA8AAFF80B8, -502, -132 },
      { 0x87625F056C7C4A8B, -475, -124 },  { 0xC9BCFF6034C13053, -449, -116 },
      { 0x964E858C91BA2655, -422, -108 },  { 0xDFF9772470297EBD, -396, -100 },
      { 0xA6DFBD9FB8E5B88F, -369, -92 },   { 0xF8A95FCF88747D94, -343, -84 },
      { 0xB94470938FA89BCF, -316, -76 },   { 0x8A08F0F8BF0F156B, -289, -68 },
      { 0xCDB02555653131B6, -263, -60 },   { 0x993FE2C6D07B7FAC, -236, -52 },
      { 0xE45C10C42A2B3B06, -210, -44 },   { 0xAA242499697392D3, -183, -36 },
      { 0xFD87B5F28300CA0E, -157, -28 },   { 0xBCE5086492111AEB, -130, -20 },
      { 0x8CBCCC096F5088CC, -103, -12 },   { 0xD1B71758E219652C, -77, -4 },
      { 0x9C40000000000000, -50, 4 },      { 0xE8D4A51000000000, -24, 12 },
      { 0xAD78EBC5AC620000, 3, 20 },       { 0x813F3978F8940984, 30, 28 },
      { 0xC097CE7BC90715B3, 56, 36 },      { 0x8F7E32CE7BEA5C70, 83, 44 },
      { 0xD5D238A4ABE98068, 109, 52 },     { 0x9F4F2726179A2245, 136, 60 },
      { 0xED63A231D4C4FB27, 162, 68 },     { 0xB0DE65388CC8ADA8, 189, 76 },
      { 0x83C7088E1AAB65DB, 216, 84 },     { 0xC45D1DF942711D9A, 242, 92 },
      { 0x924D692CA61BE758, 269, 100 },    { 0xDA01EE641A708DEA, 295, 108 },
      { 0xA26DA3999AEF774A, 322, 116 },    { 0xF209787BB47D6B85, 348, 124 },
      { 0xB454E4A179DD1877, 375, 132 },    { 0x865B86925B9BC5C2, 402, 140 },
      { 0xC83553C5C8965D3D, 428, 148 },    { 0x952AB45CFA97A0B3, 455, 156 },
      { 0xDE469FBD99A05FE3, 481, 164 },    { 0xA59BC234DB398C25, 508, 172 },
      { 0xF6C69A72A3989F5C, 534, 180 },    { 0xB7DCBF5354E9BECE, 561, 188 },
      { 0x88FCF317F22241E2, 588, 196 },    { 0xCC20CE9BD35C78A5, 614, 204 },
      { 0x98165AF37B2153DF, 641, 212 },    { 0xE2A0B5DC971F303A, 667, 220 },
      { 0xA8D9D1535CE3B396, 694, 228 },    { 0xFB9B7CD9A4A7443C, 720, 236 },
      { 0xBB764C4CA7A44410, 747, 244 },    { 0x8BAB8EEFB6409C1A, 774, 252 },
      { 0xD01FEF10A657842C, 800, 260 },    { 0x9B10A4E5E9913129, 827, 268 },
      { 0xE7109BFBA19C0C9D, 853, 276 },    { 0xAC2820D9623BF429, 880, 284 },
      { 0x80444B5E7AA7CF85, 907, 292 },    { 0xBF21E44003ACDD2D, 933, 300 },
      { 0x8E679C2F5E44FF8F, 960, 308 },    { 0xD433179D9C8CB841, 986, 316 },
      { 0x9E19DB92B4E31BA9, 1013, 324 },
   } };
 
   // This computation gives exactly the same results for k as
   //      k = ceil((kAlpha - e - 1) * 0.30102999566398114)
   // for |e| <= 1500, but doesn't require floating-point operations.
   // NB: log_10(2) ~= 78913 / 2^18
   const int f = kAlpha - e - 1;
   const int k = (f * 78913) / (1 << 18) + static_cast<int>(f > 0);
 
   const int index =
      (-kCachedPowersMinDecExp + k + (kCachedPowersDecStep - 1)) /
      kCachedPowersDecStep;
 
   const cached_power cached = kCachedPowers[static_cast<std::size_t>(index)];
 
   return cached;
}

References kAlpha.

Referenced by grisu2().

Here is the caller graph for this function:

grisu2() [1/2]

bool internal::dtoa_impl::grisu2 ( char *  buf, char *  last, int &  len, int &  decimal_exponent, diyfp  m_minus, diyfp  v, diyfp  m_plus  )

inline

v = buf * 10^decimal_exponent len is the length of the buffer (number of decimal digits) The buffer must be large enough, i.e. >= max_digits10.

Definition at line 923 of file ToChars.cpp.

{
 
   //  --------(-----------------------+-----------------------)--------    (A)
   //          m-                      v                       m+
   //
   //  --------------------(-----------+-----------------------)--------    (B)
   //                      m-          v                       m+
   //
   // First scale v (and m- and m+) such that the exponent is in the range
   // [alpha, gamma].
 
   const cached_power cached = get_cached_power_for_binary_exponent(m_plus.e);
 
   const diyfp c_minus_k(cached.f, cached.e); // = c ~= 10^-k
 
   // The exponent of the products is = v.e + c_minus_k.e + q and is in the
   // range [alpha,gamma]
   const diyfp w = diyfp::mul(v, c_minus_k);
   const diyfp w_minus = diyfp::mul(m_minus, c_minus_k);
   const diyfp w_plus = diyfp::mul(m_plus, c_minus_k);
 
   //  ----(---+---)---------------(---+---)---------------(---+---)----
   //          w-                      w                       w+
   //          = c*m-                  = c*v                   = c*m+
   //
   // diyfp::mul rounds its result and c_minus_k is approximated too. w, w- and
   // w+ are now off by a small amount.
   // In fact:
   //
   //      w - v * 10^k < 1 ulp
   //
   // To account for this inaccuracy, add resp. subtract 1 ulp.
   //
   //  --------+---[---------------(---+---)---------------]---+--------
   //          w-  M-                  w                   M+  w+
   //
   // Now any number in [M-, M+] (bounds included) will round to w when input,
   // regardless of how the input rounding algorithm breaks ties.
   //
   // And digit_gen generates the shortest possible such number in [M-, M+].
   // Note that this does not mean that Grisu2 always generates the shortest
   // possible number in the interval (m-, m+).
   const diyfp M_minus(w_minus.f + 1, w_minus.e);
   const diyfp M_plus(w_plus.f - 1, w_plus.e);
 
   decimal_exponent = -cached.k; // = -(-k) = k
 
   return grisu2_digit_gen(buf, last, len, decimal_exponent, M_minus, w, M_plus);
}

References internal::dtoa_impl::diyfp::e, internal::dtoa_impl::cached_power::e, internal::dtoa_impl::diyfp::f, internal::dtoa_impl::cached_power::f, get_cached_power_for_binary_exponent(), grisu2_digit_gen(), internal::dtoa_impl::cached_power::k, and internal::dtoa_impl::diyfp::mul().

Referenced by internal::float_to_chars(), and grisu2().

Here is the call graph for this function:

Here is the caller graph for this function:

grisu2() [2/2]

template<typename FloatType >

bool internal::dtoa_impl::grisu2	(	char *	buf,
		char *	last,
		int &	len,
		int &	decimal_exponent,
		FloatType	value
	)

v = buf * 10^decimal_exponent len is the length of the buffer (number of decimal digits) The buffer must be large enough, i.e. >= max_digits10.

Definition at line 982 of file ToChars.cpp.

{
   static_assert(
      diyfp::kPrecision >= std::numeric_limits<FloatType>::digits + 3,
      "internal error: not enough precision");
 
   // If the neighbors (and boundaries) of 'value' are always computed for
   // double-precision numbers, all float's can be recovered using strtod (and
   // strtof). However, the resulting decimal representations are not exactly
   // "short".
   //
   // The documentation for 'std::to_chars'
   // (https://en.cppreference.com/w/cpp/utility/to_chars) says "value is
   // converted to a string as if by std::sprintf in the default ("C") locale"
   // and since sprintf promotes float's to double's, I think this is exactly
   // what 'std::to_chars' does. On the other hand, the documentation for
   // 'std::to_chars' requires that "parsing the representation using the
   // corresponding std::from_chars function recovers value exactly". That
   // indicates that single precision floating-point numbers should be recovered
   // using 'std::strtof'.
   //
   // NB: If the neighbors are computed for single-precision numbers, there is a
   // single float
   //     (7.0385307e-26f) which can't be recovered using strtod. The resulting
   //     double precision value is off by 1 ulp.
#if 0
    const boundaries w = compute_boundaries(static_cast<double>(value));
#else
   const boundaries w = compute_boundaries(value);
#endif
 
   return grisu2(buf, last, len, decimal_exponent, w.minus, w.w, w.plus);
}

References compute_boundaries(), grisu2(), internal::dtoa_impl::diyfp::kPrecision, internal::dtoa_impl::boundaries::minus, internal::dtoa_impl::boundaries::plus, and internal::dtoa_impl::boundaries::w.

Here is the call graph for this function:

grisu2_digit_gen()

bool internal::dtoa_impl::grisu2_digit_gen ( char *  buffer, char *  last, int &  length, int &  decimal_exponent, diyfp  M_minus, diyfp  w, diyfp  M_plus  )

inline

Generates V = buffer * 10^decimal_exponent, such that M- <= V <= M+. M- and M+ must be normalized and share the same exponent -60 <= e <= -32.

Definition at line 669 of file ToChars.cpp.

{
   static_assert(kAlpha >= -60, "internal error");
   static_assert(kGamma <= -32, "internal error");
 
   const int max_length = static_cast<int>(last - buffer);
 
   // Generates the digits (and the exponent) of a decimal floating-point
   // number V = buffer * 10^decimal_exponent in the range [M-, M+]. The diyfp's
   // w, M- and M+ share the same exponent e, which satisfies alpha <= e <=
   // gamma.
   //
   //               <--------------------------- delta ---->
   //                                  <---- dist --------->
   // --------------[------------------+-------------------]--------------
   //               M-                 w                   M+
   //
   // Grisu2 generates the digits of M+ from left to right and stops as soon as
   // V is in [M-,M+].
 
   std::uint64_t delta =
      diyfp::sub(M_plus, M_minus)
         .f; // (significand of (M+ - M-), implicit exponent is e)
   std::uint64_t dist =
      diyfp::sub(M_plus, w)
         .f; // (significand of (M+ - w ), implicit exponent is e)
 
   // Split M+ = f * 2^e into two parts p1 and p2 (note: e < 0):
   //
   //      M+ = f * 2^e
   //         = ((f div 2^-e) * 2^-e + (f mod 2^-e)) * 2^e
   //         = ((p1        ) * 2^-e + (p2        )) * 2^e
   //         = p1 + p2 * 2^e
 
   const diyfp one(std::uint64_t { 1 } << -M_plus.e, M_plus.e);
 
   auto p1 = static_cast<std::uint32_t>(
      M_plus.f >>
      -one.e); // p1 = f div 2^-e (Since -e >= 32, p1 fits into a 32-bit int.)
   std::uint64_t p2 = M_plus.f & (one.f - 1); // p2 = f mod 2^-e
 
   // 1)
   //
   // Generate the digits of the integral part p1 = d[n-1]...d[1]d[0]
 
   std::uint32_t pow10;
   const int k = find_largest_pow10(p1, pow10);
 
   //      10^(k-1) <= p1 < 10^k, pow10 = 10^(k-1)
   //
   //      p1 = (p1 div 10^(k-1)) * 10^(k-1) + (p1 mod 10^(k-1))
   //         = (d[k-1]         ) * 10^(k-1) + (p1 mod 10^(k-1))
   //
   //      M+ = p1                                             + p2 * 2^e
   //         = d[k-1] * 10^(k-1) + (p1 mod 10^(k-1))          + p2 * 2^e
   //         = d[k-1] * 10^(k-1) + ((p1 mod 10^(k-1)) * 2^-e + p2) * 2^e
   //         = d[k-1] * 10^(k-1) + (                         rest) * 2^e
   //
   // Now generate the digits d[n] of p1 from left to right (n = k-1,...,0)
   //
   //      p1 = d[k-1]...d[n] * 10^n + d[n-1]...d[0]
   //
   // but stop as soon as
   //
   //      rest * 2^e = (d[n-1]...d[0] * 2^-e + p2) * 2^e <= delta * 2^e
 
   int n = k;
   while (n > 0)
   {
      // Check that we are able to write the next symbol into the buffer
      if (length >= max_length)
         return false;
 
      // Invariants:
      //      M+ = buffer * 10^n + (p1 + p2 * 2^e)    (buffer = 0 for n = k)
      //      pow10 = 10^(n-1) <= p1 < 10^n
      //
      const std::uint32_t d = p1 / pow10; // d = p1 div 10^(n-1)
      const std::uint32_t r = p1 % pow10; // r = p1 mod 10^(n-1)
      //
      //      M+ = buffer * 10^n + (d * 10^(n-1) + r) + p2 * 2^e
      //         = (buffer * 10 + d) * 10^(n-1) + (r + p2 * 2^e)
      //
      buffer[length++] =
         static_cast<char>('0' + d); // buffer := buffer * 10 + d
      //
      //      M+ = buffer * 10^(n-1) + (r + p2 * 2^e)
      //
      p1 = r;
      n--;
      //
      //      M+ = buffer * 10^n + (p1 + p2 * 2^e)
      //      pow10 = 10^n
      //
 
      // Now check if enough digits have been generated.
      // Compute
      //
      //      p1 + p2 * 2^e = (p1 * 2^-e + p2) * 2^e = rest * 2^e
      //
      // Note:
      // Since rest and delta share the same exponent e, it suffices to
      // compare the significands.
      const std::uint64_t rest = (std::uint64_t { p1 } << -one.e) + p2;
      if (rest <= delta)
      {
         // V = buffer * 10^n, with M- <= V <= M+.
 
         decimal_exponent += n;
 
         // We may now just stop. But instead look if the buffer could be
         // decremented to bring V closer to w.
         //
         // pow10 = 10^n is now 1 ulp in the decimal representation V.
         // The rounding procedure works with diyfp's with an implicit
         // exponent of e.
         //
         //      10^n = (10^n * 2^-e) * 2^e = ulp * 2^e
         //
         const std::uint64_t ten_n = std::uint64_t { pow10 } << -one.e;
         grisu2_round(buffer, length, dist, delta, rest, ten_n);
 
         return true;
      }
 
      pow10 /= 10;
      //
      //      pow10 = 10^(n-1) <= p1 < 10^n
      // Invariants restored.
   }
 
   // 2)
   //
   // The digits of the integral part have been generated:
   //
   //      M+ = d[k-1]...d[1]d[0] + p2 * 2^e
   //         = buffer            + p2 * 2^e
   //
   // Now generate the digits of the fractional part p2 * 2^e.
   //
   // Note:
   // No decimal point is generated: the exponent is adjusted instead.
   //
   // p2 actually represents the fraction
   //
   //      p2 * 2^e
   //          = p2 / 2^-e
   //          = d[-1] / 10^1 + d[-2] / 10^2 + ...
   //
   // Now generate the digits d[-m] of p1 from left to right (m = 1,2,...)
   //
   //      p2 * 2^e = d[-1]d[-2]...d[-m] * 10^-m
   //                      + 10^-m * (d[-m-1] / 10^1 + d[-m-2] / 10^2 + ...)
   //
   // using
   //
   //      10^m * p2 = ((10^m * p2) div 2^-e) * 2^-e + ((10^m * p2) mod 2^-e)
   //                = (                   d) * 2^-e + (                   r)
   //
   // or
   //      10^m * p2 * 2^e = d + r * 2^e
   //
   // i.e.
   //
   //      M+ = buffer + p2 * 2^e
   //         = buffer + 10^-m * (d + r * 2^e)
   //         = (buffer * 10^m + d) * 10^-m + 10^-m * r * 2^e
   //
   // and stop as soon as 10^-m * r * 2^e <= delta * 2^e
 
   int m = 0;
   for (;;)
   {
      // Check that we are able to write the next symbol into the buffer
      if (length >= max_length)
         return false;
      // Invariant:
      //      M+ = buffer * 10^-m + 10^-m * (d[-m-1] / 10 + d[-m-2] / 10^2 +
      //      ...)
      //      * 2^e
      //         = buffer * 10^-m + 10^-m * (p2 )
      //         * 2^e = buffer * 10^-m + 10^-m * (1/10 * (10 * p2) ) * 2^e =
      //         buffer * 10^-m + 10^-m * (1/10 * ((10*p2 div 2^-e) * 2^-e +
      //         (10*p2 mod 2^-e)) * 2^e
      //
      p2 *= 10;
      const std::uint64_t d = p2 >> -one.e;     // d = (10 * p2) div 2^-e
      const std::uint64_t r = p2 & (one.f - 1); // r = (10 * p2) mod 2^-e
      //
      //      M+ = buffer * 10^-m + 10^-m * (1/10 * (d * 2^-e + r) * 2^e
      //         = buffer * 10^-m + 10^-m * (1/10 * (d + r * 2^e))
      //         = (buffer * 10 + d) * 10^(-m-1) + 10^(-m-1) * r * 2^e
      //
      buffer[length++] =
         static_cast<char>('0' + d); // buffer := buffer * 10 + d
      //
      //      M+ = buffer * 10^(-m-1) + 10^(-m-1) * r * 2^e
      //
      p2 = r;
      m++;
      //
      //      M+ = buffer * 10^-m + 10^-m * p2 * 2^e
      // Invariant restored.
 
      // Check if enough digits have been generated.
      //
      //      10^-m * p2 * 2^e <= delta * 2^e
      //              p2 * 2^e <= 10^m * delta * 2^e
      //                    p2 <= 10^m * delta
      delta *= 10;
      dist *= 10;
      if (p2 <= delta)
      {
         break;
      }
   }
 
   // V = buffer * 10^-m, with M- <= V <= M+.
 
   decimal_exponent -= m;
 
   // 1 ulp in the decimal representation is now 10^-m.
   // Since delta and dist are now scaled by 10^m, we need to do the
   // same with ulp in order to keep the units in sync.
   //
   //      10^m * 10^-m = 1 = 2^-e * 2^e = ten_m * 2^e
   //
   const std::uint64_t ten_m = one.f;
   grisu2_round(buffer, length, dist, delta, p2, ten_m);
 
   // By construction this algorithm generates the shortest possible decimal
   // number (Loitsch, Theorem 6.2) which rounds back to w.
   // For an input number of precision p, at least
   //
   //      N = 1 + ceil(p * log_10(2))
   //
   // decimal digits are sufficient to identify all binary floating-point
   // numbers (Matula, "In-and-Out conversions").
   // This implies that the algorithm does not produce more than N decimal
   // digits.
   //
   //      N = 17 for p = 53 (IEEE double precision)
   //      N = 9  for p = 24 (IEEE single precision)
 
   return true;
}

References internal::dtoa_impl::diyfp::e, internal::dtoa_impl::diyfp::f, find_largest_pow10(), grisu2_round(), kAlpha, kGamma, and internal::dtoa_impl::diyfp::sub().

Referenced by grisu2().

Here is the call graph for this function:

Here is the caller graph for this function:

grisu2_round()

void internal::dtoa_impl::grisu2_round ( char *  buf, int  len, std::uint64_t  dist, std::uint64_t  delta, std::uint64_t  rest, std::uint64_t  ten_k  )

inline

Definition at line 633 of file ToChars.cpp.

{
 
   //               <--------------------------- delta ---->
   //                                  <---- dist --------->
   // --------------[------------------+-------------------]--------------
   //               M-                 w                   M+
   //
   //                                  ten_k
   //                                <------>
   //                                       <---- rest ---->
   // --------------[------------------+----+--------------]--------------
   //                                  w    V
   //                                       = buf * 10^k
   //
   // ten_k represents a unit-in-the-last-place in the decimal representation
   // stored in buf.
   // Decrement buf by ten_k while this takes buf closer to w.
 
   // The tests are written in this order to avoid overflow in unsigned
   // integer arithmetic.
 
   while (rest < dist && delta - rest >= ten_k &&
          (rest + ten_k < dist || dist - rest > rest + ten_k - dist))
   {
      buf[len - 1]--;
      rest += ten_k;
   }
}

Referenced by grisu2_digit_gen().

Here is the caller graph for this function:

reinterpret_bits()

template<typename Target , typename Source >

Target internal::dtoa_impl::reinterpret_bits

(

const Source

source

)

Definition at line 154 of file ToChars.cpp.

{
   static_assert(sizeof(Target) == sizeof(Source), "size mismatch");
 
   Target target;
   std::memcpy(&target, &source, sizeof(Source));
   return target;
}

Variable Documentation

kAlpha

constexpr int internal::dtoa_impl::kAlpha = -60

constexpr

Definition at line 443 of file ToChars.cpp.

Referenced by get_cached_power_for_binary_exponent(), and grisu2_digit_gen().

kGamma

constexpr int internal::dtoa_impl::kGamma = -32

constexpr

Definition at line 444 of file ToChars.cpp.

Referenced by grisu2_digit_gen().