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wxhelper-new/inc/nlohmann/detail/conversions/to_chars.hpp

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添加项目文件。
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// __ _____ _____ _____
// __| | __| | | | JSON for Modern C++
// | | |__ | | | | | | version 3.11.3
// |_____|_____|_____|_|___| https://github.com/nlohmann/json
//
// SPDX-FileCopyrightText: 2009 Florian Loitsch <https://florian.loitsch.com/>
// SPDX-FileCopyrightText: 2013-2023 Niels Lohmann <https://nlohmann.me>
// SPDX-License-Identifier: MIT
#pragma once
#include <array> // array
#include <cmath> // signbit, isfinite
#include <cstdint> // intN_t, uintN_t
#include <cstring> // memcpy, memmove
#include <limits> // numeric_limits
#include <type_traits> // conditional
#include <nlohmann/detail/macro_scope.hpp>
NLOHMANN_JSON_NAMESPACE_BEGIN
namespace detail
{
/*!
@brief implements the Grisu2 algorithm for binary to decimal floating-point
conversion.
This implementation is a slightly modified version of the reference
implementation which may be obtained from
http://florian.loitsch.com/publications (bench.tar.gz).
The code is distributed under the MIT license, Copyright (c) 2009 Florian Loitsch.
For a detailed description of the algorithm see:
[1] Loitsch, "Printing Floating-Point Numbers Quickly and Accurately with
Integers", Proceedings of the ACM SIGPLAN 2010 Conference on Programming
Language Design and Implementation, PLDI 2010
[2] Burger, Dybvig, "Printing Floating-Point Numbers Quickly and Accurately",
Proceedings of the ACM SIGPLAN 1996 Conference on Programming Language
Design and Implementation, PLDI 1996
*/
namespace dtoa_impl
{
template<typename Target, typename Source>
Target reinterpret_bits(const Source source)
{
static_assert(sizeof(Target) == sizeof(Source), "size mismatch");
Target target;
std::memcpy(&target, &source, sizeof(Source));
return target;
}
struct diyfp // f * 2^e
{
static constexpr int kPrecision = 64; // = q
std::uint64_t f = 0;
int e = 0;
constexpr diyfp(std::uint64_t f_, int e_) noexcept : f(f_), e(e_) {}
/*!
@brief returns x - y
@pre x.e == y.e and x.f >= y.f
*/
static diyfp sub(const diyfp& x, const diyfp& y) noexcept
{
JSON_ASSERT(x.e == y.e);
JSON_ASSERT(x.f >= y.f);
return {x.f - y.f, x.e};
}
/*!
@brief returns x * y
@note The result is rounded. (Only the upper q bits are returned.)
*/
static diyfp mul(const diyfp& x, const diyfp& y) noexcept
{
static_assert(kPrecision == 64, "internal error");
// Computes:
// f = round((x.f * y.f) / 2^q)
// e = x.e + y.e + q
// Emulate the 64-bit * 64-bit multiplication:
//
// p = u * v
// = (u_lo + 2^32 u_hi) (v_lo + 2^32 v_hi)
// = (u_lo v_lo ) + 2^32 ((u_lo v_hi ) + (u_hi v_lo )) + 2^64 (u_hi v_hi )
// = (p0 ) + 2^32 ((p1 ) + (p2 )) + 2^64 (p3 )
// = (p0_lo + 2^32 p0_hi) + 2^32 ((p1_lo + 2^32 p1_hi) + (p2_lo + 2^32 p2_hi)) + 2^64 (p3 )
// = (p0_lo ) + 2^32 (p0_hi + p1_lo + p2_lo ) + 2^64 (p1_hi + p2_hi + p3)
// = (p0_lo ) + 2^32 (Q ) + 2^64 (H )
// = (p0_lo ) + 2^32 (Q_lo + 2^32 Q_hi ) + 2^64 (H )
//
// (Since Q might be larger than 2^32 - 1)
//
// = (p0_lo + 2^32 Q_lo) + 2^64 (Q_hi + H)
//
// (Q_hi + H does not overflow a 64-bit int)
//
// = p_lo + 2^64 p_hi
const std::uint64_t u_lo = x.f & 0xFFFFFFFFu;
const std::uint64_t u_hi = x.f >> 32u;
const std::uint64_t v_lo = y.f & 0xFFFFFFFFu;
const std::uint64_t v_hi = y.f >> 32u;
const std::uint64_t p0 = u_lo * v_lo;
const std::uint64_t p1 = u_lo * v_hi;
const std::uint64_t p2 = u_hi * v_lo;
const std::uint64_t p3 = u_hi * v_hi;
const std::uint64_t p0_hi = p0 >> 32u;
const std::uint64_t p1_lo = p1 & 0xFFFFFFFFu;
const std::uint64_t p1_hi = p1 >> 32u;
const std::uint64_t p2_lo = p2 & 0xFFFFFFFFu;
const std::uint64_t p2_hi = p2 >> 32u;
std::uint64_t Q = p0_hi + p1_lo + p2_lo;
// The full product might now be computed as
//
// p_hi = p3 + p2_hi + p1_hi + (Q >> 32)
// p_lo = p0_lo + (Q << 32)
//
// But in this particular case here, the full p_lo is not required.
// Effectively we only need to add the highest bit in p_lo to p_hi (and
// Q_hi + 1 does not overflow).
Q += std::uint64_t{1} << (64u - 32u - 1u); // round, ties up
const std::uint64_t h = p3 + p2_hi + p1_hi + (Q >> 32u);
return {h, x.e + y.e + 64};
}
/*!
@brief normalize x such that the significand is >= 2^(q-1)
@pre x.f != 0
*/
static diyfp normalize(diyfp x) noexcept
{
JSON_ASSERT(x.f != 0);
while ((x.f >> 63u) == 0)
{
x.f <<= 1u;
x.e--;
}
return x;
}
/*!
@brief normalize x such that the result has the exponent E
@pre e >= x.e and the upper e - x.e bits of x.f must be zero.
*/
static diyfp normalize_to(const diyfp& x, const int target_exponent) noexcept
{
const int delta = x.e - target_exponent;
JSON_ASSERT(delta >= 0);
JSON_ASSERT(((x.f << delta) >> delta) == x.f);
return {x.f << delta, target_exponent};
}
};
struct boundaries
{
diyfp w;
diyfp minus;
diyfp plus;
};
/*!
Compute the (normalized) diyfp representing the input number 'value' and its
boundaries.
@pre value must be finite and positive
*/
template<typename FloatType>
boundaries compute_boundaries(FloatType value)
{
JSON_ASSERT(std::isfinite(value));
JSON_ASSERT(value > 0);
// Convert the IEEE representation into a diyfp.
//
// If v is denormal:
// value = 0.F * 2^(1 - bias) = ( F) * 2^(1 - bias - (p-1))
// If v is normalized:
// value = 1.F * 2^(E - bias) = (2^(p-1) + F) * 2^(E - bias - (p-1))
static_assert(std::numeric_limits<FloatType>::is_iec559,
"internal error: dtoa_short requires an IEEE-754 floating-point implementation");
constexpr int kPrecision = std::numeric_limits<FloatType>::digits; // = p (includes the hidden bit)
constexpr int kBias = std::numeric_limits<FloatType>::max_exponent - 1 + (kPrecision - 1);
constexpr int kMinExp = 1 - kBias;
constexpr std::uint64_t kHiddenBit = std::uint64_t{1} << (kPrecision - 1); // = 2^(p-1)
using bits_type = typename std::conditional<kPrecision == 24, std::uint32_t, std::uint64_t >::type;
const auto bits = static_cast<std::uint64_t>(reinterpret_bits<bits_type>(value));
const std::uint64_t E = bits >> (kPrecision - 1);
const std::uint64_t F = bits & (kHiddenBit - 1);
const bool is_denormal = E == 0;
const diyfp v = is_denormal
? diyfp(F, kMinExp)
: diyfp(F + kHiddenBit, static_cast<int>(E) - kBias);
// Compute the boundaries m- and m+ of the floating-point value
// v = f * 2^e.
//
// Determine v- and v+, the floating-point predecessor and successor if v,
// respectively.
//
// v- = v - 2^e if f != 2^(p-1) or e == e_min (A)
// = v - 2^(e-1) if f == 2^(p-1) and e > e_min (B)
//
// v+ = v + 2^e
//
// Let m- = (v- + v) / 2 and m+ = (v + v+) / 2. All real numbers _strictly_
// between m- and m+ round to v, regardless of how the input rounding
// algorithm breaks ties.
//
// ---+-------------+-------------+-------------+-------------+--- (A)
// v- m- v m+ v+
//
// -----------------+------+------+-------------+-------------+--- (B)
// v- m- v m+ v+
const bool lower_boundary_is_closer = F == 0 && E > 1;
const diyfp m_plus = diyfp(2 * v.f + 1, v.e - 1);
const diyfp m_minus = lower_boundary_is_closer
? diyfp(4 * v.f - 1, v.e - 2) // (B)
: diyfp(2 * v.f - 1, v.e - 1); // (A)
// Determine the normalized w+ = m+.
const diyfp w_plus = diyfp::normalize(m_plus);
// Determine w- = m- such that e_(w-) = e_(w+).
const diyfp w_minus = diyfp::normalize_to(m_minus, w_plus.e);
return {diyfp::normalize(v), w_minus, w_plus};
}
// Given normalized diyfp w, Grisu needs to find a (normalized) cached
// power-of-ten c, such that the exponent of the product c * w = f * 2^e lies
// within a certain range [alpha, gamma] (Definition 3.2 from [1])
//
// alpha <= e = e_c + e_w + q <= gamma
//
// or
//
// f_c * f_w * 2^alpha <= f_c 2^(e_c) * f_w 2^(e_w) * 2^q
// <= f_c * f_w * 2^gamma
//
// Since c and w are normalized, i.e. 2^(q-1) <= f < 2^q, this implies
//
// 2^(q-1) * 2^(q-1) * 2^alpha <= c * w * 2^q < 2^q * 2^q * 2^gamma
//
// or
//
// 2^(q - 2 + alpha) <= c * w < 2^(q + gamma)
//
// The choice of (alpha,gamma) determines the size of the table and the form of
// the digit generation procedure. Using (alpha,gamma)=(-60,-32) works out well
// in practice:
//
// The idea is to cut the number c * w = f * 2^e into two parts, which can be
// processed independently: An integral part p1, and a fractional part p2:
//
// f * 2^e = ( (f div 2^-e) * 2^-e + (f mod 2^-e) ) * 2^e
// = (f div 2^-e) + (f mod 2^-e) * 2^e
// = p1 + p2 * 2^e
//
// The conversion of p1 into decimal form requires a series of divisions and
// modulos by (a power of) 10. These operations are faster for 32-bit than for
// 64-bit integers, so p1 should ideally fit into a 32-bit integer. This can be
// achieved by choosing
//
// -e >= 32 or e <= -32 := gamma
//
// In order to convert the fractional part
//
// p2 * 2^e = p2 / 2^-e = d[-1] / 10^1 + d[-2] / 10^2 + ...
//
// into decimal form, the fraction is repeatedly multiplied by 10 and the digits
// d[-i] are extracted in order:
//
// (10 * p2) div 2^-e = d[-1]
// (10 * p2) mod 2^-e = d[-2] / 10^1 + ...
//
// The multiplication by 10 must not overflow. It is sufficient to choose
//
// 10 * p2 < 16 * p2 = 2^4 * p2 <= 2^64.
//
// Since p2 = f mod 2^-e < 2^-e,
//
// -e <= 60 or e >= -60 := alpha
constexpr int kAlpha = -60;
constexpr int kGamma = -32;
struct cached_power // c = f * 2^e ~= 10^k
{
std::uint64_t f;
int e;
int k;
};
/*!
For a normalized diyfp w = f * 2^e, this function returns a (normalized) cached
power-of-ten c = f_c * 2^e_c, such that the exponent of the product w * c
satisfies (Definition 3.2 from [1])
alpha <= e_c + e + q <= gamma.
*/
inline cached_power get_cached_power_for_binary_exponent(int e)
{
// Now
//
// alpha <= e_c + e + q <= gamma (1)
// ==> f_c * 2^alpha <= c * 2^e * 2^q
//
// and since the c's are normalized, 2^(q-1) <= f_c,
//
// ==> 2^(q - 1 + alpha) <= c * 2^(e + q)
// ==> 2^(alpha - e - 1) <= c
//
// If c were an exact power of ten, i.e. c = 10^k, one may determine k as
//
// k = ceil( log_10( 2^(alpha - e - 1) ) )
// = ceil( (alpha - e - 1) * log_10(2) )
//
// From the paper:
// "In theory the result of the procedure could be wrong since c is rounded,
// and the computation itself is approximated [...]. In practice, however,
// this simple function is sufficient."
//
// For IEEE double precision floating-point numbers converted into
// normalized diyfp's w = f * 2^e, with q = 64,
//
// e >= -1022 (min IEEE exponent)
// -52 (p - 1)
// -52 (p - 1, possibly normalize denormal IEEE numbers)
// -11 (normalize the diyfp)
// = -1137
//
// and
//
// e <= +1023 (max IEEE exponent)
// -52 (p - 1)
// -11 (normalize the diyfp)
// = 960
//
// This binary exponent range [-1137,960] results in a decimal exponent
// range [-307,324]. One does not need to store a cached power for each
// k in this range. For each such k it suffices to find a cached power
// such that the exponent of the product lies in [alpha,gamma].
// This implies that the difference of the decimal exponents of adjacent
// table entries must be less than or equal to
//
// floor( (gamma - alpha) * log_10(2) ) = 8.
//
// (A smaller distance gamma-alpha would require a larger table.)
// NB:
// Actually this function returns c, such that -60 <= e_c + e + 64 <= -34.
constexpr int kCachedPowersMinDecExp = -300;
constexpr int kCachedPowersDecStep = 8;
static constexpr std::array<cached_power, 79> kCachedPowers =
{
{
{ 0xAB70FE17C79AC6CA, -1060, -300 },
{ 0xFF77B1FCBEBCDC4F, -1034, -292 },
{ 0xBE5691EF416BD60C, -1007, -284 },
{ 0x8DD01FAD907FFC3C, -980, -276 },
{ 0xD3515C2831559A83, -954, -268 },
{ 0x9D71AC8FADA6C9B5, -927, -260 },
{ 0xEA9C227723EE8BCB, -901, -252 },
{ 0xAECC49914078536D, -874, -244 },
{ 0x823C12795DB6CE57, -847, -236 },
{ 0xC21094364DFB5637, -821, -228 },
{ 0x9096EA6F3848984F, -794, -220 },
{ 0xD77485CB25823AC7, -768, -212 },
{ 0xA086CFCD97BF97F4, -741, -204 },
{ 0xEF340A98172AACE5, -715, -196 },
{ 0xB23867FB2A35B28E, -688, -188 },
{ 0x84C8D4DFD2C63F3B, -661, -180 },
{ 0xC5DD44271AD3CDBA, -635, -172 },
{ 0x936B9FCEBB25C996, -608, -164 },
{ 0xDBAC6C247D62A584, -582, -156 },
{ 0xA3AB66580D5FDAF6, -555, -148 },
{ 0xF3E2F893DEC3F126, -529, -140 },
{ 0xB5B5ADA8AAFF80B8, -502, -132 },
{ 0x87625F056C7C4A8B, -475, -124 },
{ 0xC9BCFF6034C13053, -449, -116 },
{ 0x964E858C91BA2655, -422, -108 },
{ 0xDFF9772470297EBD, -396, -100 },
{ 0xA6DFBD9FB8E5B88F, -369, -92 },
{ 0xF8A95FCF88747D94, -343, -84 },
{ 0xB94470938FA89BCF, -316, -76 },
{ 0x8A08F0F8BF0F156B, -289, -68 },
{ 0xCDB02555653131B6, -263, -60 },
{ 0x993FE2C6D07B7FAC, -236, -52 },
{ 0xE45C10C42A2B3B06, -210, -44 },
{ 0xAA242499697392D3, -183, -36 },
{ 0xFD87B5F28300CA0E, -157, -28 },
{ 0xBCE5086492111AEB, -130, -20 },
{ 0x8CBCCC096F5088CC, -103, -12 },
{ 0xD1B71758E219652C, -77, -4 },
{ 0x9C40000000000000, -50, 4 },
{ 0xE8D4A51000000000, -24, 12 },
{ 0xAD78EBC5AC620000, 3, 20 },
{ 0x813F3978F8940984, 30, 28 },
{ 0xC097CE7BC90715B3, 56, 36 },
{ 0x8F7E32CE7BEA5C70, 83, 44 },
{ 0xD5D238A4ABE98068, 109, 52 },
{ 0x9F4F2726179A2245, 136, 60 },
{ 0xED63A231D4C4FB27, 162, 68 },
{ 0xB0DE65388CC8ADA8, 189, 76 },
{ 0x83C7088E1AAB65DB, 216, 84 },
{ 0xC45D1DF942711D9A, 242, 92 },
{ 0x924D692CA61BE758, 269, 100 },
{ 0xDA01EE641A708DEA, 295, 108 },
{ 0xA26DA3999AEF774A, 322, 116 },
{ 0xF209787BB47D6B85, 348, 124 },
{ 0xB454E4A179DD1877, 375, 132 },
{ 0x865B86925B9BC5C2, 402, 140 },
{ 0xC83553C5C8965D3D, 428, 148 },
{ 0x952AB45CFA97A0B3, 455, 156 },
{ 0xDE469FBD99A05FE3, 481, 164 },
{ 0xA59BC234DB398C25, 508, 172 },
{ 0xF6C69A72A3989F5C, 534, 180 },
{ 0xB7DCBF5354E9BECE, 561, 188 },
{ 0x88FCF317F22241E2, 588, 196 },
{ 0xCC20CE9BD35C78A5, 614, 204 },
{ 0x98165AF37B2153DF, 641, 212 },
{ 0xE2A0B5DC971F303A, 667, 220 },
{ 0xA8D9D1535CE3B396, 694, 228 },
{ 0xFB9B7CD9A4A7443C, 720, 236 },
{ 0xBB764C4CA7A44410, 747, 244 },
{ 0x8BAB8EEFB6409C1A, 774, 252 },
{ 0xD01FEF10A657842C, 800, 260 },
{ 0x9B10A4E5E9913129, 827, 268 },
{ 0xE7109BFBA19C0C9D, 853, 276 },
{ 0xAC2820D9623BF429, 880, 284 },
{ 0x80444B5E7AA7CF85, 907, 292 },
{ 0xBF21E44003ACDD2D, 933, 300 },
{ 0x8E679C2F5E44FF8F, 960, 308 },
{ 0xD433179D9C8CB841, 986, 316 },
{ 0x9E19DB92B4E31BA9, 1013, 324 },
}
};
// This computation gives exactly the same results for k as
// k = ceil((kAlpha - e - 1) * 0.30102999566398114)
// for |e| <= 1500, but doesn't require floating-point operations.
// NB: log_10(2) ~= 78913 / 2^18
JSON_ASSERT(e >= -1500);
JSON_ASSERT(e <= 1500);
const int f = kAlpha - e - 1;
const int k = (f * 78913) / (1 << 18) + static_cast<int>(f > 0);
const int index = (-kCachedPowersMinDecExp + k + (kCachedPowersDecStep - 1)) / kCachedPowersDecStep;
JSON_ASSERT(index >= 0);
JSON_ASSERT(static_cast<std::size_t>(index) < kCachedPowers.size());
const cached_power cached = kCachedPowers[static_cast<std::size_t>(index)];
JSON_ASSERT(kAlpha <= cached.e + e + 64);
JSON_ASSERT(kGamma >= cached.e + e + 64);
return cached;
}
/*!
For n != 0, returns k, such that pow10 := 10^(k-1) <= n < 10^k.
For n == 0, returns 1 and sets pow10 := 1.
*/
inline int find_largest_pow10(const std::uint32_t n, std::uint32_t& pow10)
{
// LCOV_EXCL_START
if (n >= 1000000000)
{
pow10 = 1000000000;
return 10;
}
// LCOV_EXCL_STOP
if (n >= 100000000)
{
pow10 = 100000000;
return 9;
}
if (n >= 10000000)
{
pow10 = 10000000;
return 8;
}
if (n >= 1000000)
{
pow10 = 1000000;
return 7;
}
if (n >= 100000)
{
pow10 = 100000;
return 6;
}
if (n >= 10000)
{
pow10 = 10000;
return 5;
}
if (n >= 1000)
{
pow10 = 1000;
return 4;
}
if (n >= 100)
{
pow10 = 100;
return 3;
}
if (n >= 10)
{
pow10 = 10;
return 2;
}
pow10 = 1;
return 1;
}
inline void grisu2_round(char* buf, int len, std::uint64_t dist, std::uint64_t delta,
std::uint64_t rest, std::uint64_t ten_k)
{
JSON_ASSERT(len >= 1);
JSON_ASSERT(dist <= delta);
JSON_ASSERT(rest <= delta);
JSON_ASSERT(ten_k > 0);
// <--------------------------- delta ---->
// <---- dist --------->
// --------------[------------------+-------------------]--------------
// M- w M+
//
// ten_k
// <------>
// <---- rest ---->
// --------------[------------------+----+--------------]--------------
// w V
// = buf * 10^k
//
// ten_k represents a unit-in-the-last-place in the decimal representation
// stored in buf.
// Decrement buf by ten_k while this takes buf closer to w.
// The tests are written in this order to avoid overflow in unsigned
// integer arithmetic.
while (rest < dist
&& delta - rest >= ten_k
&& (rest + ten_k < dist || dist - rest > rest + ten_k - dist))
{
JSON_ASSERT(buf[len - 1] != '0');
buf[len - 1]--;
rest += ten_k;
}
}
/*!
Generates V = buffer * 10^decimal_exponent, such that M- <= V <= M+.
M- and M+ must be normalized and share the same exponent -60 <= e <= -32.
*/
inline void grisu2_digit_gen(char* buffer, int& length, int& decimal_exponent,
diyfp M_minus, diyfp w, diyfp M_plus)
{
static_assert(kAlpha >= -60, "internal error");
static_assert(kGamma <= -32, "internal error");
// Generates the digits (and the exponent) of a decimal floating-point
// number V = buffer * 10^decimal_exponent in the range [M-, M+]. The diyfp's
// w, M- and M+ share the same exponent e, which satisfies alpha <= e <= gamma.
//
// <--------------------------- delta ---->
// <---- dist --------->
// --------------[------------------+-------------------]--------------
// M- w M+
//
// Grisu2 generates the digits of M+ from left to right and stops as soon as
// V is in [M-,M+].
JSON_ASSERT(M_plus.e >= kAlpha);
JSON_ASSERT(M_plus.e <= kGamma);
std::uint64_t delta = diyfp::sub(M_plus, M_minus).f; // (significand of (M+ - M-), implicit exponent is e)
std::uint64_t dist = diyfp::sub(M_plus, w ).f; // (significand of (M+ - w ), implicit exponent is e)
// Split M+ = f * 2^e into two parts p1 and p2 (note: e < 0):
//
// M+ = f * 2^e
// = ((f div 2^-e) * 2^-e + (f mod 2^-e)) * 2^e
// = ((p1 ) * 2^-e + (p2 )) * 2^e
// = p1 + p2 * 2^e
const diyfp one(std::uint64_t{1} << -M_plus.e, M_plus.e);
auto p1 = static_cast<std::uint32_t>(M_plus.f >> -one.e); // p1 = f div 2^-e (Since -e >= 32, p1 fits into a 32-bit int.)
std::uint64_t p2 = M_plus.f & (one.f - 1); // p2 = f mod 2^-e
// 1)
//
// Generate the digits of the integral part p1 = d[n-1]...d[1]d[0]
JSON_ASSERT(p1 > 0);
std::uint32_t pow10{};
const int k = find_largest_pow10(p1, pow10);
// 10^(k-1) <= p1 < 10^k, pow10 = 10^(k-1)
//
// p1 = (p1 div 10^(k-1)) * 10^(k-1) + (p1 mod 10^(k-1))
// = (d[k-1] ) * 10^(k-1) + (p1 mod 10^(k-1))
//
// M+ = p1 + p2 * 2^e
// = d[k-1] * 10^(k-1) + (p1 mod 10^(k-1)) + p2 * 2^e
// = d[k-1] * 10^(k-1) + ((p1 mod 10^(k-1)) * 2^-e + p2) * 2^e
// = d[k-1] * 10^(k-1) + ( rest) * 2^e
//
// Now generate the digits d[n] of p1 from left to right (n = k-1,...,0)
//
// p1 = d[k-1]...d[n] * 10^n + d[n-1]...d[0]
//
// but stop as soon as
//
// rest * 2^e = (d[n-1]...d[0] * 2^-e + p2) * 2^e <= delta * 2^e
int n = k;
while (n > 0)
{
// Invariants:
// M+ = buffer * 10^n + (p1 + p2 * 2^e) (buffer = 0 for n = k)
// pow10 = 10^(n-1) <= p1 < 10^n
//
const std::uint32_t d = p1 / pow10; // d = p1 div 10^(n-1)
const std::uint32_t r = p1 % pow10; // r = p1 mod 10^(n-1)
//
// M+ = buffer * 10^n + (d * 10^(n-1) + r) + p2 * 2^e
// = (buffer * 10 + d) * 10^(n-1) + (r + p2 * 2^e)
//
JSON_ASSERT(d <= 9);
buffer[length++] = static_cast<char>('0' + d); // buffer := buffer * 10 + d
//
// M+ = buffer * 10^(n-1) + (r + p2 * 2^e)
//
p1 = r;
n--;
//
// M+ = buffer * 10^n + (p1 + p2 * 2^e)
// pow10 = 10^n
//
// Now check if enough digits have been generated.
// Compute
//
// p1 + p2 * 2^e = (p1 * 2^-e + p2) * 2^e = rest * 2^e
//
// Note:
// Since rest and delta share the same exponent e, it suffices to
// compare the significands.
const std::uint64_t rest = (std::uint64_t{p1} << -one.e) + p2;
if (rest <= delta)
{
// V = buffer * 10^n, with M- <= V <= M+.
decimal_exponent += n;
// We may now just stop. But instead look if the buffer could be
// decremented to bring V closer to w.
//
// pow10 = 10^n is now 1 ulp in the decimal representation V.
// The rounding procedure works with diyfp's with an implicit
// exponent of e.
//
// 10^n = (10^n * 2^-e) * 2^e = ulp * 2^e
//
const std::uint64_t ten_n = std::uint64_t{pow10} << -one.e;
grisu2_round(buffer, length, dist, delta, rest, ten_n);
return;
}
pow10 /= 10;
//
// pow10 = 10^(n-1) <= p1 < 10^n
// Invariants restored.
}
// 2)
//
// The digits of the integral part have been generated:
//
// M+ = d[k-1]...d[1]d[0] + p2 * 2^e
// = buffer + p2 * 2^e
//
// Now generate the digits of the fractional part p2 * 2^e.
//
// Note:
// No decimal point is generated: the exponent is adjusted instead.
//
// p2 actually represents the fraction
//
// p2 * 2^e
// = p2 / 2^-e
// = d[-1] / 10^1 + d[-2] / 10^2 + ...
//
// Now generate the digits d[-m] of p1 from left to right (m = 1,2,...)
//
// p2 * 2^e = d[-1]d[-2]...d[-m] * 10^-m
// + 10^-m * (d[-m-1] / 10^1 + d[-m-2] / 10^2 + ...)
//
// using
//
// 10^m * p2 = ((10^m * p2) div 2^-e) * 2^-e + ((10^m * p2) mod 2^-e)
// = ( d) * 2^-e + ( r)
//
// or
// 10^m * p2 * 2^e = d + r * 2^e
//
// i.e.
//
// M+ = buffer + p2 * 2^e
// = buffer + 10^-m * (d + r * 2^e)
// = (buffer * 10^m + d) * 10^-m + 10^-m * r * 2^e
//
// and stop as soon as 10^-m * r * 2^e <= delta * 2^e
JSON_ASSERT(p2 > delta);
int m = 0;
for (;;)
{
// Invariant:
// M+ = buffer * 10^-m + 10^-m * (d[-m-1] / 10 + d[-m-2] / 10^2 + ...) * 2^e
// = buffer * 10^-m + 10^-m * (p2 ) * 2^e
// = buffer * 10^-m + 10^-m * (1/10 * (10 * p2) ) * 2^e
// = buffer * 10^-m + 10^-m * (1/10 * ((10*p2 div 2^-e) * 2^-e + (10*p2 mod 2^-e)) * 2^e
//
JSON_ASSERT(p2 <= (std::numeric_limits<std::uint64_t>::max)() / 10);
p2 *= 10;
const std::uint64_t d = p2 >> -one.e; // d = (10 * p2) div 2^-e
const std::uint64_t r = p2 & (one.f - 1); // r = (10 * p2) mod 2^-e
//
// M+ = buffer * 10^-m + 10^-m * (1/10 * (d * 2^-e + r) * 2^e
// = buffer * 10^-m + 10^-m * (1/10 * (d + r * 2^e))
// = (buffer * 10 + d) * 10^(-m-1) + 10^(-m-1) * r * 2^e
//
JSON_ASSERT(d <= 9);
buffer[length++] = static_cast<char>('0' + d); // buffer := buffer * 10 + d
//
// M+ = buffer * 10^(-m-1) + 10^(-m-1) * r * 2^e
//
p2 = r;
m++;
//
// M+ = buffer * 10^-m + 10^-m * p2 * 2^e
// Invariant restored.
// Check if enough digits have been generated.
//
// 10^-m * p2 * 2^e <= delta * 2^e
// p2 * 2^e <= 10^m * delta * 2^e
// p2 <= 10^m * delta
delta *= 10;
dist *= 10;
if (p2 <= delta)
{
break;
}
}
// V = buffer * 10^-m, with M- <= V <= M+.
decimal_exponent -= m;
// 1 ulp in the decimal representation is now 10^-m.
// Since delta and dist are now scaled by 10^m, we need to do the
// same with ulp in order to keep the units in sync.
//
// 10^m * 10^-m = 1 = 2^-e * 2^e = ten_m * 2^e
//
const std::uint64_t ten_m = one.f;
grisu2_round(buffer, length, dist, delta, p2, ten_m);
// By construction this algorithm generates the shortest possible decimal
// number (Loitsch, Theorem 6.2) which rounds back to w.
// For an input number of precision p, at least
//
// N = 1 + ceil(p * log_10(2))
//
// decimal digits are sufficient to identify all binary floating-point
// numbers (Matula, "In-and-Out conversions").
// This implies that the algorithm does not produce more than N decimal
// digits.
//
// N = 17 for p = 53 (IEEE double precision)
// N = 9 for p = 24 (IEEE single precision)
}
/*!
v = buf * 10^decimal_exponent
len is the length of the buffer (number of decimal digits)
The buffer must be large enough, i.e. >= max_digits10.
*/
JSON_HEDLEY_NON_NULL(1)
inline void grisu2(char* buf, int& len, int& decimal_exponent,
diyfp m_minus, diyfp v, diyfp m_plus)
{
JSON_ASSERT(m_plus.e == m_minus.e);
JSON_ASSERT(m_plus.e == v.e);
// --------(-----------------------+-----------------------)-------- (A)
// m- v m+
//
// --------------------(-----------+-----------------------)-------- (B)
// m- v m+
//
// First scale v (and m- and m+) such that the exponent is in the range
// [alpha, gamma].
const cached_power cached = get_cached_power_for_binary_exponent(m_plus.e);
const diyfp c_minus_k(cached.f, cached.e); // = c ~= 10^-k
// The exponent of the products is = v.e + c_minus_k.e + q and is in the range [alpha,gamma]
const diyfp w = diyfp::mul(v, c_minus_k);
const diyfp w_minus = diyfp::mul(m_minus, c_minus_k);
const diyfp w_plus = diyfp::mul(m_plus, c_minus_k);
// ----(---+---)---------------(---+---)---------------(---+---)----
// w- w w+
// = c*m- = c*v = c*m+
//
// diyfp::mul rounds its result and c_minus_k is approximated too. w, w- and
// w+ are now off by a small amount.
// In fact:
//
// w - v * 10^k < 1 ulp
//
// To account for this inaccuracy, add resp. subtract 1 ulp.
//
// --------+---[---------------(---+---)---------------]---+--------
// w- M- w M+ w+
//
// Now any number in [M-, M+] (bounds included) will round to w when input,
// regardless of how the input rounding algorithm breaks ties.
//
// And digit_gen generates the shortest possible such number in [M-, M+].
// Note that this does not mean that Grisu2 always generates the shortest
// possible number in the interval (m-, m+).
const diyfp M_minus(w_minus.f + 1, w_minus.e);
const diyfp M_plus (w_plus.f - 1, w_plus.e );
decimal_exponent = -cached.k; // = -(-k) = k
grisu2_digit_gen(buf, len, decimal_exponent, M_minus, w, M_plus);
}
/*!
v = buf * 10^decimal_exponent
len is the length of the buffer (number of decimal digits)
The buffer must be large enough, i.e. >= max_digits10.
*/
template<typename FloatType>
JSON_HEDLEY_NON_NULL(1)
void grisu2(char* buf, int& len, int& decimal_exponent, FloatType value)
{
static_assert(diyfp::kPrecision >= std::numeric_limits<FloatType>::digits + 3,
"internal error: not enough precision");
JSON_ASSERT(std::isfinite(value));
JSON_ASSERT(value > 0);
// If the neighbors (and boundaries) of 'value' are always computed for double-precision
// numbers, all float's can be recovered using strtod (and strtof). However, the resulting
// decimal representations are not exactly "short".
//
// The documentation for 'std::to_chars' (https://en.cppreference.com/w/cpp/utility/to_chars)
// says "value is converted to a string as if by std::sprintf in the default ("C") locale"
// and since sprintf promotes floats to doubles, 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 // NOLINT(readability-avoid-unconditional-preprocessor-if)
const boundaries w = compute_boundaries(static_cast<double>(value));
#else
const boundaries w = compute_boundaries(value);
#endif
grisu2(buf, len, decimal_exponent, w.minus, w.w, w.plus);
}
/*!
@brief appends a decimal representation of e to buf
@return a pointer to the element following the exponent.
@pre -1000 < e < 1000
*/
JSON_HEDLEY_NON_NULL(1)
JSON_HEDLEY_RETURNS_NON_NULL
inline char* append_exponent(char* buf, int e)
{
JSON_ASSERT(e > -1000);
JSON_ASSERT(e < 1000);
if (e < 0)
{
e = -e;
*buf++ = '-';
}
else
{
*buf++ = '+';
}
auto k = static_cast<std::uint32_t>(e);
if (k < 10)
{
// Always print at least two digits in the exponent.
// This is for compatibility with printf("%g").
*buf++ = '0';
*buf++ = static_cast<char>('0' + k);
}
else if (k < 100)
{
*buf++ = static_cast<char>('0' + k / 10);
k %= 10;
*buf++ = static_cast<char>('0' + k);
}
else
{
*buf++ = static_cast<char>('0' + k / 100);
k %= 100;
*buf++ = static_cast<char>('0' + k / 10);
k %= 10;
*buf++ = static_cast<char>('0' + k);
}
return buf;
}
/*!
@brief prettify v = buf * 10^decimal_exponent
If v is in the range [10^min_exp, 10^max_exp) it will be printed in fixed-point
notation. Otherwise it will be printed in exponential notation.
@pre min_exp < 0
@pre max_exp > 0
*/
JSON_HEDLEY_NON_NULL(1)
JSON_HEDLEY_RETURNS_NON_NULL
inline char* format_buffer(char* buf, int len, int decimal_exponent,
int min_exp, int max_exp)
{
JSON_ASSERT(min_exp < 0);
JSON_ASSERT(max_exp > 0);
const int k = len;
const int n = len + decimal_exponent;
// v = buf * 10^(n-k)
// k is the length of the buffer (number of decimal digits)
// n is the position of the decimal point relative to the start of the buffer.
if (k <= n && n <= max_exp)
{
// 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 buf + (static_cast<size_t>(n) + 2);
}
if (0 < n && n <= max_exp)
{
// dig.its
// len <= max_digits10 + 1
JSON_ASSERT(k > n);
std::memmove(buf + (static_cast<size_t>(n) + 1), buf + n, static_cast<size_t>(k) - static_cast<size_t>(n));
buf[n] = '.';
return buf + (static_cast<size_t>(k) + 1U);
}
if (min_exp < n && n <= 0)
{
// 0.[000]digits
// len <= 2 + (-min_exp - 1) + max_digits10
std::memmove(buf + (2 + static_cast<size_t>(-n)), buf, static_cast<size_t>(k));
buf[0] = '0';
buf[1] = '.';
std::memset(buf + 2, '0', static_cast<size_t>(-n));
return buf + (2U + static_cast<size_t>(-n) + static_cast<size_t>(k));
}
if (k == 1)
{
// dE+123
// len <= 1 + 5
buf += 1;
}
else
{
// d.igitsE+123
// len <= max_digits10 + 1 + 5
std::memmove(buf + 2, buf + 1, static_cast<size_t>(k) - 1);
buf[1] = '.';
buf += 1 + static_cast<size_t>(k);
}
*buf++ = 'e';
return append_exponent(buf, n - 1);
}
} // namespace dtoa_impl
/*!
@brief generates a decimal representation of the floating-point number value in [first, last).
The format of the resulting decimal representation is similar to printf's %g
format. Returns an iterator pointing past-the-end of the decimal representation.
@note The input number must be finite, i.e. NaN's and Inf's are not supported.
@note The buffer must be large enough.
@note The result is NOT null-terminated.
*/
template<typename FloatType>
JSON_HEDLEY_NON_NULL(1, 2)
JSON_HEDLEY_RETURNS_NON_NULL
char* to_chars(char* first, const char* last, FloatType value)
{
static_cast<void>(last); // maybe unused - fix warning
JSON_ASSERT(std::isfinite(value));
// Use signbit(value) instead of (value < 0) since signbit works for -0.
if (std::signbit(value))
{
value = -value;
*first++ = '-';
}
#ifdef __GNUC__
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wfloat-equal"
#endif
if (value == 0) // +-0
{
*first++ = '0';
// Make it look like a floating-point number (#362, #378)
*first++ = '.';
*first++ = '0';
return first;
}
#ifdef __GNUC__
#pragma GCC diagnostic pop
#endif
JSON_ASSERT(last - first >= std::numeric_limits<FloatType>::max_digits10);
// Compute v = buffer * 10^decimal_exponent.
// The decimal digits are stored in the buffer, which needs to be interpreted
// as an unsigned decimal integer.
// len is the length of the buffer, i.e. the number of decimal digits.
int len = 0;
int decimal_exponent = 0;
dtoa_impl::grisu2(first, len, decimal_exponent, value);
JSON_ASSERT(len <= std::numeric_limits<FloatType>::max_digits10);
// Format the buffer like printf("%.*g", prec, value)
constexpr int kMinExp = -4;
// Use digits10 here to increase compatibility with version 2.
constexpr int kMaxExp = std::numeric_limits<FloatType>::digits10;
JSON_ASSERT(last - first >= kMaxExp + 2);
JSON_ASSERT(last - first >= 2 + (-kMinExp - 1) + std::numeric_limits<FloatType>::max_digits10);
JSON_ASSERT(last - first >= std::numeric_limits<FloatType>::max_digits10 + 6);
return dtoa_impl::format_buffer(first, len, decimal_exponent, kMinExp, kMaxExp);
}
} // namespace detail
NLOHMANN_JSON_NAMESPACE_END
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