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+#pragma once
+
+#include <cassert> // assert
+#include <ciso646> // or, and, not
+#include <cmath> // signbit, isfinite
+#include <cstdint> // intN_t, uintN_t
+#include <cstring> // memcpy, memmove
+
+namespace nlohmann
+{
+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
+
+ uint64_t f = 0;
+ int e = 0;
+
+ constexpr diyfp(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
+ {
+ assert(x.e == y.e);
+ 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 uint64_t u_lo = x.f & 0xFFFFFFFF;
+ const uint64_t u_hi = x.f >> 32;
+ const uint64_t v_lo = y.f & 0xFFFFFFFF;
+ const uint64_t v_hi = y.f >> 32;
+
+ const uint64_t p0 = u_lo * v_lo;
+ const uint64_t p1 = u_lo * v_hi;
+ const uint64_t p2 = u_hi * v_lo;
+ const uint64_t p3 = u_hi * v_hi;
+
+ const uint64_t p0_hi = p0 >> 32;
+ const uint64_t p1_lo = p1 & 0xFFFFFFFF;
+ const uint64_t p1_hi = p1 >> 32;
+ const uint64_t p2_lo = p2 & 0xFFFFFFFF;
+ const uint64_t p2_hi = p2 >> 32;
+
+ 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 += uint64_t{1} << (64 - 32 - 1); // round, ties up
+
+ const uint64_t h = p3 + p2_hi + p1_hi + (Q >> 32);
+
+ 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
+ {
+ assert(x.f != 0);
+
+ while ((x.f >> 63) == 0)
+ {
+ x.f <<= 1;
+ 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;
+
+ assert(delta >= 0);
+ 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)
+{
+ assert(std::isfinite(value));
+ 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 uint64_t kHiddenBit = uint64_t{1} << (kPrecision - 1); // = 2^(p-1)
+
+ using bits_type = typename std::conditional< kPrecision == 24, uint32_t, uint64_t >::type;
+
+ const uint64_t bits = reinterpret_bits<bits_type>(value);
+ const uint64_t E = bits >> (kPrecision - 1);
+ const 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 and 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
+{
+ 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 exakt 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 kCachedPowersSize = 79;
+ constexpr int kCachedPowersMinDecExp = -300;
+ constexpr int kCachedPowersDecStep = 8;
+
+ static constexpr cached_power 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
+ assert(e >= -1500);
+ 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;
+ assert(index >= 0);
+ assert(index < kCachedPowersSize);
+ static_cast<void>(kCachedPowersSize); // Fix warning.
+
+ const cached_power cached = kCachedPowers[index];
+ assert(kAlpha <= cached.e + e + 64);
+ 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 uint32_t n, uint32_t& pow10)
+{
+ // LCOV_EXCL_START
+ if (n >= 1000000000)
+ {
+ pow10 = 1000000000;
+ return 10;
+ }
+ // LCOV_EXCL_STOP
+ else if (n >= 100000000)
+ {
+ pow10 = 100000000;
+ return 9;
+ }
+ else if (n >= 10000000)
+ {
+ pow10 = 10000000;
+ return 8;
+ }
+ else if (n >= 1000000)
+ {
+ pow10 = 1000000;
+ return 7;
+ }
+ else if (n >= 100000)
+ {
+ pow10 = 100000;
+ return 6;
+ }
+ else if (n >= 10000)
+ {
+ pow10 = 10000;
+ return 5;
+ }
+ else if (n >= 1000)
+ {
+ pow10 = 1000;
+ return 4;
+ }
+ else if (n >= 100)
+ {
+ pow10 = 100;
+ return 3;
+ }
+ else if (n >= 10)
+ {
+ pow10 = 10;
+ return 2;
+ }
+ else
+ {
+ pow10 = 1;
+ return 1;
+ }
+}
+
+inline void grisu2_round(char* buf, int len, uint64_t dist, uint64_t delta,
+ uint64_t rest, uint64_t ten_k)
+{
+ assert(len >= 1);
+ assert(dist <= delta);
+ assert(rest <= delta);
+ 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
+ and delta - rest >= ten_k
+ and (rest + ten_k < dist or dist - rest > rest + ten_k - dist))
+ {
+ 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+].
+
+ assert(M_plus.e >= kAlpha);
+ assert(M_plus.e <= kGamma);
+
+ uint64_t delta = diyfp::sub(M_plus, M_minus).f; // (significand of (M+ - M-), implicit exponent is e)
+ 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(uint64_t{1} << -M_plus.e, M_plus.e);
+
+ auto p1 = static_cast<uint32_t>(M_plus.f >> -one.e); // p1 = f div 2^-e (Since -e >= 32, p1 fits into a 32-bit int.)
+ 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]
+
+ assert(p1 > 0);
+
+ 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 uint32_t d = p1 / pow10; // d = p1 div 10^(n-1)
+ const 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)
+ //
+ 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 uint64_t rest = (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 uint64_t ten_n = 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
+
+ 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
+ //
+ assert(p2 <= UINT64_MAX / 10);
+ p2 *= 10;
+ const uint64_t d = p2 >> -one.e; // d = (10 * p2) div 2^-e
+ const 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
+ //
+ 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 uint64_t ten_m = one.f;
+ grisu2_round(buffer, length, dist, delta, p2, ten_m);
+
+ // By construction this algorithm generates the shortest possible decimal
+ // number (Loitsch, Theorem 6.2) which rounds back to w.
+ // For an input number of precision p, at least
+ //
+ // N = 1 + ceil(p * log_10(2))
+ //
+ // decimal digits are sufficient to identify all binary floating-point
+ // numbers (Matula, "In-and-Out conversions").
+ // This implies that the algorithm does not produce more than N decimal
+ // digits.
+ //
+ // N = 17 for p = 53 (IEEE double precision)
+ // N = 9 for p = 24 (IEEE single precision)
+}
+
+/*!
+v = buf * 10^decimal_exponent
+len is the length of the buffer (number of decimal digits)
+The buffer must be large enough, i.e. >= max_digits10.
+*/
+inline void grisu2(char* buf, int& len, int& decimal_exponent,
+ diyfp m_minus, diyfp v, diyfp m_plus)
+{
+ assert(m_plus.e == m_minus.e);
+ 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>
+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");
+
+ assert(std::isfinite(value));
+ 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 float's to double's, I think this is exactly what 'std::to_chars'
+ // does.
+ // On the other hand, the documentation for 'std::to_chars' requires that "parsing the
+ // representation using the corresponding std::from_chars function recovers value exactly". That
+ // indicates that single precision floating-point numbers should be recovered using
+ // 'std::strtof'.
+ //
+ // NB: If the neighbors are computed for single-precision numbers, there is a single float
+ // (7.0385307e-26f) which can't be recovered using strtod. The resulting double precision
+ // value is off by 1 ulp.
+#if 0
+ const boundaries w = compute_boundaries(static_cast<double>(value));
+#else
+ const boundaries w = compute_boundaries(value);
+#endif
+
+ grisu2(buf, len, decimal_exponent, w.minus, w.w, w.plus);
+}
+
+/*!
+@brief appends a decimal representation of e to buf
+@return a pointer to the element following the exponent.
+@pre -1000 < e < 1000
+*/
+inline char* append_exponent(char* buf, int e)
+{
+ assert(e > -1000);
+ assert(e < 1000);
+
+ if (e < 0)
+ {
+ e = -e;
+ *buf++ = '-';
+ }
+ else
+ {
+ *buf++ = '+';
+ }
+
+ auto k = static_cast<uint32_t>(e);
+ if (k < 10)
+ {
+ // Always print at least two digits in the exponent.
+ // This is for compatibility with printf("%g").
+ *buf++ = '0';
+ *buf++ = static_cast<char>('0' + k);
+ }
+ else if (k < 100)
+ {
+ *buf++ = static_cast<char>('0' + k / 10);
+ k %= 10;
+ *buf++ = static_cast<char>('0' + k);
+ }
+ else
+ {
+ *buf++ = static_cast<char>('0' + k / 100);
+ k %= 100;
+ *buf++ = static_cast<char>('0' + k / 10);
+ k %= 10;
+ *buf++ = static_cast<char>('0' + k);
+ }
+
+ return buf;
+}
+
+/*!
+@brief prettify v = buf * 10^decimal_exponent
+
+If v is in the range [10^min_exp, 10^max_exp) it will be printed in fixed-point
+notation. Otherwise it will be printed in exponential notation.
+
+@pre min_exp < 0
+@pre max_exp > 0
+*/
+inline char* format_buffer(char* buf, int len, int decimal_exponent,
+ int min_exp, int max_exp)
+{
+ assert(min_exp < 0);
+ 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 and n <= max_exp)
+ {
+ // digits[000]
+ // len <= max_exp + 2
+
+ std::memset(buf + k, '0', static_cast<size_t>(n - k));
+ // Make it look like a floating-point number (#362, #378)
+ buf[n + 0] = '.';
+ buf[n + 1] = '0';
+ return buf + (n + 2);
+ }
+
+ if (0 < n and n <= max_exp)
+ {
+ // dig.its
+ // len <= max_digits10 + 1
+
+ assert(k > n);
+
+ std::memmove(buf + (n + 1), buf + n, static_cast<size_t>(k - n));
+ buf[n] = '.';
+ return buf + (k + 1);
+ }
+
+ if (min_exp < n and n <= 0)
+ {
+ // 0.[000]digits
+ // len <= 2 + (-min_exp - 1) + max_digits10
+
+ std::memmove(buf + (2 + -n), buf, static_cast<size_t>(k));
+ buf[0] = '0';
+ buf[1] = '.';
+ std::memset(buf + 2, '0', static_cast<size_t>(-n));
+ return buf + (2 + (-n) + 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 + 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>
+char* to_chars(char* first, const char* last, FloatType value)
+{
+ static_cast<void>(last); // maybe unused - fix warning
+ assert(std::isfinite(value));
+
+ // Use signbit(value) instead of (value < 0) since signbit works for -0.
+ if (std::signbit(value))
+ {
+ value = -value;
+ *first++ = '-';
+ }
+
+ if (value == 0) // +-0
+ {
+ *first++ = '0';
+ // Make it look like a floating-point number (#362, #378)
+ *first++ = '.';
+ *first++ = '0';
+ return first;
+ }
+
+ 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);
+
+ 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;
+
+ assert(last - first >= kMaxExp + 2);
+ assert(last - first >= 2 + (-kMinExp - 1) + std::numeric_limits<FloatType>::max_digits10);
+ assert(last - first >= std::numeric_limits<FloatType>::max_digits10 + 6);
+
+ return dtoa_impl::format_buffer(first, len, decimal_exponent, kMinExp, kMaxExp);
+}
+
+} // namespace detail
+} // namespace nlohmann
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