| #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> |
| |
| 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 |
| |
| 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 std::uint64_t bits = reinterpret_bits<bits_type>(value); |
| const std::uint64_t E = bits >> (kPrecision - 1); |
| const std::uint64_t F = bits & (kHiddenBit - 1); |
| |
| const bool is_denormal = E == 0; |
| const diyfp v = is_denormal |
| ? diyfp(F, kMinExp) |
| : diyfp(F + kHiddenBit, static_cast<int>(E) - kBias); |
| |
| // Compute the boundaries m- and m+ of the floating-point value |
| // v = f * 2^e. |
| // |
| // Determine v- and v+, the floating-point predecessor and successor if v, |
| // respectively. |
| // |
| // v- = v - 2^e if f != 2^(p-1) or e == e_min (A) |
| // = v - 2^(e-1) if f == 2^(p-1) and e > e_min (B) |
| // |
| // v+ = v + 2^e |
| // |
| // Let m- = (v- + v) / 2 and m+ = (v + v+) / 2. All real numbers _strictly_ |
| // between m- and m+ round to v, regardless of how the input rounding |
| // algorithm breaks ties. |
| // |
| // ---+-------------+-------------+-------------+-------------+--- (A) |
| // v- m- v m+ v+ |
| // |
| // -----------------+------+------+-------------+-------------+--- (B) |
| // v- m- v m+ v+ |
| |
| const bool lower_boundary_is_closer = F == 0 && E > 1; |
| const diyfp m_plus = diyfp(2 * v.f + 1, v.e - 1); |
| const diyfp m_minus = lower_boundary_is_closer |
| ? diyfp(4 * v.f - 1, v.e - 2) // (B) |
| : diyfp(2 * v.f - 1, v.e - 1); // (A) |
| |
| // Determine the normalized w+ = m+. |
| const diyfp w_plus = diyfp::normalize(m_plus); |
| |
| // Determine w- = m- such that e_(w-) = e_(w+). |
| const diyfp w_minus = diyfp::normalize_to(m_minus, w_plus.e); |
| |
| return {diyfp::normalize(v), w_minus, w_plus}; |
| } |
| |
| // Given normalized diyfp w, Grisu needs to find a (normalized) cached |
| // power-of-ten c, such that the exponent of the product c * w = f * 2^e lies |
| // within a certain range [alpha, gamma] (Definition 3.2 from [1]) |
| // |
| // alpha <= e = e_c + e_w + q <= gamma |
| // |
| // or |
| // |
| // f_c * f_w * 2^alpha <= f_c 2^(e_c) * f_w 2^(e_w) * 2^q |
| // <= f_c * f_w * 2^gamma |
| // |
| // Since c and w are normalized, i.e. 2^(q-1) <= f < 2^q, this implies |
| // |
| // 2^(q-1) * 2^(q-1) * 2^alpha <= c * w * 2^q < 2^q * 2^q * 2^gamma |
| // |
| // or |
| // |
| // 2^(q - 2 + alpha) <= c * w < 2^(q + gamma) |
| // |
| // The choice of (alpha,gamma) determines the size of the table and the form of |
| // the digit generation procedure. Using (alpha,gamma)=(-60,-32) works out well |
| // in practice: |
| // |
| // The idea is to cut the number c * w = f * 2^e into two parts, which can be |
| // processed independently: An integral part p1, and a fractional part p2: |
| // |
| // f * 2^e = ( (f div 2^-e) * 2^-e + (f mod 2^-e) ) * 2^e |
| // = (f div 2^-e) + (f mod 2^-e) * 2^e |
| // = p1 + p2 * 2^e |
| // |
| // The conversion of p1 into decimal form requires a series of divisions and |
| // modulos by (a power of) 10. These operations are faster for 32-bit than for |
| // 64-bit integers, so p1 should ideally fit into a 32-bit integer. This can be |
| // achieved by choosing |
| // |
| // -e >= 32 or e <= -32 := gamma |
| // |
| // In order to convert the fractional part |
| // |
| // p2 * 2^e = p2 / 2^-e = d[-1] / 10^1 + d[-2] / 10^2 + ... |
| // |
| // into decimal form, the fraction is repeatedly multiplied by 10 and the digits |
| // d[-i] are extracted in order: |
| // |
| // (10 * p2) div 2^-e = d[-1] |
| // (10 * p2) mod 2^-e = d[-2] / 10^1 + ... |
| // |
| // The multiplication by 10 must not overflow. It is sufficient to choose |
| // |
| // 10 * p2 < 16 * p2 = 2^4 * p2 <= 2^64. |
| // |
| // Since p2 = f mod 2^-e < 2^-e, |
| // |
| // -e <= 60 or e >= -60 := alpha |
| |
| constexpr int kAlpha = -60; |
| constexpr int kGamma = -32; |
| |
| struct cached_power // c = f * 2^e ~= 10^k |
| { |
| std::uint64_t f; |
| int e; |
| int k; |
| }; |
| |
| /*! |
| For a normalized diyfp w = f * 2^e, this function returns a (normalized) cached |
| power-of-ten c = f_c * 2^e_c, such that the exponent of the product w * c |
| satisfies (Definition 3.2 from [1]) |
| |
| alpha <= e_c + e + q <= gamma. |
| */ |
| inline cached_power get_cached_power_for_binary_exponent(int e) |
| { |
| // Now |
| // |
| // alpha <= e_c + e + q <= gamma (1) |
| // ==> f_c * 2^alpha <= c * 2^e * 2^q |
| // |
| // and since the c's are normalized, 2^(q-1) <= f_c, |
| // |
| // ==> 2^(q - 1 + alpha) <= c * 2^(e + q) |
| // ==> 2^(alpha - e - 1) <= c |
| // |
| // If c were an exact power of ten, i.e. c = 10^k, one may determine k as |
| // |
| // k = ceil( log_10( 2^(alpha - e - 1) ) ) |
| // = ceil( (alpha - e - 1) * log_10(2) ) |
| // |
| // From the paper: |
| // "In theory the result of the procedure could be wrong since c is rounded, |
| // and the computation itself is approximated [...]. In practice, however, |
| // this simple function is sufficient." |
| // |
| // For IEEE double precision floating-point numbers converted into |
| // normalized diyfp's w = f * 2^e, with q = 64, |
| // |
| // e >= -1022 (min IEEE exponent) |
| // -52 (p - 1) |
| // -52 (p - 1, possibly normalize denormal IEEE numbers) |
| // -11 (normalize the diyfp) |
| // = -1137 |
| // |
| // and |
| // |
| // e <= +1023 (max IEEE exponent) |
| // -52 (p - 1) |
| // -11 (normalize the diyfp) |
| // = 960 |
| // |
| // This binary exponent range [-1137,960] results in a decimal exponent |
| // range [-307,324]. One does not need to store a cached power for each |
| // k in this range. For each such k it suffices to find a cached power |
| // such that the exponent of the product lies in [alpha,gamma]. |
| // This implies that the difference of the decimal exponents of adjacent |
| // table entries must be less than or equal to |
| // |
| // floor( (gamma - alpha) * log_10(2) ) = 8. |
| // |
| // (A smaller distance gamma-alpha would require a larger table.) |
| |
| // NB: |
| // Actually this function returns c, such that -60 <= e_c + e + 64 <= -34. |
| |
| constexpr int kCachedPowersMinDecExp = -300; |
| constexpr int kCachedPowersDecStep = 8; |
| |
| static constexpr std::array<cached_power, 79> kCachedPowers = |
| { |
| { |
| { 0xAB70FE17C79AC6CA, -1060, -300 }, |
| { 0xFF77B1FCBEBCDC4F, -1034, -292 }, |
| { 0xBE5691EF416BD60C, -1007, -284 }, |
| { 0x8DD01FAD907FFC3C, -980, -276 }, |
| { 0xD3515C2831559A83, -954, -268 }, |
| { 0x9D71AC8FADA6C9B5, -927, -260 }, |
| { 0xEA9C227723EE8BCB, -901, -252 }, |
| { 0xAECC49914078536D, -874, -244 }, |
| { 0x823C12795DB6CE57, -847, -236 }, |
| { 0xC21094364DFB5637, -821, -228 }, |
| { 0x9096EA6F3848984F, -794, -220 }, |
| { 0xD77485CB25823AC7, -768, -212 }, |
| { 0xA086CFCD97BF97F4, -741, -204 }, |
| { 0xEF340A98172AACE5, -715, -196 }, |
| { 0xB23867FB2A35B28E, -688, -188 }, |
| { 0x84C8D4DFD2C63F3B, -661, -180 }, |
| { 0xC5DD44271AD3CDBA, -635, -172 }, |
| { 0x936B9FCEBB25C996, -608, -164 }, |
| { 0xDBAC6C247D62A584, -582, -156 }, |
| { 0xA3AB66580D5FDAF6, -555, -148 }, |
| { 0xF3E2F893DEC3F126, -529, -140 }, |
| { 0xB5B5ADA8AAFF80B8, -502, -132 }, |
| { 0x87625F056C7C4A8B, -475, -124 }, |
| { 0xC9BCFF6034C13053, -449, -116 }, |
| { 0x964E858C91BA2655, -422, -108 }, |
| { 0xDFF9772470297EBD, -396, -100 }, |
| { 0xA6DFBD9FB8E5B88F, -369, -92 }, |
| { 0xF8A95FCF88747D94, -343, -84 }, |
| { 0xB94470938FA89BCF, -316, -76 }, |
| { 0x8A08F0F8BF0F156B, -289, -68 }, |
| { 0xCDB02555653131B6, -263, -60 }, |
| { 0x993FE2C6D07B7FAC, -236, -52 }, |
| { 0xE45C10C42A2B3B06, -210, -44 }, |
| { 0xAA242499697392D3, -183, -36 }, |
| { 0xFD87B5F28300CA0E, -157, -28 }, |
| { 0xBCE5086492111AEB, -130, -20 }, |
| { 0x8CBCCC096F5088CC, -103, -12 }, |
| { 0xD1B71758E219652C, -77, -4 }, |
| { 0x9C40000000000000, -50, 4 }, |
| { 0xE8D4A51000000000, -24, 12 }, |
| { 0xAD78EBC5AC620000, 3, 20 }, |
| { 0x813F3978F8940984, 30, 28 }, |
| { 0xC097CE7BC90715B3, 56, 36 }, |
| { 0x8F7E32CE7BEA5C70, 83, 44 }, |
| { 0xD5D238A4ABE98068, 109, 52 }, |
| { 0x9F4F2726179A2245, 136, 60 }, |
| { 0xED63A231D4C4FB27, 162, 68 }, |
| { 0xB0DE65388CC8ADA8, 189, 76 }, |
| { 0x83C7088E1AAB65DB, 216, 84 }, |
| { 0xC45D1DF942711D9A, 242, 92 }, |
| { 0x924D692CA61BE758, 269, 100 }, |
| { 0xDA01EE641A708DEA, 295, 108 }, |
| { 0xA26DA3999AEF774A, 322, 116 }, |
| { 0xF209787BB47D6B85, 348, 124 }, |
| { 0xB454E4A179DD1877, 375, 132 }, |
| { 0x865B86925B9BC5C2, 402, 140 }, |
| { 0xC83553C5C8965D3D, 428, 148 }, |
| { 0x952AB45CFA97A0B3, 455, 156 }, |
| { 0xDE469FBD99A05FE3, 481, 164 }, |
| { 0xA59BC234DB398C25, 508, 172 }, |
| { 0xF6C69A72A3989F5C, 534, 180 }, |
| { 0xB7DCBF5354E9BECE, 561, 188 }, |
| { 0x88FCF317F22241E2, 588, 196 }, |
| { 0xCC20CE9BD35C78A5, 614, 204 }, |
| { 0x98165AF37B2153DF, 641, 212 }, |
| { 0xE2A0B5DC971F303A, 667, 220 }, |
| { 0xA8D9D1535CE3B396, 694, 228 }, |
| { 0xFB9B7CD9A4A7443C, 720, 236 }, |
| { 0xBB764C4CA7A44410, 747, 244 }, |
| { 0x8BAB8EEFB6409C1A, 774, 252 }, |
| { 0xD01FEF10A657842C, 800, 260 }, |
| { 0x9B10A4E5E9913129, 827, 268 }, |
| { 0xE7109BFBA19C0C9D, 853, 276 }, |
| { 0xAC2820D9623BF429, 880, 284 }, |
| { 0x80444B5E7AA7CF85, 907, 292 }, |
| { 0xBF21E44003ACDD2D, 933, 300 }, |
| { 0x8E679C2F5E44FF8F, 960, 308 }, |
| { 0xD433179D9C8CB841, 986, 316 }, |
| { 0x9E19DB92B4E31BA9, 1013, 324 }, |
| } |
| }; |
| |
| // This computation gives exactly the same results for k as |
| // k = ceil((kAlpha - e - 1) * 0.30102999566398114) |
| // for |e| <= 1500, but doesn't require floating-point operations. |
| // NB: log_10(2) ~= 78913 / 2^18 |
| 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 |
| else if (n >= 100000000) |
| { |
| pow10 = 100000000; |
| return 9; |
| } |
| else if (n >= 10000000) |
| { |
| pow10 = 10000000; |
| return 8; |
| } |
| else if (n >= 1000000) |
| { |
| pow10 = 1000000; |
| return 7; |
| } |
| else if (n >= 100000) |
| { |
| pow10 = 100000; |
| return 6; |
| } |
| else if (n >= 10000) |
| { |
| pow10 = 10000; |
| return 5; |
| } |
| else if (n >= 1000) |
| { |
| pow10 = 1000; |
| return 4; |
| } |
| else if (n >= 100) |
| { |
| pow10 = 100; |
| return 3; |
| } |
| else if (n >= 10) |
| { |
| pow10 = 10; |
| return 2; |
| } |
| else |
| { |
| pow10 = 1; |
| return 1; |
| } |
| } |
| |
| inline void grisu2_round(char* buf, int len, std::uint64_t dist, std::uint64_t delta, |
| std::uint64_t rest, std::uint64_t ten_k) |
| { |
| 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 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 |
| */ |
| 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++ = '-'; |
| } |
| |
| if (value == 0) // +-0 |
| { |
| *first++ = '0'; |
| // Make it look like a floating-point number (#362, #378) |
| *first++ = '.'; |
| *first++ = '0'; |
| return first; |
| } |
| |
| 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 |
| } // namespace nlohmann |
