diff --git a/src/json.hpp b/src/json.hpp index a80cc36a..e1d09f22 100644 --- a/src/json.hpp +++ b/src/json.hpp @@ -6453,6 +6453,1035 @@ class binary_writer #include // string #include // is_same +// #include "detail/conversions/to_chars.hpp" + + +#include // assert +#include // or, and, not +#include // signbit, isfinite +#include // intN_t, uintN_t +#include // memcpy, memmove + +namespace nlohmann +{ +namespace detail +{ + +// 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 +Target reinterpret_bits(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; + int e; + + constexpr diyfp() : f(0), e(0) {} + constexpr diyfp(uint64_t f_, int e_) : f(f_), e(e_) {} + + // Returns x - y. + // PRE: x.e == y.e and x.f >= y.f + static diyfp sub(diyfp x, diyfp y); + + // Returns x * y. + // The result is rounded. (Only the upper q bits are returned.) + static diyfp mul(diyfp x, diyfp y); + + // Normalize x such that the significand is >= 2^(q-1). + // PRE: x.f != 0 + static diyfp normalize(diyfp x); + + // 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(diyfp x, int e); +}; + +inline diyfp diyfp::sub(diyfp x, diyfp y) +{ + assert(x.e == y.e); + assert(x.f >= y.f); + + return diyfp(x.f - y.f, x.e); +} + +inline diyfp diyfp::mul(diyfp x, diyfp y) +{ + 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 diyfp(h, x.e + y.e + 64); +} + +inline diyfp diyfp::normalize(diyfp x) +{ + assert(x.f != 0); + + while ((x.f >> 63) == 0) + { + x.f <<= 1; + x.e--; + } + + return x; +} + +inline diyfp diyfp::normalize_to(diyfp x, int target_exponent) +{ + const int delta = x.e - target_exponent; + + assert(delta >= 0); + assert(((x.f << delta) >> delta) == x.f); + + return diyfp(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 +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::is_iec559, + "internal error: dtoa_short requires an IEEE-754 floating-point implementation"); + + constexpr int kPrecision = std::numeric_limits::digits; // = p (includes the hidden bit) + constexpr int kBias = std::numeric_limits::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(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, 1 - kBias) + : diyfp(F + kHiddenBit, static_cast(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) + (f > 0); + + const int index = (-kCachedPowersMinDecExp + k + (kCachedPowersDecStep - 1)) / kCachedPowersDecStep; + assert(index >= 0); + assert(index < kCachedPowersSize); + static_cast(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(uint32_t n, uint32_t& pow10) +{ + if (n >= 1000000000) { pow10 = 1000000000; return 10; } + 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, 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); + + uint32_t p1 = static_cast(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('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('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 +void grisu2(char* buf, int& len, int& decimal_exponent, FloatType value) +{ + static_assert(diyfp::kPrecision >= std::numeric_limits::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' (http://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(value)); +#else + const boundaries w = compute_boundaries(value); +#endif + + grisu2(buf, len, decimal_exponent, w.minus, w.w, w.plus); +} + +// Appends a decimal representation of e to buf. +// Returns 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++ = '+'; + } + + uint32_t k = static_cast(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('0' + k); + } + else if (k < 100) + { + *buf++ = static_cast('0' + k / 10); + k %= 10; + *buf++ = static_cast('0' + k); + } + else + { + *buf++ = static_cast('0' + k / 100); + k %= 100; + *buf++ = static_cast('0' + k / 10); + k %= 10; + *buf++ = static_cast('0' + k); + } + + return buf; +} + +// 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(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(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(k)); + buf[0] = '0'; + buf[1] = '.'; + std::memset(buf + 2, '0', static_cast(-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(k - 1)); + buf[1] = '.'; + buf += 1 + k; + } + + *buf++ = 'e'; + return append_exponent(buf, n - 1); +} + +} // namespace dtoa_impl + +// 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 +char* to_chars(char* first, char* last, FloatType value) +{ + static_cast(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::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::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::digits10; + + assert(last - first >= kMaxExp + 2); + assert(last - first >= 2 + (-kMinExp - 1) + std::numeric_limits::max_digits10); + assert(last - first >= std::numeric_limits::max_digits10 + 6); + + return dtoa_impl::format_buffer(first, len, decimal_exponent, kMinExp, kMaxExp); +} + +} // namespace detail +} // namespace nlohmann + // #include "detail/macro_scope.hpp" // #include "detail/meta.hpp" @@ -7079,12 +8108,34 @@ class serializer void dump_float(number_float_t x) { // NaN / inf - if (not std::isfinite(x) or std::isnan(x)) + if (not std::isfinite(x)) { o->write_characters("null", 4); return; } + // If number_float_t is an IEEE-754 single or double precision number, + // use the Grisu2 algorithm to produce short numbers which are guaranteed + // to round-trip, using strtof and strtod, resp. + // + // NB: The test below works if == . + static constexpr bool is_ieee_single_or_double + = (std::numeric_limits::is_iec559 and std::numeric_limits::digits == 24 and std::numeric_limits::max_exponent == 128) or + (std::numeric_limits::is_iec559 and std::numeric_limits::digits == 53 and std::numeric_limits::max_exponent == 1024); + + dump_float(x, std::integral_constant()); + } + + void dump_float(number_float_t x, std::true_type /*is_ieee_single_or_double*/) + { + char* begin = number_buffer.data(); + char* end = ::nlohmann::detail::to_chars(begin, begin + number_buffer.size(), x); + + o->write_characters(begin, static_cast(end - begin)); + } + + void dump_float(number_float_t x, std::false_type /*is_ieee_single_or_double*/) + { // get number of digits for a text -> float -> text round-trip static constexpr auto d = std::numeric_limits::digits10;