From 332f35203398b0274677a0705398b8b0c37974ae Mon Sep 17 00:00:00 2001 From: abolz Date: Mon, 15 Jan 2018 20:45:51 +0100 Subject: [PATCH] Add an implementation of the Grisu2 algorithm for binary to decimal floating-point conversion This is an attempt to fix #360. The algorithm produces decimal representations which are guaranteed to roundtrip and in ~99.8% actually produces the shortest possible representation. So this is a nice compromise between using a precision of digits10 and max_digits10. Note 1: The implementation only works for IEEE single/double precision numbers. So the old implementation is kept for compatibility with non-IEEE implementations and 'long double'. Note 2: If number_float_t is 'float', not all serialized numbers can be recovered using strtod (strtof works, though). (There is exactly one such number and the result is off by 1 ulp.) This can be avoided by changing the implementation (the fix is trivial), but then the resulting decimal numbers are not exactly short. --- README.md | 4 + develop/detail/conversions/to_chars.hpp | 1027 +++++++++++++++++++++++ 2 files changed, 1031 insertions(+) create mode 100644 develop/detail/conversions/to_chars.hpp diff --git a/README.md b/README.md index d5935755..82f0ba02 100644 --- a/README.md +++ b/README.md @@ -813,6 +813,10 @@ THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR I The class contains the UTF-8 Decoder from Bjoern Hoehrmann which is licensed under the [MIT License](http://opensource.org/licenses/MIT) (see above). Copyright © 2008-2009 [Björn Hoehrmann](http://bjoern.hoehrmann.de/) +* * * + +The class contains a slightly modified version of the Grisu2 algorithm from Florian Loitsch which is licensed under the [MIT License](http://opensource.org/licenses/MIT) (see above). Copyright © 2009 [Florian Loitsch](http://florian.loitsch.com/) + ## Contact If you have questions regarding the library, I would like to invite you to [open an issue at GitHub](https://github.com/nlohmann/json/issues/new). Please describe your request, problem, or question as detailed as possible, and also mention the version of the library you are using as well as the version of your compiler and operating system. Opening an issue at GitHub allows other users and contributors to this library to collaborate. For instance, I have little experience with MSVC, and most issues in this regard have been solved by a growing community. If you have a look at the [closed issues](https://github.com/nlohmann/json/issues?q=is%3Aissue+is%3Aclosed), you will see that we react quite timely in most cases. diff --git a/develop/detail/conversions/to_chars.hpp b/develop/detail/conversions/to_chars.hpp new file mode 100644 index 00000000..d5e9feb0 --- /dev/null +++ b/develop/detail/conversions/to_chars.hpp @@ -0,0 +1,1027 @@ +#pragma once + +#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