3 #include <array> // array
4 #include <cmath> // signbit, isfinite
5 #include <cstdint> // intN_t, uintN_t
6 #include <cstring> // memcpy, memmove
7 #include <limits> // numeric_limits
8 #include <type_traits> // conditional
10 #include <nlohmann/detail/macro_scope.hpp>
18 @brief implements the Grisu2 algorithm for binary to decimal floating-point
21 This implementation is a slightly modified version of the reference
22 implementation which may be obtained from
23 http://florian.loitsch.com/publications (bench.tar.gz).
25 The code is distributed under the MIT license, Copyright (c) 2009 Florian Loitsch.
27 For a detailed description of the algorithm see:
29 [1] Loitsch, "Printing Floating-Point Numbers Quickly and Accurately with
30 Integers", Proceedings of the ACM SIGPLAN 2010 Conference on Programming
31 Language Design and Implementation, PLDI 2010
32 [2] Burger, Dybvig, "Printing Floating-Point Numbers Quickly and Accurately",
33 Proceedings of the ACM SIGPLAN 1996 Conference on Programming Language
34 Design and Implementation, PLDI 1996
39 template<typename Target, typename Source>
40 Target reinterpret_bits(const Source source)
42 static_assert(sizeof(Target) == sizeof(Source), "size mismatch");
45 std::memcpy(&target, &source, sizeof(Source));
49 struct diyfp // f * 2^e
51 static constexpr int kPrecision = 64; // = q
56 constexpr diyfp(std::uint64_t f_, int e_) noexcept : f(f_), e(e_) {}
60 @pre x.e == y.e and x.f >= y.f
62 static diyfp sub(const diyfp& x, const diyfp& y) noexcept
64 JSON_ASSERT(x.e == y.e);
65 JSON_ASSERT(x.f >= y.f);
67 return {x.f - y.f, x.e};
72 @note The result is rounded. (Only the upper q bits are returned.)
74 static diyfp mul(const diyfp& x, const diyfp& y) noexcept
76 static_assert(kPrecision == 64, "internal error");
79 // f = round((x.f * y.f) / 2^q)
82 // Emulate the 64-bit * 64-bit multiplication:
85 // = (u_lo + 2^32 u_hi) (v_lo + 2^32 v_hi)
86 // = (u_lo v_lo ) + 2^32 ((u_lo v_hi ) + (u_hi v_lo )) + 2^64 (u_hi v_hi )
87 // = (p0 ) + 2^32 ((p1 ) + (p2 )) + 2^64 (p3 )
88 // = (p0_lo + 2^32 p0_hi) + 2^32 ((p1_lo + 2^32 p1_hi) + (p2_lo + 2^32 p2_hi)) + 2^64 (p3 )
89 // = (p0_lo ) + 2^32 (p0_hi + p1_lo + p2_lo ) + 2^64 (p1_hi + p2_hi + p3)
90 // = (p0_lo ) + 2^32 (Q ) + 2^64 (H )
91 // = (p0_lo ) + 2^32 (Q_lo + 2^32 Q_hi ) + 2^64 (H )
93 // (Since Q might be larger than 2^32 - 1)
95 // = (p0_lo + 2^32 Q_lo) + 2^64 (Q_hi + H)
97 // (Q_hi + H does not overflow a 64-bit int)
101 const std::uint64_t u_lo = x.f & 0xFFFFFFFFu;
102 const std::uint64_t u_hi = x.f >> 32u;
103 const std::uint64_t v_lo = y.f & 0xFFFFFFFFu;
104 const std::uint64_t v_hi = y.f >> 32u;
106 const std::uint64_t p0 = u_lo * v_lo;
107 const std::uint64_t p1 = u_lo * v_hi;
108 const std::uint64_t p2 = u_hi * v_lo;
109 const std::uint64_t p3 = u_hi * v_hi;
111 const std::uint64_t p0_hi = p0 >> 32u;
112 const std::uint64_t p1_lo = p1 & 0xFFFFFFFFu;
113 const std::uint64_t p1_hi = p1 >> 32u;
114 const std::uint64_t p2_lo = p2 & 0xFFFFFFFFu;
115 const std::uint64_t p2_hi = p2 >> 32u;
117 std::uint64_t Q = p0_hi + p1_lo + p2_lo;
119 // The full product might now be computed as
121 // p_hi = p3 + p2_hi + p1_hi + (Q >> 32)
122 // p_lo = p0_lo + (Q << 32)
124 // But in this particular case here, the full p_lo is not required.
125 // Effectively we only need to add the highest bit in p_lo to p_hi (and
126 // Q_hi + 1 does not overflow).
128 Q += std::uint64_t{1} << (64u - 32u - 1u); // round, ties up
130 const std::uint64_t h = p3 + p2_hi + p1_hi + (Q >> 32u);
132 return {h, x.e + y.e + 64};
136 @brief normalize x such that the significand is >= 2^(q-1)
139 static diyfp normalize(diyfp x) noexcept
141 JSON_ASSERT(x.f != 0);
143 while ((x.f >> 63u) == 0)
153 @brief normalize x such that the result has the exponent E
154 @pre e >= x.e and the upper e - x.e bits of x.f must be zero.
156 static diyfp normalize_to(const diyfp& x, const int target_exponent) noexcept
158 const int delta = x.e - target_exponent;
160 JSON_ASSERT(delta >= 0);
161 JSON_ASSERT(((x.f << delta) >> delta) == x.f);
163 return {x.f << delta, target_exponent};
175 Compute the (normalized) diyfp representing the input number 'value' and its
178 @pre value must be finite and positive
180 template<typename FloatType>
181 boundaries compute_boundaries(FloatType value)
183 JSON_ASSERT(std::isfinite(value));
184 JSON_ASSERT(value > 0);
186 // Convert the IEEE representation into a diyfp.
189 // value = 0.F * 2^(1 - bias) = ( F) * 2^(1 - bias - (p-1))
190 // If v is normalized:
191 // value = 1.F * 2^(E - bias) = (2^(p-1) + F) * 2^(E - bias - (p-1))
193 static_assert(std::numeric_limits<FloatType>::is_iec559,
194 "internal error: dtoa_short requires an IEEE-754 floating-point implementation");
196 constexpr int kPrecision = std::numeric_limits<FloatType>::digits; // = p (includes the hidden bit)
197 constexpr int kBias = std::numeric_limits<FloatType>::max_exponent - 1 + (kPrecision - 1);
198 constexpr int kMinExp = 1 - kBias;
199 constexpr std::uint64_t kHiddenBit = std::uint64_t{1} << (kPrecision - 1); // = 2^(p-1)
201 using bits_type = typename std::conditional<kPrecision == 24, std::uint32_t, std::uint64_t >::type;
203 const auto bits = static_cast<std::uint64_t>(reinterpret_bits<bits_type>(value));
204 const std::uint64_t E = bits >> (kPrecision - 1);
205 const std::uint64_t F = bits & (kHiddenBit - 1);
207 const bool is_denormal = E == 0;
208 const diyfp v = is_denormal
210 : diyfp(F + kHiddenBit, static_cast<int>(E) - kBias);
212 // Compute the boundaries m- and m+ of the floating-point value
215 // Determine v- and v+, the floating-point predecessor and successor if v,
218 // v- = v - 2^e if f != 2^(p-1) or e == e_min (A)
219 // = v - 2^(e-1) if f == 2^(p-1) and e > e_min (B)
223 // Let m- = (v- + v) / 2 and m+ = (v + v+) / 2. All real numbers _strictly_
224 // between m- and m+ round to v, regardless of how the input rounding
225 // algorithm breaks ties.
227 // ---+-------------+-------------+-------------+-------------+--- (A)
230 // -----------------+------+------+-------------+-------------+--- (B)
233 const bool lower_boundary_is_closer = F == 0 && E > 1;
234 const diyfp m_plus = diyfp(2 * v.f + 1, v.e - 1);
235 const diyfp m_minus = lower_boundary_is_closer
236 ? diyfp(4 * v.f - 1, v.e - 2) // (B)
237 : diyfp(2 * v.f - 1, v.e - 1); // (A)
239 // Determine the normalized w+ = m+.
240 const diyfp w_plus = diyfp::normalize(m_plus);
242 // Determine w- = m- such that e_(w-) = e_(w+).
243 const diyfp w_minus = diyfp::normalize_to(m_minus, w_plus.e);
245 return {diyfp::normalize(v), w_minus, w_plus};
248 // Given normalized diyfp w, Grisu needs to find a (normalized) cached
249 // power-of-ten c, such that the exponent of the product c * w = f * 2^e lies
250 // within a certain range [alpha, gamma] (Definition 3.2 from [1])
252 // alpha <= e = e_c + e_w + q <= gamma
256 // f_c * f_w * 2^alpha <= f_c 2^(e_c) * f_w 2^(e_w) * 2^q
257 // <= f_c * f_w * 2^gamma
259 // Since c and w are normalized, i.e. 2^(q-1) <= f < 2^q, this implies
261 // 2^(q-1) * 2^(q-1) * 2^alpha <= c * w * 2^q < 2^q * 2^q * 2^gamma
265 // 2^(q - 2 + alpha) <= c * w < 2^(q + gamma)
267 // The choice of (alpha,gamma) determines the size of the table and the form of
268 // the digit generation procedure. Using (alpha,gamma)=(-60,-32) works out well
271 // The idea is to cut the number c * w = f * 2^e into two parts, which can be
272 // processed independently: An integral part p1, and a fractional part p2:
274 // f * 2^e = ( (f div 2^-e) * 2^-e + (f mod 2^-e) ) * 2^e
275 // = (f div 2^-e) + (f mod 2^-e) * 2^e
278 // The conversion of p1 into decimal form requires a series of divisions and
279 // modulos by (a power of) 10. These operations are faster for 32-bit than for
280 // 64-bit integers, so p1 should ideally fit into a 32-bit integer. This can be
281 // achieved by choosing
283 // -e >= 32 or e <= -32 := gamma
285 // In order to convert the fractional part
287 // p2 * 2^e = p2 / 2^-e = d[-1] / 10^1 + d[-2] / 10^2 + ...
289 // into decimal form, the fraction is repeatedly multiplied by 10 and the digits
290 // d[-i] are extracted in order:
292 // (10 * p2) div 2^-e = d[-1]
293 // (10 * p2) mod 2^-e = d[-2] / 10^1 + ...
295 // The multiplication by 10 must not overflow. It is sufficient to choose
297 // 10 * p2 < 16 * p2 = 2^4 * p2 <= 2^64.
299 // Since p2 = f mod 2^-e < 2^-e,
301 // -e <= 60 or e >= -60 := alpha
303 constexpr int kAlpha = -60;
304 constexpr int kGamma = -32;
306 struct cached_power // c = f * 2^e ~= 10^k
314 For a normalized diyfp w = f * 2^e, this function returns a (normalized) cached
315 power-of-ten c = f_c * 2^e_c, such that the exponent of the product w * c
316 satisfies (Definition 3.2 from [1])
318 alpha <= e_c + e + q <= gamma.
320 inline cached_power get_cached_power_for_binary_exponent(int e)
324 // alpha <= e_c + e + q <= gamma (1)
325 // ==> f_c * 2^alpha <= c * 2^e * 2^q
327 // and since the c's are normalized, 2^(q-1) <= f_c,
329 // ==> 2^(q - 1 + alpha) <= c * 2^(e + q)
330 // ==> 2^(alpha - e - 1) <= c
332 // If c were an exact power of ten, i.e. c = 10^k, one may determine k as
334 // k = ceil( log_10( 2^(alpha - e - 1) ) )
335 // = ceil( (alpha - e - 1) * log_10(2) )
338 // "In theory the result of the procedure could be wrong since c is rounded,
339 // and the computation itself is approximated [...]. In practice, however,
340 // this simple function is sufficient."
342 // For IEEE double precision floating-point numbers converted into
343 // normalized diyfp's w = f * 2^e, with q = 64,
345 // e >= -1022 (min IEEE exponent)
347 // -52 (p - 1, possibly normalize denormal IEEE numbers)
348 // -11 (normalize the diyfp)
353 // e <= +1023 (max IEEE exponent)
355 // -11 (normalize the diyfp)
358 // This binary exponent range [-1137,960] results in a decimal exponent
359 // range [-307,324]. One does not need to store a cached power for each
360 // k in this range. For each such k it suffices to find a cached power
361 // such that the exponent of the product lies in [alpha,gamma].
362 // This implies that the difference of the decimal exponents of adjacent
363 // table entries must be less than or equal to
365 // floor( (gamma - alpha) * log_10(2) ) = 8.
367 // (A smaller distance gamma-alpha would require a larger table.)
370 // Actually this function returns c, such that -60 <= e_c + e + 64 <= -34.
372 constexpr int kCachedPowersMinDecExp = -300;
373 constexpr int kCachedPowersDecStep = 8;
375 static constexpr std::array<cached_power, 79> kCachedPowers =
378 { 0xAB70FE17C79AC6CA, -1060, -300 },
379 { 0xFF77B1FCBEBCDC4F, -1034, -292 },
380 { 0xBE5691EF416BD60C, -1007, -284 },
381 { 0x8DD01FAD907FFC3C, -980, -276 },
382 { 0xD3515C2831559A83, -954, -268 },
383 { 0x9D71AC8FADA6C9B5, -927, -260 },
384 { 0xEA9C227723EE8BCB, -901, -252 },
385 { 0xAECC49914078536D, -874, -244 },
386 { 0x823C12795DB6CE57, -847, -236 },
387 { 0xC21094364DFB5637, -821, -228 },
388 { 0x9096EA6F3848984F, -794, -220 },
389 { 0xD77485CB25823AC7, -768, -212 },
390 { 0xA086CFCD97BF97F4, -741, -204 },
391 { 0xEF340A98172AACE5, -715, -196 },
392 { 0xB23867FB2A35B28E, -688, -188 },
393 { 0x84C8D4DFD2C63F3B, -661, -180 },
394 { 0xC5DD44271AD3CDBA, -635, -172 },
395 { 0x936B9FCEBB25C996, -608, -164 },
396 { 0xDBAC6C247D62A584, -582, -156 },
397 { 0xA3AB66580D5FDAF6, -555, -148 },
398 { 0xF3E2F893DEC3F126, -529, -140 },
399 { 0xB5B5ADA8AAFF80B8, -502, -132 },
400 { 0x87625F056C7C4A8B, -475, -124 },
401 { 0xC9BCFF6034C13053, -449, -116 },
402 { 0x964E858C91BA2655, -422, -108 },
403 { 0xDFF9772470297EBD, -396, -100 },
404 { 0xA6DFBD9FB8E5B88F, -369, -92 },
405 { 0xF8A95FCF88747D94, -343, -84 },
406 { 0xB94470938FA89BCF, -316, -76 },
407 { 0x8A08F0F8BF0F156B, -289, -68 },
408 { 0xCDB02555653131B6, -263, -60 },
409 { 0x993FE2C6D07B7FAC, -236, -52 },
410 { 0xE45C10C42A2B3B06, -210, -44 },
411 { 0xAA242499697392D3, -183, -36 },
412 { 0xFD87B5F28300CA0E, -157, -28 },
413 { 0xBCE5086492111AEB, -130, -20 },
414 { 0x8CBCCC096F5088CC, -103, -12 },
415 { 0xD1B71758E219652C, -77, -4 },
416 { 0x9C40000000000000, -50, 4 },
417 { 0xE8D4A51000000000, -24, 12 },
418 { 0xAD78EBC5AC620000, 3, 20 },
419 { 0x813F3978F8940984, 30, 28 },
420 { 0xC097CE7BC90715B3, 56, 36 },
421 { 0x8F7E32CE7BEA5C70, 83, 44 },
422 { 0xD5D238A4ABE98068, 109, 52 },
423 { 0x9F4F2726179A2245, 136, 60 },
424 { 0xED63A231D4C4FB27, 162, 68 },
425 { 0xB0DE65388CC8ADA8, 189, 76 },
426 { 0x83C7088E1AAB65DB, 216, 84 },
427 { 0xC45D1DF942711D9A, 242, 92 },
428 { 0x924D692CA61BE758, 269, 100 },
429 { 0xDA01EE641A708DEA, 295, 108 },
430 { 0xA26DA3999AEF774A, 322, 116 },
431 { 0xF209787BB47D6B85, 348, 124 },
432 { 0xB454E4A179DD1877, 375, 132 },
433 { 0x865B86925B9BC5C2, 402, 140 },
434 { 0xC83553C5C8965D3D, 428, 148 },
435 { 0x952AB45CFA97A0B3, 455, 156 },
436 { 0xDE469FBD99A05FE3, 481, 164 },
437 { 0xA59BC234DB398C25, 508, 172 },
438 { 0xF6C69A72A3989F5C, 534, 180 },
439 { 0xB7DCBF5354E9BECE, 561, 188 },
440 { 0x88FCF317F22241E2, 588, 196 },
441 { 0xCC20CE9BD35C78A5, 614, 204 },
442 { 0x98165AF37B2153DF, 641, 212 },
443 { 0xE2A0B5DC971F303A, 667, 220 },
444 { 0xA8D9D1535CE3B396, 694, 228 },
445 { 0xFB9B7CD9A4A7443C, 720, 236 },
446 { 0xBB764C4CA7A44410, 747, 244 },
447 { 0x8BAB8EEFB6409C1A, 774, 252 },
448 { 0xD01FEF10A657842C, 800, 260 },
449 { 0x9B10A4E5E9913129, 827, 268 },
450 { 0xE7109BFBA19C0C9D, 853, 276 },
451 { 0xAC2820D9623BF429, 880, 284 },
452 { 0x80444B5E7AA7CF85, 907, 292 },
453 { 0xBF21E44003ACDD2D, 933, 300 },
454 { 0x8E679C2F5E44FF8F, 960, 308 },
455 { 0xD433179D9C8CB841, 986, 316 },
456 { 0x9E19DB92B4E31BA9, 1013, 324 },
460 // This computation gives exactly the same results for k as
461 // k = ceil((kAlpha - e - 1) * 0.30102999566398114)
462 // for |e| <= 1500, but doesn't require floating-point operations.
463 // NB: log_10(2) ~= 78913 / 2^18
464 JSON_ASSERT(e >= -1500);
465 JSON_ASSERT(e <= 1500);
466 const int f = kAlpha - e - 1;
467 const int k = (f * 78913) / (1 << 18) + static_cast<int>(f > 0);
469 const int index = (-kCachedPowersMinDecExp + k + (kCachedPowersDecStep - 1)) / kCachedPowersDecStep;
470 JSON_ASSERT(index >= 0);
471 JSON_ASSERT(static_cast<std::size_t>(index) < kCachedPowers.size());
473 const cached_power cached = kCachedPowers[static_cast<std::size_t>(index)];
474 JSON_ASSERT(kAlpha <= cached.e + e + 64);
475 JSON_ASSERT(kGamma >= cached.e + e + 64);
481 For n != 0, returns k, such that pow10 := 10^(k-1) <= n < 10^k.
482 For n == 0, returns 1 and sets pow10 := 1.
484 inline int find_largest_pow10(const std::uint32_t n, std::uint32_t& pow10)
538 inline void grisu2_round(char* buf, int len, std::uint64_t dist, std::uint64_t delta,
539 std::uint64_t rest, std::uint64_t ten_k)
541 JSON_ASSERT(len >= 1);
542 JSON_ASSERT(dist <= delta);
543 JSON_ASSERT(rest <= delta);
544 JSON_ASSERT(ten_k > 0);
546 // <--------------------------- delta ---->
547 // <---- dist --------->
548 // --------------[------------------+-------------------]--------------
554 // --------------[------------------+----+--------------]--------------
558 // ten_k represents a unit-in-the-last-place in the decimal representation
560 // Decrement buf by ten_k while this takes buf closer to w.
562 // The tests are written in this order to avoid overflow in unsigned
563 // integer arithmetic.
566 && delta - rest >= ten_k
567 && (rest + ten_k < dist || dist - rest > rest + ten_k - dist))
569 JSON_ASSERT(buf[len - 1] != '0');
576 Generates V = buffer * 10^decimal_exponent, such that M- <= V <= M+.
577 M- and M+ must be normalized and share the same exponent -60 <= e <= -32.
579 inline void grisu2_digit_gen(char* buffer, int& length, int& decimal_exponent,
580 diyfp M_minus, diyfp w, diyfp M_plus)
582 static_assert(kAlpha >= -60, "internal error");
583 static_assert(kGamma <= -32, "internal error");
585 // Generates the digits (and the exponent) of a decimal floating-point
586 // number V = buffer * 10^decimal_exponent in the range [M-, M+]. The diyfp's
587 // w, M- and M+ share the same exponent e, which satisfies alpha <= e <= gamma.
589 // <--------------------------- delta ---->
590 // <---- dist --------->
591 // --------------[------------------+-------------------]--------------
594 // Grisu2 generates the digits of M+ from left to right and stops as soon as
597 JSON_ASSERT(M_plus.e >= kAlpha);
598 JSON_ASSERT(M_plus.e <= kGamma);
600 std::uint64_t delta = diyfp::sub(M_plus, M_minus).f; // (significand of (M+ - M-), implicit exponent is e)
601 std::uint64_t dist = diyfp::sub(M_plus, w ).f; // (significand of (M+ - w ), implicit exponent is e)
603 // Split M+ = f * 2^e into two parts p1 and p2 (note: e < 0):
606 // = ((f div 2^-e) * 2^-e + (f mod 2^-e)) * 2^e
607 // = ((p1 ) * 2^-e + (p2 )) * 2^e
610 const diyfp one(std::uint64_t{1} << -M_plus.e, M_plus.e);
612 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.)
613 std::uint64_t p2 = M_plus.f & (one.f - 1); // p2 = f mod 2^-e
617 // Generate the digits of the integral part p1 = d[n-1]...d[1]d[0]
621 std::uint32_t pow10{};
622 const int k = find_largest_pow10(p1, pow10);
624 // 10^(k-1) <= p1 < 10^k, pow10 = 10^(k-1)
626 // p1 = (p1 div 10^(k-1)) * 10^(k-1) + (p1 mod 10^(k-1))
627 // = (d[k-1] ) * 10^(k-1) + (p1 mod 10^(k-1))
629 // M+ = p1 + p2 * 2^e
630 // = d[k-1] * 10^(k-1) + (p1 mod 10^(k-1)) + p2 * 2^e
631 // = d[k-1] * 10^(k-1) + ((p1 mod 10^(k-1)) * 2^-e + p2) * 2^e
632 // = d[k-1] * 10^(k-1) + ( rest) * 2^e
634 // Now generate the digits d[n] of p1 from left to right (n = k-1,...,0)
636 // p1 = d[k-1]...d[n] * 10^n + d[n-1]...d[0]
638 // but stop as soon as
640 // rest * 2^e = (d[n-1]...d[0] * 2^-e + p2) * 2^e <= delta * 2^e
646 // M+ = buffer * 10^n + (p1 + p2 * 2^e) (buffer = 0 for n = k)
647 // pow10 = 10^(n-1) <= p1 < 10^n
649 const std::uint32_t d = p1 / pow10; // d = p1 div 10^(n-1)
650 const std::uint32_t r = p1 % pow10; // r = p1 mod 10^(n-1)
652 // M+ = buffer * 10^n + (d * 10^(n-1) + r) + p2 * 2^e
653 // = (buffer * 10 + d) * 10^(n-1) + (r + p2 * 2^e)
656 buffer[length++] = static_cast<char>('0' + d); // buffer := buffer * 10 + d
658 // M+ = buffer * 10^(n-1) + (r + p2 * 2^e)
663 // M+ = buffer * 10^n + (p1 + p2 * 2^e)
667 // Now check if enough digits have been generated.
670 // p1 + p2 * 2^e = (p1 * 2^-e + p2) * 2^e = rest * 2^e
673 // Since rest and delta share the same exponent e, it suffices to
674 // compare the significands.
675 const std::uint64_t rest = (std::uint64_t{p1} << -one.e) + p2;
678 // V = buffer * 10^n, with M- <= V <= M+.
680 decimal_exponent += n;
682 // We may now just stop. But instead look if the buffer could be
683 // decremented to bring V closer to w.
685 // pow10 = 10^n is now 1 ulp in the decimal representation V.
686 // The rounding procedure works with diyfp's with an implicit
689 // 10^n = (10^n * 2^-e) * 2^e = ulp * 2^e
691 const std::uint64_t ten_n = std::uint64_t{pow10} << -one.e;
692 grisu2_round(buffer, length, dist, delta, rest, ten_n);
699 // pow10 = 10^(n-1) <= p1 < 10^n
700 // Invariants restored.
705 // The digits of the integral part have been generated:
707 // M+ = d[k-1]...d[1]d[0] + p2 * 2^e
708 // = buffer + p2 * 2^e
710 // Now generate the digits of the fractional part p2 * 2^e.
713 // No decimal point is generated: the exponent is adjusted instead.
715 // p2 actually represents the fraction
719 // = d[-1] / 10^1 + d[-2] / 10^2 + ...
721 // Now generate the digits d[-m] of p1 from left to right (m = 1,2,...)
723 // p2 * 2^e = d[-1]d[-2]...d[-m] * 10^-m
724 // + 10^-m * (d[-m-1] / 10^1 + d[-m-2] / 10^2 + ...)
728 // 10^m * p2 = ((10^m * p2) div 2^-e) * 2^-e + ((10^m * p2) mod 2^-e)
729 // = ( d) * 2^-e + ( r)
732 // 10^m * p2 * 2^e = d + r * 2^e
736 // M+ = buffer + p2 * 2^e
737 // = buffer + 10^-m * (d + r * 2^e)
738 // = (buffer * 10^m + d) * 10^-m + 10^-m * r * 2^e
740 // and stop as soon as 10^-m * r * 2^e <= delta * 2^e
742 JSON_ASSERT(p2 > delta);
748 // M+ = buffer * 10^-m + 10^-m * (d[-m-1] / 10 + d[-m-2] / 10^2 + ...) * 2^e
749 // = buffer * 10^-m + 10^-m * (p2 ) * 2^e
750 // = buffer * 10^-m + 10^-m * (1/10 * (10 * p2) ) * 2^e
751 // = buffer * 10^-m + 10^-m * (1/10 * ((10*p2 div 2^-e) * 2^-e + (10*p2 mod 2^-e)) * 2^e
753 JSON_ASSERT(p2 <= (std::numeric_limits<std::uint64_t>::max)() / 10);
755 const std::uint64_t d = p2 >> -one.e; // d = (10 * p2) div 2^-e
756 const std::uint64_t r = p2 & (one.f - 1); // r = (10 * p2) mod 2^-e
758 // M+ = buffer * 10^-m + 10^-m * (1/10 * (d * 2^-e + r) * 2^e
759 // = buffer * 10^-m + 10^-m * (1/10 * (d + r * 2^e))
760 // = (buffer * 10 + d) * 10^(-m-1) + 10^(-m-1) * r * 2^e
763 buffer[length++] = static_cast<char>('0' + d); // buffer := buffer * 10 + d
765 // M+ = buffer * 10^(-m-1) + 10^(-m-1) * r * 2^e
770 // M+ = buffer * 10^-m + 10^-m * p2 * 2^e
771 // Invariant restored.
773 // Check if enough digits have been generated.
775 // 10^-m * p2 * 2^e <= delta * 2^e
776 // p2 * 2^e <= 10^m * delta * 2^e
777 // p2 <= 10^m * delta
786 // V = buffer * 10^-m, with M- <= V <= M+.
788 decimal_exponent -= m;
790 // 1 ulp in the decimal representation is now 10^-m.
791 // Since delta and dist are now scaled by 10^m, we need to do the
792 // same with ulp in order to keep the units in sync.
794 // 10^m * 10^-m = 1 = 2^-e * 2^e = ten_m * 2^e
796 const std::uint64_t ten_m = one.f;
797 grisu2_round(buffer, length, dist, delta, p2, ten_m);
799 // By construction this algorithm generates the shortest possible decimal
800 // number (Loitsch, Theorem 6.2) which rounds back to w.
801 // For an input number of precision p, at least
803 // N = 1 + ceil(p * log_10(2))
805 // decimal digits are sufficient to identify all binary floating-point
806 // numbers (Matula, "In-and-Out conversions").
807 // This implies that the algorithm does not produce more than N decimal
810 // N = 17 for p = 53 (IEEE double precision)
811 // N = 9 for p = 24 (IEEE single precision)
815 v = buf * 10^decimal_exponent
816 len is the length of the buffer (number of decimal digits)
817 The buffer must be large enough, i.e. >= max_digits10.
819 JSON_HEDLEY_NON_NULL(1)
820 inline void grisu2(char* buf, int& len, int& decimal_exponent,
821 diyfp m_minus, diyfp v, diyfp m_plus)
823 JSON_ASSERT(m_plus.e == m_minus.e);
824 JSON_ASSERT(m_plus.e == v.e);
826 // --------(-----------------------+-----------------------)-------- (A)
829 // --------------------(-----------+-----------------------)-------- (B)
832 // First scale v (and m- and m+) such that the exponent is in the range
835 const cached_power cached = get_cached_power_for_binary_exponent(m_plus.e);
837 const diyfp c_minus_k(cached.f, cached.e); // = c ~= 10^-k
839 // The exponent of the products is = v.e + c_minus_k.e + q and is in the range [alpha,gamma]
840 const diyfp w = diyfp::mul(v, c_minus_k);
841 const diyfp w_minus = diyfp::mul(m_minus, c_minus_k);
842 const diyfp w_plus = diyfp::mul(m_plus, c_minus_k);
844 // ----(---+---)---------------(---+---)---------------(---+---)----
846 // = c*m- = c*v = c*m+
848 // diyfp::mul rounds its result and c_minus_k is approximated too. w, w- and
849 // w+ are now off by a small amount.
852 // w - v * 10^k < 1 ulp
854 // To account for this inaccuracy, add resp. subtract 1 ulp.
856 // --------+---[---------------(---+---)---------------]---+--------
859 // Now any number in [M-, M+] (bounds included) will round to w when input,
860 // regardless of how the input rounding algorithm breaks ties.
862 // And digit_gen generates the shortest possible such number in [M-, M+].
863 // Note that this does not mean that Grisu2 always generates the shortest
864 // possible number in the interval (m-, m+).
865 const diyfp M_minus(w_minus.f + 1, w_minus.e);
866 const diyfp M_plus (w_plus.f - 1, w_plus.e );
868 decimal_exponent = -cached.k; // = -(-k) = k
870 grisu2_digit_gen(buf, len, decimal_exponent, M_minus, w, M_plus);
874 v = buf * 10^decimal_exponent
875 len is the length of the buffer (number of decimal digits)
876 The buffer must be large enough, i.e. >= max_digits10.
878 template<typename FloatType>
879 JSON_HEDLEY_NON_NULL(1)
880 void grisu2(char* buf, int& len, int& decimal_exponent, FloatType value)
882 static_assert(diyfp::kPrecision >= std::numeric_limits<FloatType>::digits + 3,
883 "internal error: not enough precision");
885 JSON_ASSERT(std::isfinite(value));
886 JSON_ASSERT(value > 0);
888 // If the neighbors (and boundaries) of 'value' are always computed for double-precision
889 // numbers, all float's can be recovered using strtod (and strtof). However, the resulting
890 // decimal representations are not exactly "short".
892 // The documentation for 'std::to_chars' (https://en.cppreference.com/w/cpp/utility/to_chars)
893 // says "value is converted to a string as if by std::sprintf in the default ("C") locale"
894 // and since sprintf promotes floats to doubles, I think this is exactly what 'std::to_chars'
896 // On the other hand, the documentation for 'std::to_chars' requires that "parsing the
897 // representation using the corresponding std::from_chars function recovers value exactly". That
898 // indicates that single precision floating-point numbers should be recovered using
901 // NB: If the neighbors are computed for single-precision numbers, there is a single float
902 // (7.0385307e-26f) which can't be recovered using strtod. The resulting double precision
903 // value is off by 1 ulp.
905 const boundaries w = compute_boundaries(static_cast<double>(value));
907 const boundaries w = compute_boundaries(value);
910 grisu2(buf, len, decimal_exponent, w.minus, w.w, w.plus);
914 @brief appends a decimal representation of e to buf
915 @return a pointer to the element following the exponent.
916 @pre -1000 < e < 1000
918 JSON_HEDLEY_NON_NULL(1)
919 JSON_HEDLEY_RETURNS_NON_NULL
920 inline char* append_exponent(char* buf, int e)
922 JSON_ASSERT(e > -1000);
923 JSON_ASSERT(e < 1000);
935 auto k = static_cast<std::uint32_t>(e);
938 // Always print at least two digits in the exponent.
939 // This is for compatibility with printf("%g").
941 *buf++ = static_cast<char>('0' + k);
945 *buf++ = static_cast<char>('0' + k / 10);
947 *buf++ = static_cast<char>('0' + k);
951 *buf++ = static_cast<char>('0' + k / 100);
953 *buf++ = static_cast<char>('0' + k / 10);
955 *buf++ = static_cast<char>('0' + k);
962 @brief prettify v = buf * 10^decimal_exponent
964 If v is in the range [10^min_exp, 10^max_exp) it will be printed in fixed-point
965 notation. Otherwise it will be printed in exponential notation.
970 JSON_HEDLEY_NON_NULL(1)
971 JSON_HEDLEY_RETURNS_NON_NULL
972 inline char* format_buffer(char* buf, int len, int decimal_exponent,
973 int min_exp, int max_exp)
975 JSON_ASSERT(min_exp < 0);
976 JSON_ASSERT(max_exp > 0);
979 const int n = len + decimal_exponent;
981 // v = buf * 10^(n-k)
982 // k is the length of the buffer (number of decimal digits)
983 // n is the position of the decimal point relative to the start of the buffer.
985 if (k <= n && n <= max_exp)
988 // len <= max_exp + 2
990 std::memset(buf + k, '0', static_cast<size_t>(n) - static_cast<size_t>(k));
991 // Make it look like a floating-point number (#362, #378)
994 return buf + (static_cast<size_t>(n) + 2);
997 if (0 < n && n <= max_exp)
1000 // len <= max_digits10 + 1
1004 std::memmove(buf + (static_cast<size_t>(n) + 1), buf + n, static_cast<size_t>(k) - static_cast<size_t>(n));
1006 return buf + (static_cast<size_t>(k) + 1U);
1009 if (min_exp < n && n <= 0)
1012 // len <= 2 + (-min_exp - 1) + max_digits10
1014 std::memmove(buf + (2 + static_cast<size_t>(-n)), buf, static_cast<size_t>(k));
1017 std::memset(buf + 2, '0', static_cast<size_t>(-n));
1018 return buf + (2U + static_cast<size_t>(-n) + static_cast<size_t>(k));
1031 // len <= max_digits10 + 1 + 5
1033 std::memmove(buf + 2, buf + 1, static_cast<size_t>(k) - 1);
1035 buf += 1 + static_cast<size_t>(k);
1039 return append_exponent(buf, n - 1);
1042 } // namespace dtoa_impl
1045 @brief generates a decimal representation of the floating-point number value in [first, last).
1047 The format of the resulting decimal representation is similar to printf's %g
1048 format. Returns an iterator pointing past-the-end of the decimal representation.
1050 @note The input number must be finite, i.e. NaN's and Inf's are not supported.
1051 @note The buffer must be large enough.
1052 @note The result is NOT null-terminated.
1054 template<typename FloatType>
1055 JSON_HEDLEY_NON_NULL(1, 2)
1056 JSON_HEDLEY_RETURNS_NON_NULL
1057 char* to_chars(char* first, const char* last, FloatType value)
1059 static_cast<void>(last); // maybe unused - fix warning
1060 JSON_ASSERT(std::isfinite(value));
1062 // Use signbit(value) instead of (value < 0) since signbit works for -0.
1063 if (std::signbit(value))
1070 #pragma GCC diagnostic push
1071 #pragma GCC diagnostic ignored "-Wfloat-equal"
1073 if (value == 0) // +-0
1076 // Make it look like a floating-point number (#362, #378)
1082 #pragma GCC diagnostic pop
1085 JSON_ASSERT(last - first >= std::numeric_limits<FloatType>::max_digits10);
1087 // Compute v = buffer * 10^decimal_exponent.
1088 // The decimal digits are stored in the buffer, which needs to be interpreted
1089 // as an unsigned decimal integer.
1090 // len is the length of the buffer, i.e. the number of decimal digits.
1092 int decimal_exponent = 0;
1093 dtoa_impl::grisu2(first, len, decimal_exponent, value);
1095 JSON_ASSERT(len <= std::numeric_limits<FloatType>::max_digits10);
1097 // Format the buffer like printf("%.*g", prec, value)
1098 constexpr int kMinExp = -4;
1099 // Use digits10 here to increase compatibility with version 2.
1100 constexpr int kMaxExp = std::numeric_limits<FloatType>::digits10;
1102 JSON_ASSERT(last - first >= kMaxExp + 2);
1103 JSON_ASSERT(last - first >= 2 + (-kMinExp - 1) + std::numeric_limits<FloatType>::max_digits10);
1104 JSON_ASSERT(last - first >= std::numeric_limits<FloatType>::max_digits10 + 6);
1106 return dtoa_impl::format_buffer(first, len, decimal_exponent, kMinExp, kMaxExp);
1109 } // namespace detail
1110 } // namespace nlohmann