Horizon
to_chars.hpp
1 #pragma once
2 
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
9 
10 #include <nlohmann/detail/macro_scope.hpp>
11 
12 namespace nlohmann
13 {
14 namespace detail
15 {
16 
36 namespace dtoa_impl
37 {
38 
39 template<typename Target, typename Source>
40 Target reinterpret_bits(const Source source)
41 {
42  static_assert(sizeof(Target) == sizeof(Source), "size mismatch");
43 
44  Target target;
45  std::memcpy(&target, &source, sizeof(Source));
46  return target;
47 }
48 
49 struct diyfp // f * 2^e
50 {
51  static constexpr int kPrecision = 64; // = q
52 
53  std::uint64_t f = 0;
54  int e = 0;
55 
56  constexpr diyfp(std::uint64_t f_, int e_) noexcept : f(f_), e(e_) {}
57 
62  static diyfp sub(const diyfp& x, const diyfp& y) noexcept
63  {
64  JSON_ASSERT(x.e == y.e);
65  JSON_ASSERT(x.f >= y.f);
66 
67  return {x.f - y.f, x.e};
68  }
69 
74  static diyfp mul(const diyfp& x, const diyfp& y) noexcept
75  {
76  static_assert(kPrecision == 64, "internal error");
77 
78  // Computes:
79  // f = round((x.f * y.f) / 2^q)
80  // e = x.e + y.e + q
81 
82  // Emulate the 64-bit * 64-bit multiplication:
83  //
84  // p = u * v
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 )
92  //
93  // (Since Q might be larger than 2^32 - 1)
94  //
95  // = (p0_lo + 2^32 Q_lo) + 2^64 (Q_hi + H)
96  //
97  // (Q_hi + H does not overflow a 64-bit int)
98  //
99  // = p_lo + 2^64 p_hi
100 
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;
105 
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;
110 
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;
116 
117  std::uint64_t Q = p0_hi + p1_lo + p2_lo;
118 
119  // The full product might now be computed as
120  //
121  // p_hi = p3 + p2_hi + p1_hi + (Q >> 32)
122  // p_lo = p0_lo + (Q << 32)
123  //
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).
127 
128  Q += std::uint64_t{1} << (64u - 32u - 1u); // round, ties up
129 
130  const std::uint64_t h = p3 + p2_hi + p1_hi + (Q >> 32u);
131 
132  return {h, x.e + y.e + 64};
133  }
134 
139  static diyfp normalize(diyfp x) noexcept
140  {
141  JSON_ASSERT(x.f != 0);
142 
143  while ((x.f >> 63u) == 0)
144  {
145  x.f <<= 1u;
146  x.e--;
147  }
148 
149  return x;
150  }
151 
156  static diyfp normalize_to(const diyfp& x, const int target_exponent) noexcept
157  {
158  const int delta = x.e - target_exponent;
159 
160  JSON_ASSERT(delta >= 0);
161  JSON_ASSERT(((x.f << delta) >> delta) == x.f);
162 
163  return {x.f << delta, target_exponent};
164  }
165 };
166 
168 {
169  diyfp w;
170  diyfp minus;
171  diyfp plus;
172 };
173 
180 template<typename FloatType>
182 {
183  JSON_ASSERT(std::isfinite(value));
184  JSON_ASSERT(value > 0);
185 
186  // Convert the IEEE representation into a diyfp.
187  //
188  // If v is denormal:
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))
192 
193  static_assert(std::numeric_limits<FloatType>::is_iec559,
194  "internal error: dtoa_short requires an IEEE-754 floating-point implementation");
195 
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)
200 
201  using bits_type = typename std::conditional<kPrecision == 24, std::uint32_t, std::uint64_t >::type;
202 
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);
206 
207  const bool is_denormal = E == 0;
208  const diyfp v = is_denormal
209  ? diyfp(F, kMinExp)
210  : diyfp(F + kHiddenBit, static_cast<int>(E) - kBias);
211 
212  // Compute the boundaries m- and m+ of the floating-point value
213  // v = f * 2^e.
214  //
215  // Determine v- and v+, the floating-point predecessor and successor if v,
216  // respectively.
217  //
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)
220  //
221  // v+ = v + 2^e
222  //
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.
226  //
227  // ---+-------------+-------------+-------------+-------------+--- (A)
228  // v- m- v m+ v+
229  //
230  // -----------------+------+------+-------------+-------------+--- (B)
231  // v- m- v m+ v+
232 
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)
238 
239  // Determine the normalized w+ = m+.
240  const diyfp w_plus = diyfp::normalize(m_plus);
241 
242  // Determine w- = m- such that e_(w-) = e_(w+).
243  const diyfp w_minus = diyfp::normalize_to(m_minus, w_plus.e);
244 
245  return {diyfp::normalize(v), w_minus, w_plus};
246 }
247 
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])
251 //
252 // alpha <= e = e_c + e_w + q <= gamma
253 //
254 // or
255 //
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
258 //
259 // Since c and w are normalized, i.e. 2^(q-1) <= f < 2^q, this implies
260 //
261 // 2^(q-1) * 2^(q-1) * 2^alpha <= c * w * 2^q < 2^q * 2^q * 2^gamma
262 //
263 // or
264 //
265 // 2^(q - 2 + alpha) <= c * w < 2^(q + gamma)
266 //
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
269 // in practice:
270 //
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:
273 //
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
276 // = p1 + p2 * 2^e
277 //
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
282 //
283 // -e >= 32 or e <= -32 := gamma
284 //
285 // In order to convert the fractional part
286 //
287 // p2 * 2^e = p2 / 2^-e = d[-1] / 10^1 + d[-2] / 10^2 + ...
288 //
289 // into decimal form, the fraction is repeatedly multiplied by 10 and the digits
290 // d[-i] are extracted in order:
291 //
292 // (10 * p2) div 2^-e = d[-1]
293 // (10 * p2) mod 2^-e = d[-2] / 10^1 + ...
294 //
295 // The multiplication by 10 must not overflow. It is sufficient to choose
296 //
297 // 10 * p2 < 16 * p2 = 2^4 * p2 <= 2^64.
298 //
299 // Since p2 = f mod 2^-e < 2^-e,
300 //
301 // -e <= 60 or e >= -60 := alpha
302 
303 constexpr int kAlpha = -60;
304 constexpr int kGamma = -32;
305 
306 struct cached_power // c = f * 2^e ~= 10^k
307 {
308  std::uint64_t f;
309  int e;
310  int k;
311 };
312 
321 {
322  // Now
323  //
324  // alpha <= e_c + e + q <= gamma (1)
325  // ==> f_c * 2^alpha <= c * 2^e * 2^q
326  //
327  // and since the c's are normalized, 2^(q-1) <= f_c,
328  //
329  // ==> 2^(q - 1 + alpha) <= c * 2^(e + q)
330  // ==> 2^(alpha - e - 1) <= c
331  //
332  // If c were an exact power of ten, i.e. c = 10^k, one may determine k as
333  //
334  // k = ceil( log_10( 2^(alpha - e - 1) ) )
335  // = ceil( (alpha - e - 1) * log_10(2) )
336  //
337  // From the paper:
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."
341  //
342  // For IEEE double precision floating-point numbers converted into
343  // normalized diyfp's w = f * 2^e, with q = 64,
344  //
345  // e >= -1022 (min IEEE exponent)
346  // -52 (p - 1)
347  // -52 (p - 1, possibly normalize denormal IEEE numbers)
348  // -11 (normalize the diyfp)
349  // = -1137
350  //
351  // and
352  //
353  // e <= +1023 (max IEEE exponent)
354  // -52 (p - 1)
355  // -11 (normalize the diyfp)
356  // = 960
357  //
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
364  //
365  // floor( (gamma - alpha) * log_10(2) ) = 8.
366  //
367  // (A smaller distance gamma-alpha would require a larger table.)
368 
369  // NB:
370  // Actually this function returns c, such that -60 <= e_c + e + 64 <= -34.
371 
372  constexpr int kCachedPowersMinDecExp = -300;
373  constexpr int kCachedPowersDecStep = 8;
374 
375  static constexpr std::array<cached_power, 79> kCachedPowers =
376  {
377  {
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 },
457  }
458  };
459 
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);
468 
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());
472 
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);
476 
477  return cached;
478 }
479 
484 inline int find_largest_pow10(const std::uint32_t n, std::uint32_t& pow10)
485 {
486  // LCOV_EXCL_START
487  if (n >= 1000000000)
488  {
489  pow10 = 1000000000;
490  return 10;
491  }
492  // LCOV_EXCL_STOP
493  if (n >= 100000000)
494  {
495  pow10 = 100000000;
496  return 9;
497  }
498  if (n >= 10000000)
499  {
500  pow10 = 10000000;
501  return 8;
502  }
503  if (n >= 1000000)
504  {
505  pow10 = 1000000;
506  return 7;
507  }
508  if (n >= 100000)
509  {
510  pow10 = 100000;
511  return 6;
512  }
513  if (n >= 10000)
514  {
515  pow10 = 10000;
516  return 5;
517  }
518  if (n >= 1000)
519  {
520  pow10 = 1000;
521  return 4;
522  }
523  if (n >= 100)
524  {
525  pow10 = 100;
526  return 3;
527  }
528  if (n >= 10)
529  {
530  pow10 = 10;
531  return 2;
532  }
533 
534  pow10 = 1;
535  return 1;
536 }
537 
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)
540 {
541  JSON_ASSERT(len >= 1);
542  JSON_ASSERT(dist <= delta);
543  JSON_ASSERT(rest <= delta);
544  JSON_ASSERT(ten_k > 0);
545 
546  // <--------------------------- delta ---->
547  // <---- dist --------->
548  // --------------[------------------+-------------------]--------------
549  // M- w M+
550  //
551  // ten_k
552  // <------>
553  // <---- rest ---->
554  // --------------[------------------+----+--------------]--------------
555  // w V
556  // = buf * 10^k
557  //
558  // ten_k represents a unit-in-the-last-place in the decimal representation
559  // stored in buf.
560  // Decrement buf by ten_k while this takes buf closer to w.
561 
562  // The tests are written in this order to avoid overflow in unsigned
563  // integer arithmetic.
564 
565  while (rest < dist
566  && delta - rest >= ten_k
567  && (rest + ten_k < dist || dist - rest > rest + ten_k - dist))
568  {
569  JSON_ASSERT(buf[len - 1] != '0');
570  buf[len - 1]--;
571  rest += ten_k;
572  }
573 }
574 
579 inline void grisu2_digit_gen(char* buffer, int& length, int& decimal_exponent,
580  diyfp M_minus, diyfp w, diyfp M_plus)
581 {
582  static_assert(kAlpha >= -60, "internal error");
583  static_assert(kGamma <= -32, "internal error");
584 
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.
588  //
589  // <--------------------------- delta ---->
590  // <---- dist --------->
591  // --------------[------------------+-------------------]--------------
592  // M- w M+
593  //
594  // Grisu2 generates the digits of M+ from left to right and stops as soon as
595  // V is in [M-,M+].
596 
597  JSON_ASSERT(M_plus.e >= kAlpha);
598  JSON_ASSERT(M_plus.e <= kGamma);
599 
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)
602 
603  // Split M+ = f * 2^e into two parts p1 and p2 (note: e < 0):
604  //
605  // M+ = f * 2^e
606  // = ((f div 2^-e) * 2^-e + (f mod 2^-e)) * 2^e
607  // = ((p1 ) * 2^-e + (p2 )) * 2^e
608  // = p1 + p2 * 2^e
609 
610  const diyfp one(std::uint64_t{1} << -M_plus.e, M_plus.e);
611 
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
614 
615  // 1)
616  //
617  // Generate the digits of the integral part p1 = d[n-1]...d[1]d[0]
618 
619  JSON_ASSERT(p1 > 0);
620 
621  std::uint32_t pow10{};
622  const int k = find_largest_pow10(p1, pow10);
623 
624  // 10^(k-1) <= p1 < 10^k, pow10 = 10^(k-1)
625  //
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))
628  //
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
633  //
634  // Now generate the digits d[n] of p1 from left to right (n = k-1,...,0)
635  //
636  // p1 = d[k-1]...d[n] * 10^n + d[n-1]...d[0]
637  //
638  // but stop as soon as
639  //
640  // rest * 2^e = (d[n-1]...d[0] * 2^-e + p2) * 2^e <= delta * 2^e
641 
642  int n = k;
643  while (n > 0)
644  {
645  // Invariants:
646  // M+ = buffer * 10^n + (p1 + p2 * 2^e) (buffer = 0 for n = k)
647  // pow10 = 10^(n-1) <= p1 < 10^n
648  //
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)
651  //
652  // M+ = buffer * 10^n + (d * 10^(n-1) + r) + p2 * 2^e
653  // = (buffer * 10 + d) * 10^(n-1) + (r + p2 * 2^e)
654  //
655  JSON_ASSERT(d <= 9);
656  buffer[length++] = static_cast<char>('0' + d); // buffer := buffer * 10 + d
657  //
658  // M+ = buffer * 10^(n-1) + (r + p2 * 2^e)
659  //
660  p1 = r;
661  n--;
662  //
663  // M+ = buffer * 10^n + (p1 + p2 * 2^e)
664  // pow10 = 10^n
665  //
666 
667  // Now check if enough digits have been generated.
668  // Compute
669  //
670  // p1 + p2 * 2^e = (p1 * 2^-e + p2) * 2^e = rest * 2^e
671  //
672  // Note:
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;
676  if (rest <= delta)
677  {
678  // V = buffer * 10^n, with M- <= V <= M+.
679 
680  decimal_exponent += n;
681 
682  // We may now just stop. But instead look if the buffer could be
683  // decremented to bring V closer to w.
684  //
685  // pow10 = 10^n is now 1 ulp in the decimal representation V.
686  // The rounding procedure works with diyfp's with an implicit
687  // exponent of e.
688  //
689  // 10^n = (10^n * 2^-e) * 2^e = ulp * 2^e
690  //
691  const std::uint64_t ten_n = std::uint64_t{pow10} << -one.e;
692  grisu2_round(buffer, length, dist, delta, rest, ten_n);
693 
694  return;
695  }
696 
697  pow10 /= 10;
698  //
699  // pow10 = 10^(n-1) <= p1 < 10^n
700  // Invariants restored.
701  }
702 
703  // 2)
704  //
705  // The digits of the integral part have been generated:
706  //
707  // M+ = d[k-1]...d[1]d[0] + p2 * 2^e
708  // = buffer + p2 * 2^e
709  //
710  // Now generate the digits of the fractional part p2 * 2^e.
711  //
712  // Note:
713  // No decimal point is generated: the exponent is adjusted instead.
714  //
715  // p2 actually represents the fraction
716  //
717  // p2 * 2^e
718  // = p2 / 2^-e
719  // = d[-1] / 10^1 + d[-2] / 10^2 + ...
720  //
721  // Now generate the digits d[-m] of p1 from left to right (m = 1,2,...)
722  //
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 + ...)
725  //
726  // using
727  //
728  // 10^m * p2 = ((10^m * p2) div 2^-e) * 2^-e + ((10^m * p2) mod 2^-e)
729  // = ( d) * 2^-e + ( r)
730  //
731  // or
732  // 10^m * p2 * 2^e = d + r * 2^e
733  //
734  // i.e.
735  //
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
739  //
740  // and stop as soon as 10^-m * r * 2^e <= delta * 2^e
741 
742  JSON_ASSERT(p2 > delta);
743 
744  int m = 0;
745  for (;;)
746  {
747  // Invariant:
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
752  //
753  JSON_ASSERT(p2 <= (std::numeric_limits<std::uint64_t>::max)() / 10);
754  p2 *= 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
757  //
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
761  //
762  JSON_ASSERT(d <= 9);
763  buffer[length++] = static_cast<char>('0' + d); // buffer := buffer * 10 + d
764  //
765  // M+ = buffer * 10^(-m-1) + 10^(-m-1) * r * 2^e
766  //
767  p2 = r;
768  m++;
769  //
770  // M+ = buffer * 10^-m + 10^-m * p2 * 2^e
771  // Invariant restored.
772 
773  // Check if enough digits have been generated.
774  //
775  // 10^-m * p2 * 2^e <= delta * 2^e
776  // p2 * 2^e <= 10^m * delta * 2^e
777  // p2 <= 10^m * delta
778  delta *= 10;
779  dist *= 10;
780  if (p2 <= delta)
781  {
782  break;
783  }
784  }
785 
786  // V = buffer * 10^-m, with M- <= V <= M+.
787 
788  decimal_exponent -= m;
789 
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.
793  //
794  // 10^m * 10^-m = 1 = 2^-e * 2^e = ten_m * 2^e
795  //
796  const std::uint64_t ten_m = one.f;
797  grisu2_round(buffer, length, dist, delta, p2, ten_m);
798 
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
802  //
803  // N = 1 + ceil(p * log_10(2))
804  //
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
808  // digits.
809  //
810  // N = 17 for p = 53 (IEEE double precision)
811  // N = 9 for p = 24 (IEEE single precision)
812 }
813 
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)
822 {
823  JSON_ASSERT(m_plus.e == m_minus.e);
824  JSON_ASSERT(m_plus.e == v.e);
825 
826  // --------(-----------------------+-----------------------)-------- (A)
827  // m- v m+
828  //
829  // --------------------(-----------+-----------------------)-------- (B)
830  // m- v m+
831  //
832  // First scale v (and m- and m+) such that the exponent is in the range
833  // [alpha, gamma].
834 
835  const cached_power cached = get_cached_power_for_binary_exponent(m_plus.e);
836 
837  const diyfp c_minus_k(cached.f, cached.e); // = c ~= 10^-k
838 
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);
843 
844  // ----(---+---)---------------(---+---)---------------(---+---)----
845  // w- w w+
846  // = c*m- = c*v = c*m+
847  //
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.
850  // In fact:
851  //
852  // w - v * 10^k < 1 ulp
853  //
854  // To account for this inaccuracy, add resp. subtract 1 ulp.
855  //
856  // --------+---[---------------(---+---)---------------]---+--------
857  // w- M- w M+ w+
858  //
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.
861  //
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 );
867 
868  decimal_exponent = -cached.k; // = -(-k) = k
869 
870  grisu2_digit_gen(buf, len, decimal_exponent, M_minus, w, M_plus);
871 }
872 
878 template<typename FloatType>
879 JSON_HEDLEY_NON_NULL(1)
880 void grisu2(char* buf, int& len, int& decimal_exponent, FloatType value)
881 {
882  static_assert(diyfp::kPrecision >= std::numeric_limits<FloatType>::digits + 3,
883  "internal error: not enough precision");
884 
885  JSON_ASSERT(std::isfinite(value));
886  JSON_ASSERT(value > 0);
887 
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".
891  //
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 float's to double's, I think this is exactly what 'std::to_chars'
895  // does.
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
899  // 'std::strtof'.
900  //
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.
904 #if 0
905  const boundaries w = compute_boundaries(static_cast<double>(value));
906 #else
908 #endif
909 
910  grisu2(buf, len, decimal_exponent, w.minus, w.w, w.plus);
911 }
912 
918 JSON_HEDLEY_NON_NULL(1)
919 JSON_HEDLEY_RETURNS_NON_NULL
920 inline char* append_exponent(char* buf, int e)
921 {
922  JSON_ASSERT(e > -1000);
923  JSON_ASSERT(e < 1000);
924 
925  if (e < 0)
926  {
927  e = -e;
928  *buf++ = '-';
929  }
930  else
931  {
932  *buf++ = '+';
933  }
934 
935  auto k = static_cast<std::uint32_t>(e);
936  if (k < 10)
937  {
938  // Always print at least two digits in the exponent.
939  // This is for compatibility with printf("%g").
940  *buf++ = '0';
941  *buf++ = static_cast<char>('0' + k);
942  }
943  else if (k < 100)
944  {
945  *buf++ = static_cast<char>('0' + k / 10);
946  k %= 10;
947  *buf++ = static_cast<char>('0' + k);
948  }
949  else
950  {
951  *buf++ = static_cast<char>('0' + k / 100);
952  k %= 100;
953  *buf++ = static_cast<char>('0' + k / 10);
954  k %= 10;
955  *buf++ = static_cast<char>('0' + k);
956  }
957 
958  return buf;
959 }
960 
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)
974 {
975  JSON_ASSERT(min_exp < 0);
976  JSON_ASSERT(max_exp > 0);
977 
978  const int k = len;
979  const int n = len + decimal_exponent;
980 
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.
984 
985  if (k <= n && n <= max_exp)
986  {
987  // digits[000]
988  // len <= max_exp + 2
989 
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)
992  buf[n + 0] = '.';
993  buf[n + 1] = '0';
994  return buf + (static_cast<size_t>(n) + 2);
995  }
996 
997  if (0 < n && n <= max_exp)
998  {
999  // dig.its
1000  // len <= max_digits10 + 1
1001 
1002  JSON_ASSERT(k > n);
1003 
1004  std::memmove(buf + (static_cast<size_t>(n) + 1), buf + n, static_cast<size_t>(k) - static_cast<size_t>(n));
1005  buf[n] = '.';
1006  return buf + (static_cast<size_t>(k) + 1U);
1007  }
1008 
1009  if (min_exp < n && n <= 0)
1010  {
1011  // 0.[000]digits
1012  // len <= 2 + (-min_exp - 1) + max_digits10
1013 
1014  std::memmove(buf + (2 + static_cast<size_t>(-n)), buf, static_cast<size_t>(k));
1015  buf[0] = '0';
1016  buf[1] = '.';
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));
1019  }
1020 
1021  if (k == 1)
1022  {
1023  // dE+123
1024  // len <= 1 + 5
1025 
1026  buf += 1;
1027  }
1028  else
1029  {
1030  // d.igitsE+123
1031  // len <= max_digits10 + 1 + 5
1032 
1033  std::memmove(buf + 2, buf + 1, static_cast<size_t>(k) - 1);
1034  buf[1] = '.';
1035  buf += 1 + static_cast<size_t>(k);
1036  }
1037 
1038  *buf++ = 'e';
1039  return append_exponent(buf, n - 1);
1040 }
1041 
1042 } // namespace dtoa_impl
1043 
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)
1058 {
1059  static_cast<void>(last); // maybe unused - fix warning
1060  JSON_ASSERT(std::isfinite(value));
1061 
1062  // Use signbit(value) instead of (value < 0) since signbit works for -0.
1063  if (std::signbit(value))
1064  {
1065  value = -value;
1066  *first++ = '-';
1067  }
1068 
1069 #ifdef __GNUC__
1070 #pragma GCC diagnostic push
1071 #pragma GCC diagnostic ignored "-Wfloat-equal"
1072 #endif
1073  if (value == 0) // +-0
1074  {
1075  *first++ = '0';
1076  // Make it look like a floating-point number (#362, #378)
1077  *first++ = '.';
1078  *first++ = '0';
1079  return first;
1080  }
1081 #ifdef __GNUC__
1082 #pragma GCC diagnostic pop
1083 #endif
1084 
1085  JSON_ASSERT(last - first >= std::numeric_limits<FloatType>::max_digits10);
1086 
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.
1091  int len = 0;
1092  int decimal_exponent = 0;
1093  dtoa_impl::grisu2(first, len, decimal_exponent, value);
1094 
1095  JSON_ASSERT(len <= std::numeric_limits<FloatType>::max_digits10);
1096 
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;
1101 
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);
1105 
1106  return dtoa_impl::format_buffer(first, len, decimal_exponent, kMinExp, kMaxExp);
1107 }
1108 
1109 } // namespace detail
1110 } // namespace nlohmann
zip_uint64_t uint64_t
zip_uint64_t_t typedef.
Definition: zip.hpp:108
zip_uint32_t uint32_t
zip_uint32_t typedef.
Definition: zip.hpp:98
std::function< struct zip_source *(struct zip *)> source
Source creation for adding files.
Definition: zip.hpp:122
void grisu2(char *buf, int &len, int &decimal_exponent, diyfp m_minus, diyfp v, diyfp m_plus)
Definition: to_chars.hpp:820
boundaries compute_boundaries(FloatType value)
Definition: to_chars.hpp:181
int find_largest_pow10(const std::uint32_t n, std::uint32_t &pow10)
Definition: to_chars.hpp:484
void grisu2_digit_gen(char *buffer, int &length, int &decimal_exponent, diyfp M_minus, diyfp w, diyfp M_plus)
Definition: to_chars.hpp:579
JSON_HEDLEY_RETURNS_NON_NULL char * format_buffer(char *buf, int len, int decimal_exponent, int min_exp, int max_exp)
prettify v = buf * 10^decimal_exponent
Definition: to_chars.hpp:972
JSON_HEDLEY_RETURNS_NON_NULL char * append_exponent(char *buf, int e)
appends a decimal representation of e to buf
Definition: to_chars.hpp:920
cached_power get_cached_power_for_binary_exponent(int e)
Definition: to_chars.hpp:320
@ value
the parser finished reading a JSON value
JSON_HEDLEY_RETURNS_NON_NULL char * to_chars(char *first, const char *last, FloatType value)
generates a decimal representation of the floating-point number value in [first, last).
Definition: to_chars.hpp:1057
namespace for Niels Lohmann
Definition: adl_serializer.hpp:12
Definition: to_chars.hpp:168
Definition: to_chars.hpp:307
Definition: to_chars.hpp:50
static diyfp normalize(diyfp x) noexcept
normalize x such that the significand is >= 2^(q-1)
Definition: to_chars.hpp:139
static diyfp normalize_to(const diyfp &x, const int target_exponent) noexcept
normalize x such that the result has the exponent E
Definition: to_chars.hpp:156
static diyfp mul(const diyfp &x, const diyfp &y) noexcept
returns x * y
Definition: to_chars.hpp:74
static diyfp sub(const diyfp &x, const diyfp &y) noexcept
returns x - y
Definition: to_chars.hpp:62