Horizon
to_chars.hpp
1 #pragma once
2 
3 #include <array> // array
4 #include <cassert> // assert
5 #include <ciso646> // or, and, not
6 #include <cmath> // signbit, isfinite
7 #include <cstdint> // intN_t, uintN_t
8 #include <cstring> // memcpy, memmove
9 #include <limits> // numeric_limits
10 #include <type_traits> // conditional
11 #include <nlohmann/detail/macro_scope.hpp>
12 
13 namespace nlohmann
14 {
15 namespace detail
16 {
17 
37 namespace dtoa_impl
38 {
39 
40 template <typename Target, typename Source>
41 Target reinterpret_bits(const Source source)
42 {
43  static_assert(sizeof(Target) == sizeof(Source), "size mismatch");
44 
45  Target target;
46  std::memcpy(&target, &source, sizeof(Source));
47  return target;
48 }
49 
50 struct diyfp // f * 2^e
51 {
52  static constexpr int kPrecision = 64; // = q
53 
54  std::uint64_t f = 0;
55  int e = 0;
56 
57  constexpr diyfp(std::uint64_t f_, int e_) noexcept : f(f_), e(e_) {}
58 
63  static diyfp sub(const diyfp& x, const diyfp& y) noexcept
64  {
65  assert(x.e == y.e);
66  assert(x.f >= y.f);
67 
68  return {x.f - y.f, x.e};
69  }
70 
75  static diyfp mul(const diyfp& x, const diyfp& y) noexcept
76  {
77  static_assert(kPrecision == 64, "internal error");
78 
79  // Computes:
80  // f = round((x.f * y.f) / 2^q)
81  // e = x.e + y.e + q
82 
83  // Emulate the 64-bit * 64-bit multiplication:
84  //
85  // p = u * v
86  // = (u_lo + 2^32 u_hi) (v_lo + 2^32 v_hi)
87  // = (u_lo v_lo ) + 2^32 ((u_lo v_hi ) + (u_hi v_lo )) + 2^64 (u_hi v_hi )
88  // = (p0 ) + 2^32 ((p1 ) + (p2 )) + 2^64 (p3 )
89  // = (p0_lo + 2^32 p0_hi) + 2^32 ((p1_lo + 2^32 p1_hi) + (p2_lo + 2^32 p2_hi)) + 2^64 (p3 )
90  // = (p0_lo ) + 2^32 (p0_hi + p1_lo + p2_lo ) + 2^64 (p1_hi + p2_hi + p3)
91  // = (p0_lo ) + 2^32 (Q ) + 2^64 (H )
92  // = (p0_lo ) + 2^32 (Q_lo + 2^32 Q_hi ) + 2^64 (H )
93  //
94  // (Since Q might be larger than 2^32 - 1)
95  //
96  // = (p0_lo + 2^32 Q_lo) + 2^64 (Q_hi + H)
97  //
98  // (Q_hi + H does not overflow a 64-bit int)
99  //
100  // = p_lo + 2^64 p_hi
101 
102  const std::uint64_t u_lo = x.f & 0xFFFFFFFFu;
103  const std::uint64_t u_hi = x.f >> 32u;
104  const std::uint64_t v_lo = y.f & 0xFFFFFFFFu;
105  const std::uint64_t v_hi = y.f >> 32u;
106 
107  const std::uint64_t p0 = u_lo * v_lo;
108  const std::uint64_t p1 = u_lo * v_hi;
109  const std::uint64_t p2 = u_hi * v_lo;
110  const std::uint64_t p3 = u_hi * v_hi;
111 
112  const std::uint64_t p0_hi = p0 >> 32u;
113  const std::uint64_t p1_lo = p1 & 0xFFFFFFFFu;
114  const std::uint64_t p1_hi = p1 >> 32u;
115  const std::uint64_t p2_lo = p2 & 0xFFFFFFFFu;
116  const std::uint64_t p2_hi = p2 >> 32u;
117 
118  std::uint64_t Q = p0_hi + p1_lo + p2_lo;
119 
120  // The full product might now be computed as
121  //
122  // p_hi = p3 + p2_hi + p1_hi + (Q >> 32)
123  // p_lo = p0_lo + (Q << 32)
124  //
125  // But in this particular case here, the full p_lo is not required.
126  // Effectively we only need to add the highest bit in p_lo to p_hi (and
127  // Q_hi + 1 does not overflow).
128 
129  Q += std::uint64_t{1} << (64u - 32u - 1u); // round, ties up
130 
131  const std::uint64_t h = p3 + p2_hi + p1_hi + (Q >> 32u);
132 
133  return {h, x.e + y.e + 64};
134  }
135 
140  static diyfp normalize(diyfp x) noexcept
141  {
142  assert(x.f != 0);
143 
144  while ((x.f >> 63u) == 0)
145  {
146  x.f <<= 1u;
147  x.e--;
148  }
149 
150  return x;
151  }
152 
157  static diyfp normalize_to(const diyfp& x, const int target_exponent) noexcept
158  {
159  const int delta = x.e - target_exponent;
160 
161  assert(delta >= 0);
162  assert(((x.f << delta) >> delta) == x.f);
163 
164  return {x.f << delta, target_exponent};
165  }
166 };
167 
169 {
170  diyfp w;
171  diyfp minus;
172  diyfp plus;
173 };
174 
181 template <typename FloatType>
183 {
184  assert(std::isfinite(value));
185  assert(value > 0);
186 
187  // Convert the IEEE representation into a diyfp.
188  //
189  // If v is denormal:
190  // value = 0.F * 2^(1 - bias) = ( F) * 2^(1 - bias - (p-1))
191  // If v is normalized:
192  // value = 1.F * 2^(E - bias) = (2^(p-1) + F) * 2^(E - bias - (p-1))
193 
194  static_assert(std::numeric_limits<FloatType>::is_iec559,
195  "internal error: dtoa_short requires an IEEE-754 floating-point implementation");
196 
197  constexpr int kPrecision = std::numeric_limits<FloatType>::digits; // = p (includes the hidden bit)
198  constexpr int kBias = std::numeric_limits<FloatType>::max_exponent - 1 + (kPrecision - 1);
199  constexpr int kMinExp = 1 - kBias;
200  constexpr std::uint64_t kHiddenBit = std::uint64_t{1} << (kPrecision - 1); // = 2^(p-1)
201 
202  using bits_type = typename std::conditional<kPrecision == 24, std::uint32_t, std::uint64_t >::type;
203 
204  const std::uint64_t bits = reinterpret_bits<bits_type>(value);
205  const std::uint64_t E = bits >> (kPrecision - 1);
206  const std::uint64_t F = bits & (kHiddenBit - 1);
207 
208  const bool is_denormal = E == 0;
209  const diyfp v = is_denormal
210  ? diyfp(F, kMinExp)
211  : diyfp(F + kHiddenBit, static_cast<int>(E) - kBias);
212 
213  // Compute the boundaries m- and m+ of the floating-point value
214  // v = f * 2^e.
215  //
216  // Determine v- and v+, the floating-point predecessor and successor if v,
217  // respectively.
218  //
219  // v- = v - 2^e if f != 2^(p-1) or e == e_min (A)
220  // = v - 2^(e-1) if f == 2^(p-1) and e > e_min (B)
221  //
222  // v+ = v + 2^e
223  //
224  // Let m- = (v- + v) / 2 and m+ = (v + v+) / 2. All real numbers _strictly_
225  // between m- and m+ round to v, regardless of how the input rounding
226  // algorithm breaks ties.
227  //
228  // ---+-------------+-------------+-------------+-------------+--- (A)
229  // v- m- v m+ v+
230  //
231  // -----------------+------+------+-------------+-------------+--- (B)
232  // v- m- v m+ v+
233 
234  const bool lower_boundary_is_closer = F == 0 and E > 1;
235  const diyfp m_plus = diyfp(2 * v.f + 1, v.e - 1);
236  const diyfp m_minus = lower_boundary_is_closer
237  ? diyfp(4 * v.f - 1, v.e - 2) // (B)
238  : diyfp(2 * v.f - 1, v.e - 1); // (A)
239 
240  // Determine the normalized w+ = m+.
241  const diyfp w_plus = diyfp::normalize(m_plus);
242 
243  // Determine w- = m- such that e_(w-) = e_(w+).
244  const diyfp w_minus = diyfp::normalize_to(m_minus, w_plus.e);
245 
246  return {diyfp::normalize(v), w_minus, w_plus};
247 }
248 
249 // Given normalized diyfp w, Grisu needs to find a (normalized) cached
250 // power-of-ten c, such that the exponent of the product c * w = f * 2^e lies
251 // within a certain range [alpha, gamma] (Definition 3.2 from [1])
252 //
253 // alpha <= e = e_c + e_w + q <= gamma
254 //
255 // or
256 //
257 // f_c * f_w * 2^alpha <= f_c 2^(e_c) * f_w 2^(e_w) * 2^q
258 // <= f_c * f_w * 2^gamma
259 //
260 // Since c and w are normalized, i.e. 2^(q-1) <= f < 2^q, this implies
261 //
262 // 2^(q-1) * 2^(q-1) * 2^alpha <= c * w * 2^q < 2^q * 2^q * 2^gamma
263 //
264 // or
265 //
266 // 2^(q - 2 + alpha) <= c * w < 2^(q + gamma)
267 //
268 // The choice of (alpha,gamma) determines the size of the table and the form of
269 // the digit generation procedure. Using (alpha,gamma)=(-60,-32) works out well
270 // in practice:
271 //
272 // The idea is to cut the number c * w = f * 2^e into two parts, which can be
273 // processed independently: An integral part p1, and a fractional part p2:
274 //
275 // f * 2^e = ( (f div 2^-e) * 2^-e + (f mod 2^-e) ) * 2^e
276 // = (f div 2^-e) + (f mod 2^-e) * 2^e
277 // = p1 + p2 * 2^e
278 //
279 // The conversion of p1 into decimal form requires a series of divisions and
280 // modulos by (a power of) 10. These operations are faster for 32-bit than for
281 // 64-bit integers, so p1 should ideally fit into a 32-bit integer. This can be
282 // achieved by choosing
283 //
284 // -e >= 32 or e <= -32 := gamma
285 //
286 // In order to convert the fractional part
287 //
288 // p2 * 2^e = p2 / 2^-e = d[-1] / 10^1 + d[-2] / 10^2 + ...
289 //
290 // into decimal form, the fraction is repeatedly multiplied by 10 and the digits
291 // d[-i] are extracted in order:
292 //
293 // (10 * p2) div 2^-e = d[-1]
294 // (10 * p2) mod 2^-e = d[-2] / 10^1 + ...
295 //
296 // The multiplication by 10 must not overflow. It is sufficient to choose
297 //
298 // 10 * p2 < 16 * p2 = 2^4 * p2 <= 2^64.
299 //
300 // Since p2 = f mod 2^-e < 2^-e,
301 //
302 // -e <= 60 or e >= -60 := alpha
303 
304 constexpr int kAlpha = -60;
305 constexpr int kGamma = -32;
306 
307 struct cached_power // c = f * 2^e ~= 10^k
308 {
309  std::uint64_t f;
310  int e;
311  int k;
312 };
313 
322 {
323  // Now
324  //
325  // alpha <= e_c + e + q <= gamma (1)
326  // ==> f_c * 2^alpha <= c * 2^e * 2^q
327  //
328  // and since the c's are normalized, 2^(q-1) <= f_c,
329  //
330  // ==> 2^(q - 1 + alpha) <= c * 2^(e + q)
331  // ==> 2^(alpha - e - 1) <= c
332  //
333  // If c were an exact power of ten, i.e. c = 10^k, one may determine k as
334  //
335  // k = ceil( log_10( 2^(alpha - e - 1) ) )
336  // = ceil( (alpha - e - 1) * log_10(2) )
337  //
338  // From the paper:
339  // "In theory the result of the procedure could be wrong since c is rounded,
340  // and the computation itself is approximated [...]. In practice, however,
341  // this simple function is sufficient."
342  //
343  // For IEEE double precision floating-point numbers converted into
344  // normalized diyfp's w = f * 2^e, with q = 64,
345  //
346  // e >= -1022 (min IEEE exponent)
347  // -52 (p - 1)
348  // -52 (p - 1, possibly normalize denormal IEEE numbers)
349  // -11 (normalize the diyfp)
350  // = -1137
351  //
352  // and
353  //
354  // e <= +1023 (max IEEE exponent)
355  // -52 (p - 1)
356  // -11 (normalize the diyfp)
357  // = 960
358  //
359  // This binary exponent range [-1137,960] results in a decimal exponent
360  // range [-307,324]. One does not need to store a cached power for each
361  // k in this range. For each such k it suffices to find a cached power
362  // such that the exponent of the product lies in [alpha,gamma].
363  // This implies that the difference of the decimal exponents of adjacent
364  // table entries must be less than or equal to
365  //
366  // floor( (gamma - alpha) * log_10(2) ) = 8.
367  //
368  // (A smaller distance gamma-alpha would require a larger table.)
369 
370  // NB:
371  // Actually this function returns c, such that -60 <= e_c + e + 64 <= -34.
372 
373  constexpr int kCachedPowersMinDecExp = -300;
374  constexpr int kCachedPowersDecStep = 8;
375 
376  static constexpr std::array<cached_power, 79> kCachedPowers =
377  {
378  {
379  { 0xAB70FE17C79AC6CA, -1060, -300 },
380  { 0xFF77B1FCBEBCDC4F, -1034, -292 },
381  { 0xBE5691EF416BD60C, -1007, -284 },
382  { 0x8DD01FAD907FFC3C, -980, -276 },
383  { 0xD3515C2831559A83, -954, -268 },
384  { 0x9D71AC8FADA6C9B5, -927, -260 },
385  { 0xEA9C227723EE8BCB, -901, -252 },
386  { 0xAECC49914078536D, -874, -244 },
387  { 0x823C12795DB6CE57, -847, -236 },
388  { 0xC21094364DFB5637, -821, -228 },
389  { 0x9096EA6F3848984F, -794, -220 },
390  { 0xD77485CB25823AC7, -768, -212 },
391  { 0xA086CFCD97BF97F4, -741, -204 },
392  { 0xEF340A98172AACE5, -715, -196 },
393  { 0xB23867FB2A35B28E, -688, -188 },
394  { 0x84C8D4DFD2C63F3B, -661, -180 },
395  { 0xC5DD44271AD3CDBA, -635, -172 },
396  { 0x936B9FCEBB25C996, -608, -164 },
397  { 0xDBAC6C247D62A584, -582, -156 },
398  { 0xA3AB66580D5FDAF6, -555, -148 },
399  { 0xF3E2F893DEC3F126, -529, -140 },
400  { 0xB5B5ADA8AAFF80B8, -502, -132 },
401  { 0x87625F056C7C4A8B, -475, -124 },
402  { 0xC9BCFF6034C13053, -449, -116 },
403  { 0x964E858C91BA2655, -422, -108 },
404  { 0xDFF9772470297EBD, -396, -100 },
405  { 0xA6DFBD9FB8E5B88F, -369, -92 },
406  { 0xF8A95FCF88747D94, -343, -84 },
407  { 0xB94470938FA89BCF, -316, -76 },
408  { 0x8A08F0F8BF0F156B, -289, -68 },
409  { 0xCDB02555653131B6, -263, -60 },
410  { 0x993FE2C6D07B7FAC, -236, -52 },
411  { 0xE45C10C42A2B3B06, -210, -44 },
412  { 0xAA242499697392D3, -183, -36 },
413  { 0xFD87B5F28300CA0E, -157, -28 },
414  { 0xBCE5086492111AEB, -130, -20 },
415  { 0x8CBCCC096F5088CC, -103, -12 },
416  { 0xD1B71758E219652C, -77, -4 },
417  { 0x9C40000000000000, -50, 4 },
418  { 0xE8D4A51000000000, -24, 12 },
419  { 0xAD78EBC5AC620000, 3, 20 },
420  { 0x813F3978F8940984, 30, 28 },
421  { 0xC097CE7BC90715B3, 56, 36 },
422  { 0x8F7E32CE7BEA5C70, 83, 44 },
423  { 0xD5D238A4ABE98068, 109, 52 },
424  { 0x9F4F2726179A2245, 136, 60 },
425  { 0xED63A231D4C4FB27, 162, 68 },
426  { 0xB0DE65388CC8ADA8, 189, 76 },
427  { 0x83C7088E1AAB65DB, 216, 84 },
428  { 0xC45D1DF942711D9A, 242, 92 },
429  { 0x924D692CA61BE758, 269, 100 },
430  { 0xDA01EE641A708DEA, 295, 108 },
431  { 0xA26DA3999AEF774A, 322, 116 },
432  { 0xF209787BB47D6B85, 348, 124 },
433  { 0xB454E4A179DD1877, 375, 132 },
434  { 0x865B86925B9BC5C2, 402, 140 },
435  { 0xC83553C5C8965D3D, 428, 148 },
436  { 0x952AB45CFA97A0B3, 455, 156 },
437  { 0xDE469FBD99A05FE3, 481, 164 },
438  { 0xA59BC234DB398C25, 508, 172 },
439  { 0xF6C69A72A3989F5C, 534, 180 },
440  { 0xB7DCBF5354E9BECE, 561, 188 },
441  { 0x88FCF317F22241E2, 588, 196 },
442  { 0xCC20CE9BD35C78A5, 614, 204 },
443  { 0x98165AF37B2153DF, 641, 212 },
444  { 0xE2A0B5DC971F303A, 667, 220 },
445  { 0xA8D9D1535CE3B396, 694, 228 },
446  { 0xFB9B7CD9A4A7443C, 720, 236 },
447  { 0xBB764C4CA7A44410, 747, 244 },
448  { 0x8BAB8EEFB6409C1A, 774, 252 },
449  { 0xD01FEF10A657842C, 800, 260 },
450  { 0x9B10A4E5E9913129, 827, 268 },
451  { 0xE7109BFBA19C0C9D, 853, 276 },
452  { 0xAC2820D9623BF429, 880, 284 },
453  { 0x80444B5E7AA7CF85, 907, 292 },
454  { 0xBF21E44003ACDD2D, 933, 300 },
455  { 0x8E679C2F5E44FF8F, 960, 308 },
456  { 0xD433179D9C8CB841, 986, 316 },
457  { 0x9E19DB92B4E31BA9, 1013, 324 },
458  }
459  };
460 
461  // This computation gives exactly the same results for k as
462  // k = ceil((kAlpha - e - 1) * 0.30102999566398114)
463  // for |e| <= 1500, but doesn't require floating-point operations.
464  // NB: log_10(2) ~= 78913 / 2^18
465  assert(e >= -1500);
466  assert(e <= 1500);
467  const int f = kAlpha - e - 1;
468  const int k = (f * 78913) / (1 << 18) + static_cast<int>(f > 0);
469 
470  const int index = (-kCachedPowersMinDecExp + k + (kCachedPowersDecStep - 1)) / kCachedPowersDecStep;
471  assert(index >= 0);
472  assert(static_cast<std::size_t>(index) < kCachedPowers.size());
473 
474  const cached_power cached = kCachedPowers[static_cast<std::size_t>(index)];
475  assert(kAlpha <= cached.e + e + 64);
476  assert(kGamma >= cached.e + e + 64);
477 
478  return cached;
479 }
480 
485 inline int find_largest_pow10(const std::uint32_t n, std::uint32_t& pow10)
486 {
487  // LCOV_EXCL_START
488  if (n >= 1000000000)
489  {
490  pow10 = 1000000000;
491  return 10;
492  }
493  // LCOV_EXCL_STOP
494  else if (n >= 100000000)
495  {
496  pow10 = 100000000;
497  return 9;
498  }
499  else if (n >= 10000000)
500  {
501  pow10 = 10000000;
502  return 8;
503  }
504  else if (n >= 1000000)
505  {
506  pow10 = 1000000;
507  return 7;
508  }
509  else if (n >= 100000)
510  {
511  pow10 = 100000;
512  return 6;
513  }
514  else if (n >= 10000)
515  {
516  pow10 = 10000;
517  return 5;
518  }
519  else if (n >= 1000)
520  {
521  pow10 = 1000;
522  return 4;
523  }
524  else if (n >= 100)
525  {
526  pow10 = 100;
527  return 3;
528  }
529  else if (n >= 10)
530  {
531  pow10 = 10;
532  return 2;
533  }
534  else
535  {
536  pow10 = 1;
537  return 1;
538  }
539 }
540 
541 inline void grisu2_round(char* buf, int len, std::uint64_t dist, std::uint64_t delta,
542  std::uint64_t rest, std::uint64_t ten_k)
543 {
544  assert(len >= 1);
545  assert(dist <= delta);
546  assert(rest <= delta);
547  assert(ten_k > 0);
548 
549  // <--------------------------- delta ---->
550  // <---- dist --------->
551  // --------------[------------------+-------------------]--------------
552  // M- w M+
553  //
554  // ten_k
555  // <------>
556  // <---- rest ---->
557  // --------------[------------------+----+--------------]--------------
558  // w V
559  // = buf * 10^k
560  //
561  // ten_k represents a unit-in-the-last-place in the decimal representation
562  // stored in buf.
563  // Decrement buf by ten_k while this takes buf closer to w.
564 
565  // The tests are written in this order to avoid overflow in unsigned
566  // integer arithmetic.
567 
568  while (rest < dist
569  and delta - rest >= ten_k
570  and (rest + ten_k < dist or dist - rest > rest + ten_k - dist))
571  {
572  assert(buf[len - 1] != '0');
573  buf[len - 1]--;
574  rest += ten_k;
575  }
576 }
577 
582 inline void grisu2_digit_gen(char* buffer, int& length, int& decimal_exponent,
583  diyfp M_minus, diyfp w, diyfp M_plus)
584 {
585  static_assert(kAlpha >= -60, "internal error");
586  static_assert(kGamma <= -32, "internal error");
587 
588  // Generates the digits (and the exponent) of a decimal floating-point
589  // number V = buffer * 10^decimal_exponent in the range [M-, M+]. The diyfp's
590  // w, M- and M+ share the same exponent e, which satisfies alpha <= e <= gamma.
591  //
592  // <--------------------------- delta ---->
593  // <---- dist --------->
594  // --------------[------------------+-------------------]--------------
595  // M- w M+
596  //
597  // Grisu2 generates the digits of M+ from left to right and stops as soon as
598  // V is in [M-,M+].
599 
600  assert(M_plus.e >= kAlpha);
601  assert(M_plus.e <= kGamma);
602 
603  std::uint64_t delta = diyfp::sub(M_plus, M_minus).f; // (significand of (M+ - M-), implicit exponent is e)
604  std::uint64_t dist = diyfp::sub(M_plus, w ).f; // (significand of (M+ - w ), implicit exponent is e)
605 
606  // Split M+ = f * 2^e into two parts p1 and p2 (note: e < 0):
607  //
608  // M+ = f * 2^e
609  // = ((f div 2^-e) * 2^-e + (f mod 2^-e)) * 2^e
610  // = ((p1 ) * 2^-e + (p2 )) * 2^e
611  // = p1 + p2 * 2^e
612 
613  const diyfp one(std::uint64_t{1} << -M_plus.e, M_plus.e);
614 
615  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.)
616  std::uint64_t p2 = M_plus.f & (one.f - 1); // p2 = f mod 2^-e
617 
618  // 1)
619  //
620  // Generate the digits of the integral part p1 = d[n-1]...d[1]d[0]
621 
622  assert(p1 > 0);
623 
624  std::uint32_t pow10;
625  const int k = find_largest_pow10(p1, pow10);
626 
627  // 10^(k-1) <= p1 < 10^k, pow10 = 10^(k-1)
628  //
629  // p1 = (p1 div 10^(k-1)) * 10^(k-1) + (p1 mod 10^(k-1))
630  // = (d[k-1] ) * 10^(k-1) + (p1 mod 10^(k-1))
631  //
632  // M+ = p1 + p2 * 2^e
633  // = d[k-1] * 10^(k-1) + (p1 mod 10^(k-1)) + p2 * 2^e
634  // = d[k-1] * 10^(k-1) + ((p1 mod 10^(k-1)) * 2^-e + p2) * 2^e
635  // = d[k-1] * 10^(k-1) + ( rest) * 2^e
636  //
637  // Now generate the digits d[n] of p1 from left to right (n = k-1,...,0)
638  //
639  // p1 = d[k-1]...d[n] * 10^n + d[n-1]...d[0]
640  //
641  // but stop as soon as
642  //
643  // rest * 2^e = (d[n-1]...d[0] * 2^-e + p2) * 2^e <= delta * 2^e
644 
645  int n = k;
646  while (n > 0)
647  {
648  // Invariants:
649  // M+ = buffer * 10^n + (p1 + p2 * 2^e) (buffer = 0 for n = k)
650  // pow10 = 10^(n-1) <= p1 < 10^n
651  //
652  const std::uint32_t d = p1 / pow10; // d = p1 div 10^(n-1)
653  const std::uint32_t r = p1 % pow10; // r = p1 mod 10^(n-1)
654  //
655  // M+ = buffer * 10^n + (d * 10^(n-1) + r) + p2 * 2^e
656  // = (buffer * 10 + d) * 10^(n-1) + (r + p2 * 2^e)
657  //
658  assert(d <= 9);
659  buffer[length++] = static_cast<char>('0' + d); // buffer := buffer * 10 + d
660  //
661  // M+ = buffer * 10^(n-1) + (r + p2 * 2^e)
662  //
663  p1 = r;
664  n--;
665  //
666  // M+ = buffer * 10^n + (p1 + p2 * 2^e)
667  // pow10 = 10^n
668  //
669 
670  // Now check if enough digits have been generated.
671  // Compute
672  //
673  // p1 + p2 * 2^e = (p1 * 2^-e + p2) * 2^e = rest * 2^e
674  //
675  // Note:
676  // Since rest and delta share the same exponent e, it suffices to
677  // compare the significands.
678  const std::uint64_t rest = (std::uint64_t{p1} << -one.e) + p2;
679  if (rest <= delta)
680  {
681  // V = buffer * 10^n, with M- <= V <= M+.
682 
683  decimal_exponent += n;
684 
685  // We may now just stop. But instead look if the buffer could be
686  // decremented to bring V closer to w.
687  //
688  // pow10 = 10^n is now 1 ulp in the decimal representation V.
689  // The rounding procedure works with diyfp's with an implicit
690  // exponent of e.
691  //
692  // 10^n = (10^n * 2^-e) * 2^e = ulp * 2^e
693  //
694  const std::uint64_t ten_n = std::uint64_t{pow10} << -one.e;
695  grisu2_round(buffer, length, dist, delta, rest, ten_n);
696 
697  return;
698  }
699 
700  pow10 /= 10;
701  //
702  // pow10 = 10^(n-1) <= p1 < 10^n
703  // Invariants restored.
704  }
705 
706  // 2)
707  //
708  // The digits of the integral part have been generated:
709  //
710  // M+ = d[k-1]...d[1]d[0] + p2 * 2^e
711  // = buffer + p2 * 2^e
712  //
713  // Now generate the digits of the fractional part p2 * 2^e.
714  //
715  // Note:
716  // No decimal point is generated: the exponent is adjusted instead.
717  //
718  // p2 actually represents the fraction
719  //
720  // p2 * 2^e
721  // = p2 / 2^-e
722  // = d[-1] / 10^1 + d[-2] / 10^2 + ...
723  //
724  // Now generate the digits d[-m] of p1 from left to right (m = 1,2,...)
725  //
726  // p2 * 2^e = d[-1]d[-2]...d[-m] * 10^-m
727  // + 10^-m * (d[-m-1] / 10^1 + d[-m-2] / 10^2 + ...)
728  //
729  // using
730  //
731  // 10^m * p2 = ((10^m * p2) div 2^-e) * 2^-e + ((10^m * p2) mod 2^-e)
732  // = ( d) * 2^-e + ( r)
733  //
734  // or
735  // 10^m * p2 * 2^e = d + r * 2^e
736  //
737  // i.e.
738  //
739  // M+ = buffer + p2 * 2^e
740  // = buffer + 10^-m * (d + r * 2^e)
741  // = (buffer * 10^m + d) * 10^-m + 10^-m * r * 2^e
742  //
743  // and stop as soon as 10^-m * r * 2^e <= delta * 2^e
744 
745  assert(p2 > delta);
746 
747  int m = 0;
748  for (;;)
749  {
750  // Invariant:
751  // M+ = buffer * 10^-m + 10^-m * (d[-m-1] / 10 + d[-m-2] / 10^2 + ...) * 2^e
752  // = buffer * 10^-m + 10^-m * (p2 ) * 2^e
753  // = buffer * 10^-m + 10^-m * (1/10 * (10 * p2) ) * 2^e
754  // = buffer * 10^-m + 10^-m * (1/10 * ((10*p2 div 2^-e) * 2^-e + (10*p2 mod 2^-e)) * 2^e
755  //
756  assert(p2 <= (std::numeric_limits<std::uint64_t>::max)() / 10);
757  p2 *= 10;
758  const std::uint64_t d = p2 >> -one.e; // d = (10 * p2) div 2^-e
759  const std::uint64_t r = p2 & (one.f - 1); // r = (10 * p2) mod 2^-e
760  //
761  // M+ = buffer * 10^-m + 10^-m * (1/10 * (d * 2^-e + r) * 2^e
762  // = buffer * 10^-m + 10^-m * (1/10 * (d + r * 2^e))
763  // = (buffer * 10 + d) * 10^(-m-1) + 10^(-m-1) * r * 2^e
764  //
765  assert(d <= 9);
766  buffer[length++] = static_cast<char>('0' + d); // buffer := buffer * 10 + d
767  //
768  // M+ = buffer * 10^(-m-1) + 10^(-m-1) * r * 2^e
769  //
770  p2 = r;
771  m++;
772  //
773  // M+ = buffer * 10^-m + 10^-m * p2 * 2^e
774  // Invariant restored.
775 
776  // Check if enough digits have been generated.
777  //
778  // 10^-m * p2 * 2^e <= delta * 2^e
779  // p2 * 2^e <= 10^m * delta * 2^e
780  // p2 <= 10^m * delta
781  delta *= 10;
782  dist *= 10;
783  if (p2 <= delta)
784  {
785  break;
786  }
787  }
788 
789  // V = buffer * 10^-m, with M- <= V <= M+.
790 
791  decimal_exponent -= m;
792 
793  // 1 ulp in the decimal representation is now 10^-m.
794  // Since delta and dist are now scaled by 10^m, we need to do the
795  // same with ulp in order to keep the units in sync.
796  //
797  // 10^m * 10^-m = 1 = 2^-e * 2^e = ten_m * 2^e
798  //
799  const std::uint64_t ten_m = one.f;
800  grisu2_round(buffer, length, dist, delta, p2, ten_m);
801 
802  // By construction this algorithm generates the shortest possible decimal
803  // number (Loitsch, Theorem 6.2) which rounds back to w.
804  // For an input number of precision p, at least
805  //
806  // N = 1 + ceil(p * log_10(2))
807  //
808  // decimal digits are sufficient to identify all binary floating-point
809  // numbers (Matula, "In-and-Out conversions").
810  // This implies that the algorithm does not produce more than N decimal
811  // digits.
812  //
813  // N = 17 for p = 53 (IEEE double precision)
814  // N = 9 for p = 24 (IEEE single precision)
815 }
816 
822 JSON_HEDLEY_NON_NULL(1)
823 inline void grisu2(char* buf, int& len, int& decimal_exponent,
824  diyfp m_minus, diyfp v, diyfp m_plus)
825 {
826  assert(m_plus.e == m_minus.e);
827  assert(m_plus.e == v.e);
828 
829  // --------(-----------------------+-----------------------)-------- (A)
830  // m- v m+
831  //
832  // --------------------(-----------+-----------------------)-------- (B)
833  // m- v m+
834  //
835  // First scale v (and m- and m+) such that the exponent is in the range
836  // [alpha, gamma].
837 
838  const cached_power cached = get_cached_power_for_binary_exponent(m_plus.e);
839 
840  const diyfp c_minus_k(cached.f, cached.e); // = c ~= 10^-k
841 
842  // The exponent of the products is = v.e + c_minus_k.e + q and is in the range [alpha,gamma]
843  const diyfp w = diyfp::mul(v, c_minus_k);
844  const diyfp w_minus = diyfp::mul(m_minus, c_minus_k);
845  const diyfp w_plus = diyfp::mul(m_plus, c_minus_k);
846 
847  // ----(---+---)---------------(---+---)---------------(---+---)----
848  // w- w w+
849  // = c*m- = c*v = c*m+
850  //
851  // diyfp::mul rounds its result and c_minus_k is approximated too. w, w- and
852  // w+ are now off by a small amount.
853  // In fact:
854  //
855  // w - v * 10^k < 1 ulp
856  //
857  // To account for this inaccuracy, add resp. subtract 1 ulp.
858  //
859  // --------+---[---------------(---+---)---------------]---+--------
860  // w- M- w M+ w+
861  //
862  // Now any number in [M-, M+] (bounds included) will round to w when input,
863  // regardless of how the input rounding algorithm breaks ties.
864  //
865  // And digit_gen generates the shortest possible such number in [M-, M+].
866  // Note that this does not mean that Grisu2 always generates the shortest
867  // possible number in the interval (m-, m+).
868  const diyfp M_minus(w_minus.f + 1, w_minus.e);
869  const diyfp M_plus (w_plus.f - 1, w_plus.e );
870 
871  decimal_exponent = -cached.k; // = -(-k) = k
872 
873  grisu2_digit_gen(buf, len, decimal_exponent, M_minus, w, M_plus);
874 }
875 
881 template <typename FloatType>
882 JSON_HEDLEY_NON_NULL(1)
883 void grisu2(char* buf, int& len, int& decimal_exponent, FloatType value)
884 {
885  static_assert(diyfp::kPrecision >= std::numeric_limits<FloatType>::digits + 3,
886  "internal error: not enough precision");
887 
888  assert(std::isfinite(value));
889  assert(value > 0);
890 
891  // If the neighbors (and boundaries) of 'value' are always computed for double-precision
892  // numbers, all float's can be recovered using strtod (and strtof). However, the resulting
893  // decimal representations are not exactly "short".
894  //
895  // The documentation for 'std::to_chars' (https://en.cppreference.com/w/cpp/utility/to_chars)
896  // says "value is converted to a string as if by std::sprintf in the default ("C") locale"
897  // and since sprintf promotes float's to double's, I think this is exactly what 'std::to_chars'
898  // does.
899  // On the other hand, the documentation for 'std::to_chars' requires that "parsing the
900  // representation using the corresponding std::from_chars function recovers value exactly". That
901  // indicates that single precision floating-point numbers should be recovered using
902  // 'std::strtof'.
903  //
904  // NB: If the neighbors are computed for single-precision numbers, there is a single float
905  // (7.0385307e-26f) which can't be recovered using strtod. The resulting double precision
906  // value is off by 1 ulp.
907 #if 0
908  const boundaries w = compute_boundaries(static_cast<double>(value));
909 #else
910  const boundaries w = compute_boundaries(value);
911 #endif
912 
913  grisu2(buf, len, decimal_exponent, w.minus, w.w, w.plus);
914 }
915 
921 JSON_HEDLEY_NON_NULL(1)
922 JSON_HEDLEY_RETURNS_NON_NULL
923 inline char* append_exponent(char* buf, int e)
924 {
925  assert(e > -1000);
926  assert(e < 1000);
927 
928  if (e < 0)
929  {
930  e = -e;
931  *buf++ = '-';
932  }
933  else
934  {
935  *buf++ = '+';
936  }
937 
938  auto k = static_cast<std::uint32_t>(e);
939  if (k < 10)
940  {
941  // Always print at least two digits in the exponent.
942  // This is for compatibility with printf("%g").
943  *buf++ = '0';
944  *buf++ = static_cast<char>('0' + k);
945  }
946  else if (k < 100)
947  {
948  *buf++ = static_cast<char>('0' + k / 10);
949  k %= 10;
950  *buf++ = static_cast<char>('0' + k);
951  }
952  else
953  {
954  *buf++ = static_cast<char>('0' + k / 100);
955  k %= 100;
956  *buf++ = static_cast<char>('0' + k / 10);
957  k %= 10;
958  *buf++ = static_cast<char>('0' + k);
959  }
960 
961  return buf;
962 }
963 
973 JSON_HEDLEY_NON_NULL(1)
974 JSON_HEDLEY_RETURNS_NON_NULL
975 inline char* format_buffer(char* buf, int len, int decimal_exponent,
976  int min_exp, int max_exp)
977 {
978  assert(min_exp < 0);
979  assert(max_exp > 0);
980 
981  const int k = len;
982  const int n = len + decimal_exponent;
983 
984  // v = buf * 10^(n-k)
985  // k is the length of the buffer (number of decimal digits)
986  // n is the position of the decimal point relative to the start of the buffer.
987 
988  if (k <= n and n <= max_exp)
989  {
990  // digits[000]
991  // len <= max_exp + 2
992 
993  std::memset(buf + k, '0', static_cast<size_t>(n - k));
994  // Make it look like a floating-point number (#362, #378)
995  buf[n + 0] = '.';
996  buf[n + 1] = '0';
997  return buf + (n + 2);
998  }
999 
1000  if (0 < n and n <= max_exp)
1001  {
1002  // dig.its
1003  // len <= max_digits10 + 1
1004 
1005  assert(k > n);
1006 
1007  std::memmove(buf + (n + 1), buf + n, static_cast<size_t>(k - n));
1008  buf[n] = '.';
1009  return buf + (k + 1);
1010  }
1011 
1012  if (min_exp < n and n <= 0)
1013  {
1014  // 0.[000]digits
1015  // len <= 2 + (-min_exp - 1) + max_digits10
1016 
1017  std::memmove(buf + (2 + -n), buf, static_cast<size_t>(k));
1018  buf[0] = '0';
1019  buf[1] = '.';
1020  std::memset(buf + 2, '0', static_cast<size_t>(-n));
1021  return buf + (2 + (-n) + k);
1022  }
1023 
1024  if (k == 1)
1025  {
1026  // dE+123
1027  // len <= 1 + 5
1028 
1029  buf += 1;
1030  }
1031  else
1032  {
1033  // d.igitsE+123
1034  // len <= max_digits10 + 1 + 5
1035 
1036  std::memmove(buf + 2, buf + 1, static_cast<size_t>(k - 1));
1037  buf[1] = '.';
1038  buf += 1 + k;
1039  }
1040 
1041  *buf++ = 'e';
1042  return append_exponent(buf, n - 1);
1043 }
1044 
1045 } // namespace dtoa_impl
1046 
1057 template <typename FloatType>
1058 JSON_HEDLEY_NON_NULL(1, 2)
1059 JSON_HEDLEY_RETURNS_NON_NULL
1060 char* to_chars(char* first, const char* last, FloatType value)
1061 {
1062  static_cast<void>(last); // maybe unused - fix warning
1063  assert(std::isfinite(value));
1064 
1065  // Use signbit(value) instead of (value < 0) since signbit works for -0.
1066  if (std::signbit(value))
1067  {
1068  value = -value;
1069  *first++ = '-';
1070  }
1071 
1072  if (value == 0) // +-0
1073  {
1074  *first++ = '0';
1075  // Make it look like a floating-point number (#362, #378)
1076  *first++ = '.';
1077  *first++ = '0';
1078  return first;
1079  }
1080 
1081  assert(last - first >= std::numeric_limits<FloatType>::max_digits10);
1082 
1083  // Compute v = buffer * 10^decimal_exponent.
1084  // The decimal digits are stored in the buffer, which needs to be interpreted
1085  // as an unsigned decimal integer.
1086  // len is the length of the buffer, i.e. the number of decimal digits.
1087  int len = 0;
1088  int decimal_exponent = 0;
1089  dtoa_impl::grisu2(first, len, decimal_exponent, value);
1090 
1091  assert(len <= std::numeric_limits<FloatType>::max_digits10);
1092 
1093  // Format the buffer like printf("%.*g", prec, value)
1094  constexpr int kMinExp = -4;
1095  // Use digits10 here to increase compatibility with version 2.
1096  constexpr int kMaxExp = std::numeric_limits<FloatType>::digits10;
1097 
1098  assert(last - first >= kMaxExp + 2);
1099  assert(last - first >= 2 + (-kMinExp - 1) + std::numeric_limits<FloatType>::max_digits10);
1100  assert(last - first >= std::numeric_limits<FloatType>::max_digits10 + 6);
1101 
1102  return dtoa_impl::format_buffer(first, len, decimal_exponent, kMinExp, kMaxExp);
1103 }
1104 
1105 } // namespace detail
1106 } // 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:823
boundaries compute_boundaries(FloatType value)
Definition: to_chars.hpp:182
int find_largest_pow10(const std::uint32_t n, std::uint32_t &pow10)
Definition: to_chars.hpp:485
void grisu2_digit_gen(char *buffer, int &length, int &decimal_exponent, diyfp M_minus, diyfp w, diyfp M_plus)
Definition: to_chars.hpp:582
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:975
JSON_HEDLEY_RETURNS_NON_NULL char * append_exponent(char *buf, int e)
appends a decimal representation of e to buf
Definition: to_chars.hpp:923
cached_power get_cached_power_for_binary_exponent(int e)
Definition: to_chars.hpp:321
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:1060
namespace for Niels Lohmann
Definition: adl_serializer.hpp:9
Definition: to_chars.hpp:169
Definition: to_chars.hpp:308
Definition: to_chars.hpp:51
static diyfp normalize(diyfp x) noexcept
normalize x such that the significand is >= 2^(q-1)
Definition: to_chars.hpp:140
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:157
static diyfp mul(const diyfp &x, const diyfp &y) noexcept
returns x * y
Definition: to_chars.hpp:75
static diyfp sub(const diyfp &x, const diyfp &y) noexcept
returns x - y
Definition: to_chars.hpp:63