1//===- llvm/Support/ScaledNumber.h - Support for scaled numbers -*- C++ -*-===//
2//
3//                     The LLVM Compiler Infrastructure
4//
5// This file is distributed under the University of Illinois Open Source
6// License. See LICENSE.TXT for details.
7//
8//===----------------------------------------------------------------------===//
9//
10// This file contains functions (and a class) useful for working with scaled
11// numbers -- in particular, pairs of integers where one represents digits and
12// another represents a scale.  The functions are helpers and live in the
13// namespace ScaledNumbers.  The class ScaledNumber is useful for modelling
14// certain cost metrics that need simple, integer-like semantics that are easy
15// to reason about.
16//
17// These might remind you of soft-floats.  If you want one of those, you're in
18// the wrong place.  Look at include/llvm/ADT/APFloat.h instead.
19//
20//===----------------------------------------------------------------------===//
21
22#ifndef LLVM_SUPPORT_SCALEDNUMBER_H
23#define LLVM_SUPPORT_SCALEDNUMBER_H
24
25#include "llvm/Support/MathExtras.h"
26#include <algorithm>
27#include <cstdint>
28#include <limits>
29#include <string>
30#include <tuple>
31#include <utility>
32
33namespace llvm {
34namespace ScaledNumbers {
35
36/// \brief Maximum scale; same as APFloat for easy debug printing.
37const int32_t MaxScale = 16383;
38
39/// \brief Maximum scale; same as APFloat for easy debug printing.
40const int32_t MinScale = -16382;
41
42/// \brief Get the width of a number.
43template <class DigitsT> inline int getWidth() { return sizeof(DigitsT) * 8; }
44
45/// \brief Conditionally round up a scaled number.
46///
47/// Given \c Digits and \c Scale, round up iff \c ShouldRound is \c true.
48/// Always returns \c Scale unless there's an overflow, in which case it
49/// returns \c 1+Scale.
50///
51/// \pre adding 1 to \c Scale will not overflow INT16_MAX.
52template <class DigitsT>
53inline std::pair<DigitsT, int16_t> getRounded(DigitsT Digits, int16_t Scale,
54                                              bool ShouldRound) {
55  static_assert(!std::numeric_limits<DigitsT>::is_signed, "expected unsigned");
56
57  if (ShouldRound)
58    if (!++Digits)
59      // Overflow.
60      return std::make_pair(DigitsT(1) << (getWidth<DigitsT>() - 1), Scale + 1);
61  return std::make_pair(Digits, Scale);
62}
63
64/// \brief Convenience helper for 32-bit rounding.
65inline std::pair<uint32_t, int16_t> getRounded32(uint32_t Digits, int16_t Scale,
66                                                 bool ShouldRound) {
67  return getRounded(Digits, Scale, ShouldRound);
68}
69
70/// \brief Convenience helper for 64-bit rounding.
71inline std::pair<uint64_t, int16_t> getRounded64(uint64_t Digits, int16_t Scale,
72                                                 bool ShouldRound) {
73  return getRounded(Digits, Scale, ShouldRound);
74}
75
76/// \brief Adjust a 64-bit scaled number down to the appropriate width.
77///
78/// \pre Adding 64 to \c Scale will not overflow INT16_MAX.
79template <class DigitsT>
80inline std::pair<DigitsT, int16_t> getAdjusted(uint64_t Digits,
81                                               int16_t Scale = 0) {
82  static_assert(!std::numeric_limits<DigitsT>::is_signed, "expected unsigned");
83
84  const int Width = getWidth<DigitsT>();
85  if (Width == 64 || Digits <= std::numeric_limits<DigitsT>::max())
86    return std::make_pair(Digits, Scale);
87
88  // Shift right and round.
89  int Shift = 64 - Width - countLeadingZeros(Digits);
90  return getRounded<DigitsT>(Digits >> Shift, Scale + Shift,
91                             Digits & (UINT64_C(1) << (Shift - 1)));
92}
93
94/// \brief Convenience helper for adjusting to 32 bits.
95inline std::pair<uint32_t, int16_t> getAdjusted32(uint64_t Digits,
96                                                  int16_t Scale = 0) {
97  return getAdjusted<uint32_t>(Digits, Scale);
98}
99
100/// \brief Convenience helper for adjusting to 64 bits.
101inline std::pair<uint64_t, int16_t> getAdjusted64(uint64_t Digits,
102                                                  int16_t Scale = 0) {
103  return getAdjusted<uint64_t>(Digits, Scale);
104}
105
106/// \brief Multiply two 64-bit integers to create a 64-bit scaled number.
107///
108/// Implemented with four 64-bit integer multiplies.
109std::pair<uint64_t, int16_t> multiply64(uint64_t LHS, uint64_t RHS);
110
111/// \brief Multiply two 32-bit integers to create a 32-bit scaled number.
112///
113/// Implemented with one 64-bit integer multiply.
114template <class DigitsT>
115inline std::pair<DigitsT, int16_t> getProduct(DigitsT LHS, DigitsT RHS) {
116  static_assert(!std::numeric_limits<DigitsT>::is_signed, "expected unsigned");
117
118  if (getWidth<DigitsT>() <= 32 || (LHS <= UINT32_MAX && RHS <= UINT32_MAX))
119    return getAdjusted<DigitsT>(uint64_t(LHS) * RHS);
120
121  return multiply64(LHS, RHS);
122}
123
124/// \brief Convenience helper for 32-bit product.
125inline std::pair<uint32_t, int16_t> getProduct32(uint32_t LHS, uint32_t RHS) {
126  return getProduct(LHS, RHS);
127}
128
129/// \brief Convenience helper for 64-bit product.
130inline std::pair<uint64_t, int16_t> getProduct64(uint64_t LHS, uint64_t RHS) {
131  return getProduct(LHS, RHS);
132}
133
134/// \brief Divide two 64-bit integers to create a 64-bit scaled number.
135///
136/// Implemented with long division.
137///
138/// \pre \c Dividend and \c Divisor are non-zero.
139std::pair<uint64_t, int16_t> divide64(uint64_t Dividend, uint64_t Divisor);
140
141/// \brief Divide two 32-bit integers to create a 32-bit scaled number.
142///
143/// Implemented with one 64-bit integer divide/remainder pair.
144///
145/// \pre \c Dividend and \c Divisor are non-zero.
146std::pair<uint32_t, int16_t> divide32(uint32_t Dividend, uint32_t Divisor);
147
148/// \brief Divide two 32-bit numbers to create a 32-bit scaled number.
149///
150/// Implemented with one 64-bit integer divide/remainder pair.
151///
152/// Returns \c (DigitsT_MAX, MaxScale) for divide-by-zero (0 for 0/0).
153template <class DigitsT>
154std::pair<DigitsT, int16_t> getQuotient(DigitsT Dividend, DigitsT Divisor) {
155  static_assert(!std::numeric_limits<DigitsT>::is_signed, "expected unsigned");
156  static_assert(sizeof(DigitsT) == 4 || sizeof(DigitsT) == 8,
157                "expected 32-bit or 64-bit digits");
158
159  // Check for zero.
160  if (!Dividend)
161    return std::make_pair(0, 0);
162  if (!Divisor)
163    return std::make_pair(std::numeric_limits<DigitsT>::max(), MaxScale);
164
165  if (getWidth<DigitsT>() == 64)
166    return divide64(Dividend, Divisor);
167  return divide32(Dividend, Divisor);
168}
169
170/// \brief Convenience helper for 32-bit quotient.
171inline std::pair<uint32_t, int16_t> getQuotient32(uint32_t Dividend,
172                                                  uint32_t Divisor) {
173  return getQuotient(Dividend, Divisor);
174}
175
176/// \brief Convenience helper for 64-bit quotient.
177inline std::pair<uint64_t, int16_t> getQuotient64(uint64_t Dividend,
178                                                  uint64_t Divisor) {
179  return getQuotient(Dividend, Divisor);
180}
181
182/// \brief Implementation of getLg() and friends.
183///
184/// Returns the rounded lg of \c Digits*2^Scale and an int specifying whether
185/// this was rounded up (1), down (-1), or exact (0).
186///
187/// Returns \c INT32_MIN when \c Digits is zero.
188template <class DigitsT>
189inline std::pair<int32_t, int> getLgImpl(DigitsT Digits, int16_t Scale) {
190  static_assert(!std::numeric_limits<DigitsT>::is_signed, "expected unsigned");
191
192  if (!Digits)
193    return std::make_pair(INT32_MIN, 0);
194
195  // Get the floor of the lg of Digits.
196  int32_t LocalFloor = sizeof(Digits) * 8 - countLeadingZeros(Digits) - 1;
197
198  // Get the actual floor.
199  int32_t Floor = Scale + LocalFloor;
200  if (Digits == UINT64_C(1) << LocalFloor)
201    return std::make_pair(Floor, 0);
202
203  // Round based on the next digit.
204  assert(LocalFloor >= 1);
205  bool Round = Digits & UINT64_C(1) << (LocalFloor - 1);
206  return std::make_pair(Floor + Round, Round ? 1 : -1);
207}
208
209/// \brief Get the lg (rounded) of a scaled number.
210///
211/// Get the lg of \c Digits*2^Scale.
212///
213/// Returns \c INT32_MIN when \c Digits is zero.
214template <class DigitsT> int32_t getLg(DigitsT Digits, int16_t Scale) {
215  return getLgImpl(Digits, Scale).first;
216}
217
218/// \brief Get the lg floor of a scaled number.
219///
220/// Get the floor of the lg of \c Digits*2^Scale.
221///
222/// Returns \c INT32_MIN when \c Digits is zero.
223template <class DigitsT> int32_t getLgFloor(DigitsT Digits, int16_t Scale) {
224  auto Lg = getLgImpl(Digits, Scale);
225  return Lg.first - (Lg.second > 0);
226}
227
228/// \brief Get the lg ceiling of a scaled number.
229///
230/// Get the ceiling of the lg of \c Digits*2^Scale.
231///
232/// Returns \c INT32_MIN when \c Digits is zero.
233template <class DigitsT> int32_t getLgCeiling(DigitsT Digits, int16_t Scale) {
234  auto Lg = getLgImpl(Digits, Scale);
235  return Lg.first + (Lg.second < 0);
236}
237
238/// \brief Implementation for comparing scaled numbers.
239///
240/// Compare two 64-bit numbers with different scales.  Given that the scale of
241/// \c L is higher than that of \c R by \c ScaleDiff, compare them.  Return -1,
242/// 1, and 0 for less than, greater than, and equal, respectively.
243///
244/// \pre 0 <= ScaleDiff < 64.
245int compareImpl(uint64_t L, uint64_t R, int ScaleDiff);
246
247/// \brief Compare two scaled numbers.
248///
249/// Compare two scaled numbers.  Returns 0 for equal, -1 for less than, and 1
250/// for greater than.
251template <class DigitsT>
252int compare(DigitsT LDigits, int16_t LScale, DigitsT RDigits, int16_t RScale) {
253  static_assert(!std::numeric_limits<DigitsT>::is_signed, "expected unsigned");
254
255  // Check for zero.
256  if (!LDigits)
257    return RDigits ? -1 : 0;
258  if (!RDigits)
259    return 1;
260
261  // Check for the scale.  Use getLgFloor to be sure that the scale difference
262  // is always lower than 64.
263  int32_t lgL = getLgFloor(LDigits, LScale), lgR = getLgFloor(RDigits, RScale);
264  if (lgL != lgR)
265    return lgL < lgR ? -1 : 1;
266
267  // Compare digits.
268  if (LScale < RScale)
269    return compareImpl(LDigits, RDigits, RScale - LScale);
270
271  return -compareImpl(RDigits, LDigits, LScale - RScale);
272}
273
274/// \brief Match scales of two numbers.
275///
276/// Given two scaled numbers, match up their scales.  Change the digits and
277/// scales in place.  Shift the digits as necessary to form equivalent numbers,
278/// losing precision only when necessary.
279///
280/// If the output value of \c LDigits (\c RDigits) is \c 0, the output value of
281/// \c LScale (\c RScale) is unspecified.
282///
283/// As a convenience, returns the matching scale.  If the output value of one
284/// number is zero, returns the scale of the other.  If both are zero, which
285/// scale is returned is unspecified.
286template <class DigitsT>
287int16_t matchScales(DigitsT &LDigits, int16_t &LScale, DigitsT &RDigits,
288                    int16_t &RScale) {
289  static_assert(!std::numeric_limits<DigitsT>::is_signed, "expected unsigned");
290
291  if (LScale < RScale)
292    // Swap arguments.
293    return matchScales(RDigits, RScale, LDigits, LScale);
294  if (!LDigits)
295    return RScale;
296  if (!RDigits || LScale == RScale)
297    return LScale;
298
299  // Now LScale > RScale.  Get the difference.
300  int32_t ScaleDiff = int32_t(LScale) - RScale;
301  if (ScaleDiff >= 2 * getWidth<DigitsT>()) {
302    // Don't bother shifting.  RDigits will get zero-ed out anyway.
303    RDigits = 0;
304    return LScale;
305  }
306
307  // Shift LDigits left as much as possible, then shift RDigits right.
308  int32_t ShiftL = std::min<int32_t>(countLeadingZeros(LDigits), ScaleDiff);
309  assert(ShiftL < getWidth<DigitsT>() && "can't shift more than width");
310
311  int32_t ShiftR = ScaleDiff - ShiftL;
312  if (ShiftR >= getWidth<DigitsT>()) {
313    // Don't bother shifting.  RDigits will get zero-ed out anyway.
314    RDigits = 0;
315    return LScale;
316  }
317
318  LDigits <<= ShiftL;
319  RDigits >>= ShiftR;
320
321  LScale -= ShiftL;
322  RScale += ShiftR;
323  assert(LScale == RScale && "scales should match");
324  return LScale;
325}
326
327/// \brief Get the sum of two scaled numbers.
328///
329/// Get the sum of two scaled numbers with as much precision as possible.
330///
331/// \pre Adding 1 to \c LScale (or \c RScale) will not overflow INT16_MAX.
332template <class DigitsT>
333std::pair<DigitsT, int16_t> getSum(DigitsT LDigits, int16_t LScale,
334                                   DigitsT RDigits, int16_t RScale) {
335  static_assert(!std::numeric_limits<DigitsT>::is_signed, "expected unsigned");
336
337  // Check inputs up front.  This is only relevant if addition overflows, but
338  // testing here should catch more bugs.
339  assert(LScale < INT16_MAX && "scale too large");
340  assert(RScale < INT16_MAX && "scale too large");
341
342  // Normalize digits to match scales.
343  int16_t Scale = matchScales(LDigits, LScale, RDigits, RScale);
344
345  // Compute sum.
346  DigitsT Sum = LDigits + RDigits;
347  if (Sum >= RDigits)
348    return std::make_pair(Sum, Scale);
349
350  // Adjust sum after arithmetic overflow.
351  DigitsT HighBit = DigitsT(1) << (getWidth<DigitsT>() - 1);
352  return std::make_pair(HighBit | Sum >> 1, Scale + 1);
353}
354
355/// \brief Convenience helper for 32-bit sum.
356inline std::pair<uint32_t, int16_t> getSum32(uint32_t LDigits, int16_t LScale,
357                                             uint32_t RDigits, int16_t RScale) {
358  return getSum(LDigits, LScale, RDigits, RScale);
359}
360
361/// \brief Convenience helper for 64-bit sum.
362inline std::pair<uint64_t, int16_t> getSum64(uint64_t LDigits, int16_t LScale,
363                                             uint64_t RDigits, int16_t RScale) {
364  return getSum(LDigits, LScale, RDigits, RScale);
365}
366
367/// \brief Get the difference of two scaled numbers.
368///
369/// Get LHS minus RHS with as much precision as possible.
370///
371/// Returns \c (0, 0) if the RHS is larger than the LHS.
372template <class DigitsT>
373std::pair<DigitsT, int16_t> getDifference(DigitsT LDigits, int16_t LScale,
374                                          DigitsT RDigits, int16_t RScale) {
375  static_assert(!std::numeric_limits<DigitsT>::is_signed, "expected unsigned");
376
377  // Normalize digits to match scales.
378  const DigitsT SavedRDigits = RDigits;
379  const int16_t SavedRScale = RScale;
380  matchScales(LDigits, LScale, RDigits, RScale);
381
382  // Compute difference.
383  if (LDigits <= RDigits)
384    return std::make_pair(0, 0);
385  if (RDigits || !SavedRDigits)
386    return std::make_pair(LDigits - RDigits, LScale);
387
388  // Check if RDigits just barely lost its last bit.  E.g., for 32-bit:
389  //
390  //   1*2^32 - 1*2^0 == 0xffffffff != 1*2^32
391  const auto RLgFloor = getLgFloor(SavedRDigits, SavedRScale);
392  if (!compare(LDigits, LScale, DigitsT(1), RLgFloor + getWidth<DigitsT>()))
393    return std::make_pair(std::numeric_limits<DigitsT>::max(), RLgFloor);
394
395  return std::make_pair(LDigits, LScale);
396}
397
398/// \brief Convenience helper for 32-bit difference.
399inline std::pair<uint32_t, int16_t> getDifference32(uint32_t LDigits,
400                                                    int16_t LScale,
401                                                    uint32_t RDigits,
402                                                    int16_t RScale) {
403  return getDifference(LDigits, LScale, RDigits, RScale);
404}
405
406/// \brief Convenience helper for 64-bit difference.
407inline std::pair<uint64_t, int16_t> getDifference64(uint64_t LDigits,
408                                                    int16_t LScale,
409                                                    uint64_t RDigits,
410                                                    int16_t RScale) {
411  return getDifference(LDigits, LScale, RDigits, RScale);
412}
413
414} // end namespace ScaledNumbers
415} // end namespace llvm
416
417namespace llvm {
418
419class raw_ostream;
420class ScaledNumberBase {
421public:
422  static const int DefaultPrecision = 10;
423
424  static void dump(uint64_t D, int16_t E, int Width);
425  static raw_ostream &print(raw_ostream &OS, uint64_t D, int16_t E, int Width,
426                            unsigned Precision);
427  static std::string toString(uint64_t D, int16_t E, int Width,
428                              unsigned Precision);
429  static int countLeadingZeros32(uint32_t N) { return countLeadingZeros(N); }
430  static int countLeadingZeros64(uint64_t N) { return countLeadingZeros(N); }
431  static uint64_t getHalf(uint64_t N) { return (N >> 1) + (N & 1); }
432
433  static std::pair<uint64_t, bool> splitSigned(int64_t N) {
434    if (N >= 0)
435      return std::make_pair(N, false);
436    uint64_t Unsigned = N == INT64_MIN ? UINT64_C(1) << 63 : uint64_t(-N);
437    return std::make_pair(Unsigned, true);
438  }
439  static int64_t joinSigned(uint64_t U, bool IsNeg) {
440    if (U > uint64_t(INT64_MAX))
441      return IsNeg ? INT64_MIN : INT64_MAX;
442    return IsNeg ? -int64_t(U) : int64_t(U);
443  }
444};
445
446/// \brief Simple representation of a scaled number.
447///
448/// ScaledNumber is a number represented by digits and a scale.  It uses simple
449/// saturation arithmetic and every operation is well-defined for every value.
450/// It's somewhat similar in behaviour to a soft-float, but is *not* a
451/// replacement for one.  If you're doing numerics, look at \a APFloat instead.
452/// Nevertheless, we've found these semantics useful for modelling certain cost
453/// metrics.
454///
455/// The number is split into a signed scale and unsigned digits.  The number
456/// represented is \c getDigits()*2^getScale().  In this way, the digits are
457/// much like the mantissa in the x87 long double, but there is no canonical
458/// form so the same number can be represented by many bit representations.
459///
460/// ScaledNumber is templated on the underlying integer type for digits, which
461/// is expected to be unsigned.
462///
463/// Unlike APFloat, ScaledNumber does not model architecture floating point
464/// behaviour -- while this might make it a little faster and easier to reason
465/// about, it certainly makes it more dangerous for general numerics.
466///
467/// ScaledNumber is totally ordered.  However, there is no canonical form, so
468/// there are multiple representations of most scalars.  E.g.:
469///
470///     ScaledNumber(8u, 0) == ScaledNumber(4u, 1)
471///     ScaledNumber(4u, 1) == ScaledNumber(2u, 2)
472///     ScaledNumber(2u, 2) == ScaledNumber(1u, 3)
473///
474/// ScaledNumber implements most arithmetic operations.  Precision is kept
475/// where possible.  Uses simple saturation arithmetic, so that operations
476/// saturate to 0.0 or getLargest() rather than under or overflowing.  It has
477/// some extra arithmetic for unit inversion.  0.0/0.0 is defined to be 0.0.
478/// Any other division by 0.0 is defined to be getLargest().
479///
480/// As a convenience for modifying the exponent, left and right shifting are
481/// both implemented, and both interpret negative shifts as positive shifts in
482/// the opposite direction.
483///
484/// Scales are limited to the range accepted by x87 long double.  This makes
485/// it trivial to add functionality to convert to APFloat (this is already
486/// relied on for the implementation of printing).
487///
488/// Possible (and conflicting) future directions:
489///
490///  1. Turn this into a wrapper around \a APFloat.
491///  2. Share the algorithm implementations with \a APFloat.
492///  3. Allow \a ScaledNumber to represent a signed number.
493template <class DigitsT> class ScaledNumber : ScaledNumberBase {
494public:
495  static_assert(!std::numeric_limits<DigitsT>::is_signed,
496                "only unsigned floats supported");
497
498  typedef DigitsT DigitsType;
499
500private:
501  typedef std::numeric_limits<DigitsType> DigitsLimits;
502
503  static const int Width = sizeof(DigitsType) * 8;
504  static_assert(Width <= 64, "invalid integer width for digits");
505
506private:
507  DigitsType Digits;
508  int16_t Scale;
509
510public:
511  ScaledNumber() : Digits(0), Scale(0) {}
512
513  ScaledNumber(DigitsType Digits, int16_t Scale)
514      : Digits(Digits), Scale(Scale) {}
515
516private:
517  ScaledNumber(const std::pair<DigitsT, int16_t> &X)
518      : Digits(X.first), Scale(X.second) {}
519
520public:
521  static ScaledNumber getZero() { return ScaledNumber(0, 0); }
522  static ScaledNumber getOne() { return ScaledNumber(1, 0); }
523  static ScaledNumber getLargest() {
524    return ScaledNumber(DigitsLimits::max(), ScaledNumbers::MaxScale);
525  }
526  static ScaledNumber get(uint64_t N) { return adjustToWidth(N, 0); }
527  static ScaledNumber getInverse(uint64_t N) {
528    return get(N).invert();
529  }
530  static ScaledNumber getFraction(DigitsType N, DigitsType D) {
531    return getQuotient(N, D);
532  }
533
534  int16_t getScale() const { return Scale; }
535  DigitsType getDigits() const { return Digits; }
536
537  /// \brief Convert to the given integer type.
538  ///
539  /// Convert to \c IntT using simple saturating arithmetic, truncating if
540  /// necessary.
541  template <class IntT> IntT toInt() const;
542
543  bool isZero() const { return !Digits; }
544  bool isLargest() const { return *this == getLargest(); }
545  bool isOne() const {
546    if (Scale > 0 || Scale <= -Width)
547      return false;
548    return Digits == DigitsType(1) << -Scale;
549  }
550
551  /// \brief The log base 2, rounded.
552  ///
553  /// Get the lg of the scalar.  lg 0 is defined to be INT32_MIN.
554  int32_t lg() const { return ScaledNumbers::getLg(Digits, Scale); }
555
556  /// \brief The log base 2, rounded towards INT32_MIN.
557  ///
558  /// Get the lg floor.  lg 0 is defined to be INT32_MIN.
559  int32_t lgFloor() const { return ScaledNumbers::getLgFloor(Digits, Scale); }
560
561  /// \brief The log base 2, rounded towards INT32_MAX.
562  ///
563  /// Get the lg ceiling.  lg 0 is defined to be INT32_MIN.
564  int32_t lgCeiling() const {
565    return ScaledNumbers::getLgCeiling(Digits, Scale);
566  }
567
568  bool operator==(const ScaledNumber &X) const { return compare(X) == 0; }
569  bool operator<(const ScaledNumber &X) const { return compare(X) < 0; }
570  bool operator!=(const ScaledNumber &X) const { return compare(X) != 0; }
571  bool operator>(const ScaledNumber &X) const { return compare(X) > 0; }
572  bool operator<=(const ScaledNumber &X) const { return compare(X) <= 0; }
573  bool operator>=(const ScaledNumber &X) const { return compare(X) >= 0; }
574
575  bool operator!() const { return isZero(); }
576
577  /// \brief Convert to a decimal representation in a string.
578  ///
579  /// Convert to a string.  Uses scientific notation for very large/small
580  /// numbers.  Scientific notation is used roughly for numbers outside of the
581  /// range 2^-64 through 2^64.
582  ///
583  /// \c Precision indicates the number of decimal digits of precision to use;
584  /// 0 requests the maximum available.
585  ///
586  /// As a special case to make debugging easier, if the number is small enough
587  /// to convert without scientific notation and has more than \c Precision
588  /// digits before the decimal place, it's printed accurately to the first
589  /// digit past zero.  E.g., assuming 10 digits of precision:
590  ///
591  ///     98765432198.7654... => 98765432198.8
592  ///      8765432198.7654... =>  8765432198.8
593  ///       765432198.7654... =>   765432198.8
594  ///        65432198.7654... =>    65432198.77
595  ///         5432198.7654... =>     5432198.765
596  std::string toString(unsigned Precision = DefaultPrecision) {
597    return ScaledNumberBase::toString(Digits, Scale, Width, Precision);
598  }
599
600  /// \brief Print a decimal representation.
601  ///
602  /// Print a string.  See toString for documentation.
603  raw_ostream &print(raw_ostream &OS,
604                     unsigned Precision = DefaultPrecision) const {
605    return ScaledNumberBase::print(OS, Digits, Scale, Width, Precision);
606  }
607  void dump() const { return ScaledNumberBase::dump(Digits, Scale, Width); }
608
609  ScaledNumber &operator+=(const ScaledNumber &X) {
610    std::tie(Digits, Scale) =
611        ScaledNumbers::getSum(Digits, Scale, X.Digits, X.Scale);
612    // Check for exponent past MaxScale.
613    if (Scale > ScaledNumbers::MaxScale)
614      *this = getLargest();
615    return *this;
616  }
617  ScaledNumber &operator-=(const ScaledNumber &X) {
618    std::tie(Digits, Scale) =
619        ScaledNumbers::getDifference(Digits, Scale, X.Digits, X.Scale);
620    return *this;
621  }
622  ScaledNumber &operator*=(const ScaledNumber &X);
623  ScaledNumber &operator/=(const ScaledNumber &X);
624  ScaledNumber &operator<<=(int16_t Shift) {
625    shiftLeft(Shift);
626    return *this;
627  }
628  ScaledNumber &operator>>=(int16_t Shift) {
629    shiftRight(Shift);
630    return *this;
631  }
632
633private:
634  void shiftLeft(int32_t Shift);
635  void shiftRight(int32_t Shift);
636
637  /// \brief Adjust two floats to have matching exponents.
638  ///
639  /// Adjust \c this and \c X to have matching exponents.  Returns the new \c X
640  /// by value.  Does nothing if \a isZero() for either.
641  ///
642  /// The value that compares smaller will lose precision, and possibly become
643  /// \a isZero().
644  ScaledNumber matchScales(ScaledNumber X) {
645    ScaledNumbers::matchScales(Digits, Scale, X.Digits, X.Scale);
646    return X;
647  }
648
649public:
650  /// \brief Scale a large number accurately.
651  ///
652  /// Scale N (multiply it by this).  Uses full precision multiplication, even
653  /// if Width is smaller than 64, so information is not lost.
654  uint64_t scale(uint64_t N) const;
655  uint64_t scaleByInverse(uint64_t N) const {
656    // TODO: implement directly, rather than relying on inverse.  Inverse is
657    // expensive.
658    return inverse().scale(N);
659  }
660  int64_t scale(int64_t N) const {
661    std::pair<uint64_t, bool> Unsigned = splitSigned(N);
662    return joinSigned(scale(Unsigned.first), Unsigned.second);
663  }
664  int64_t scaleByInverse(int64_t N) const {
665    std::pair<uint64_t, bool> Unsigned = splitSigned(N);
666    return joinSigned(scaleByInverse(Unsigned.first), Unsigned.second);
667  }
668
669  int compare(const ScaledNumber &X) const {
670    return ScaledNumbers::compare(Digits, Scale, X.Digits, X.Scale);
671  }
672  int compareTo(uint64_t N) const {
673    return ScaledNumbers::compare<uint64_t>(Digits, Scale, N, 0);
674  }
675  int compareTo(int64_t N) const { return N < 0 ? 1 : compareTo(uint64_t(N)); }
676
677  ScaledNumber &invert() { return *this = ScaledNumber::get(1) / *this; }
678  ScaledNumber inverse() const { return ScaledNumber(*this).invert(); }
679
680private:
681  static ScaledNumber getProduct(DigitsType LHS, DigitsType RHS) {
682    return ScaledNumbers::getProduct(LHS, RHS);
683  }
684  static ScaledNumber getQuotient(DigitsType Dividend, DigitsType Divisor) {
685    return ScaledNumbers::getQuotient(Dividend, Divisor);
686  }
687
688  static int countLeadingZerosWidth(DigitsType Digits) {
689    if (Width == 64)
690      return countLeadingZeros64(Digits);
691    if (Width == 32)
692      return countLeadingZeros32(Digits);
693    return countLeadingZeros32(Digits) + Width - 32;
694  }
695
696  /// \brief Adjust a number to width, rounding up if necessary.
697  ///
698  /// Should only be called for \c Shift close to zero.
699  ///
700  /// \pre Shift >= MinScale && Shift + 64 <= MaxScale.
701  static ScaledNumber adjustToWidth(uint64_t N, int32_t Shift) {
702    assert(Shift >= ScaledNumbers::MinScale && "Shift should be close to 0");
703    assert(Shift <= ScaledNumbers::MaxScale - 64 &&
704           "Shift should be close to 0");
705    auto Adjusted = ScaledNumbers::getAdjusted<DigitsT>(N, Shift);
706    return Adjusted;
707  }
708
709  static ScaledNumber getRounded(ScaledNumber P, bool Round) {
710    // Saturate.
711    if (P.isLargest())
712      return P;
713
714    return ScaledNumbers::getRounded(P.Digits, P.Scale, Round);
715  }
716};
717
718#define SCALED_NUMBER_BOP(op, base)                                            \
719  template <class DigitsT>                                                     \
720  ScaledNumber<DigitsT> operator op(const ScaledNumber<DigitsT> &L,            \
721                                    const ScaledNumber<DigitsT> &R) {          \
722    return ScaledNumber<DigitsT>(L) base R;                                    \
723  }
724SCALED_NUMBER_BOP(+, += )
725SCALED_NUMBER_BOP(-, -= )
726SCALED_NUMBER_BOP(*, *= )
727SCALED_NUMBER_BOP(/, /= )
728#undef SCALED_NUMBER_BOP
729
730template <class DigitsT>
731ScaledNumber<DigitsT> operator<<(const ScaledNumber<DigitsT> &L,
732                                 int16_t Shift) {
733  return ScaledNumber<DigitsT>(L) <<= Shift;
734}
735
736template <class DigitsT>
737ScaledNumber<DigitsT> operator>>(const ScaledNumber<DigitsT> &L,
738                                 int16_t Shift) {
739  return ScaledNumber<DigitsT>(L) >>= Shift;
740}
741
742template <class DigitsT>
743raw_ostream &operator<<(raw_ostream &OS, const ScaledNumber<DigitsT> &X) {
744  return X.print(OS, 10);
745}
746
747#define SCALED_NUMBER_COMPARE_TO_TYPE(op, T1, T2)                              \
748  template <class DigitsT>                                                     \
749  bool operator op(const ScaledNumber<DigitsT> &L, T1 R) {                     \
750    return L.compareTo(T2(R)) op 0;                                            \
751  }                                                                            \
752  template <class DigitsT>                                                     \
753  bool operator op(T1 L, const ScaledNumber<DigitsT> &R) {                     \
754    return 0 op R.compareTo(T2(L));                                            \
755  }
756#define SCALED_NUMBER_COMPARE_TO(op)                                           \
757  SCALED_NUMBER_COMPARE_TO_TYPE(op, uint64_t, uint64_t)                        \
758  SCALED_NUMBER_COMPARE_TO_TYPE(op, uint32_t, uint64_t)                        \
759  SCALED_NUMBER_COMPARE_TO_TYPE(op, int64_t, int64_t)                          \
760  SCALED_NUMBER_COMPARE_TO_TYPE(op, int32_t, int64_t)
761SCALED_NUMBER_COMPARE_TO(< )
762SCALED_NUMBER_COMPARE_TO(> )
763SCALED_NUMBER_COMPARE_TO(== )
764SCALED_NUMBER_COMPARE_TO(!= )
765SCALED_NUMBER_COMPARE_TO(<= )
766SCALED_NUMBER_COMPARE_TO(>= )
767#undef SCALED_NUMBER_COMPARE_TO
768#undef SCALED_NUMBER_COMPARE_TO_TYPE
769
770template <class DigitsT>
771uint64_t ScaledNumber<DigitsT>::scale(uint64_t N) const {
772  if (Width == 64 || N <= DigitsLimits::max())
773    return (get(N) * *this).template toInt<uint64_t>();
774
775  // Defer to the 64-bit version.
776  return ScaledNumber<uint64_t>(Digits, Scale).scale(N);
777}
778
779template <class DigitsT>
780template <class IntT>
781IntT ScaledNumber<DigitsT>::toInt() const {
782  typedef std::numeric_limits<IntT> Limits;
783  if (*this < 1)
784    return 0;
785  if (*this >= Limits::max())
786    return Limits::max();
787
788  IntT N = Digits;
789  if (Scale > 0) {
790    assert(size_t(Scale) < sizeof(IntT) * 8);
791    return N << Scale;
792  }
793  if (Scale < 0) {
794    assert(size_t(-Scale) < sizeof(IntT) * 8);
795    return N >> -Scale;
796  }
797  return N;
798}
799
800template <class DigitsT>
801ScaledNumber<DigitsT> &ScaledNumber<DigitsT>::
802operator*=(const ScaledNumber &X) {
803  if (isZero())
804    return *this;
805  if (X.isZero())
806    return *this = X;
807
808  // Save the exponents.
809  int32_t Scales = int32_t(Scale) + int32_t(X.Scale);
810
811  // Get the raw product.
812  *this = getProduct(Digits, X.Digits);
813
814  // Combine with exponents.
815  return *this <<= Scales;
816}
817template <class DigitsT>
818ScaledNumber<DigitsT> &ScaledNumber<DigitsT>::
819operator/=(const ScaledNumber &X) {
820  if (isZero())
821    return *this;
822  if (X.isZero())
823    return *this = getLargest();
824
825  // Save the exponents.
826  int32_t Scales = int32_t(Scale) - int32_t(X.Scale);
827
828  // Get the raw quotient.
829  *this = getQuotient(Digits, X.Digits);
830
831  // Combine with exponents.
832  return *this <<= Scales;
833}
834template <class DigitsT> void ScaledNumber<DigitsT>::shiftLeft(int32_t Shift) {
835  if (!Shift || isZero())
836    return;
837  assert(Shift != INT32_MIN);
838  if (Shift < 0) {
839    shiftRight(-Shift);
840    return;
841  }
842
843  // Shift as much as we can in the exponent.
844  int32_t ScaleShift = std::min(Shift, ScaledNumbers::MaxScale - Scale);
845  Scale += ScaleShift;
846  if (ScaleShift == Shift)
847    return;
848
849  // Check this late, since it's rare.
850  if (isLargest())
851    return;
852
853  // Shift the digits themselves.
854  Shift -= ScaleShift;
855  if (Shift > countLeadingZerosWidth(Digits)) {
856    // Saturate.
857    *this = getLargest();
858    return;
859  }
860
861  Digits <<= Shift;
862  return;
863}
864
865template <class DigitsT> void ScaledNumber<DigitsT>::shiftRight(int32_t Shift) {
866  if (!Shift || isZero())
867    return;
868  assert(Shift != INT32_MIN);
869  if (Shift < 0) {
870    shiftLeft(-Shift);
871    return;
872  }
873
874  // Shift as much as we can in the exponent.
875  int32_t ScaleShift = std::min(Shift, Scale - ScaledNumbers::MinScale);
876  Scale -= ScaleShift;
877  if (ScaleShift == Shift)
878    return;
879
880  // Shift the digits themselves.
881  Shift -= ScaleShift;
882  if (Shift >= Width) {
883    // Saturate.
884    *this = getZero();
885    return;
886  }
887
888  Digits >>= Shift;
889  return;
890}
891
892template <typename T> struct isPodLike;
893template <typename T> struct isPodLike<ScaledNumber<T>> {
894  static const bool value = true;
895};
896
897} // end namespace llvm
898
899#endif
900