src/fast-dtoa.cc

// Copyright 2010 the V8 project authors. All rights reserved.
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#include "v8.h"

#include "fast-dtoa.h"

#include "cached-powers.h"
#include "diy-fp.h"
#include "double.h"

namespace v8 {
namespace internal {

// The minimal and maximal target exponent define the range of w's binary
// exponent, where 'w' is the result of multiplying the input by a cached power
// of ten.
//
// A different range might be chosen on a different platform, to optimize digit
// generation, but a smaller range requires more powers of ten to be cached.
static const int minimal_target_exponent = -60;
static const int maximal_target_exponent = -32;


// Adjusts the last digit of the generated number, and screens out generated
// solutions that may be inaccurate. A solution may be inaccurate if it is
// outside the safe interval, or if we ctannot prove that it is closer to the
// input than a neighboring representation of the same length.
//
// Input: * buffer containing the digits of too_high / 10^kappa
//        * the buffer's length
//        * distance_too_high_w == (too_high - w).f() * unit
//        * unsafe_interval == (too_high - too_low).f() * unit
//        * rest = (too_high - buffer * 10^kappa).f() * unit
//        * ten_kappa = 10^kappa * unit
//        * unit = the common multiplier
// Output: returns true if the buffer is guaranteed to contain the closest
//    representable number to the input.
//  Modifies the generated digits in the buffer to approach (round towards) w.
bool RoundWeed(Vector<char> buffer,
               int length,
               uint64_t distance_too_high_w,
               uint64_t unsafe_interval,
               uint64_t rest,
               uint64_t ten_kappa,
               uint64_t unit) {
  uint64_t small_distance = distance_too_high_w - unit;
  uint64_t big_distance = distance_too_high_w + unit;
  // Let w_low  = too_high - big_distance, and
  //     w_high = too_high - small_distance.
  // Note: w_low < w < w_high
  //
  // The real w (* unit) must lie somewhere inside the interval
  // ]w_low; w_low[ (often written as "(w_low; w_low)")

  // Basically the buffer currently contains a number in the unsafe interval
  // ]too_low; too_high[ with too_low < w < too_high
  //
  //  too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  //                     ^v 1 unit            ^      ^                 ^      ^
  //  boundary_high ---------------------     .      .                 .      .
  //                     ^v 1 unit            .      .                 .      .
  //   - - - - - - - - - - - - - - - - - - -  +  - - + - - - - - -     .      .
  //                                          .      .         ^       .      .
  //                                          .  big_distance  .       .      .
  //                                          .      .         .       .    rest
  //                              small_distance     .         .       .      .
  //                                          v      .         .       .      .
  //  w_high - - - - - - - - - - - - - - - - - -     .         .       .      .
  //                     ^v 1 unit                   .         .       .      .
  //  w ----------------------------------------     .         .       .      .
  //                     ^v 1 unit                   v         .       .      .
  //  w_low  - - - - - - - - - - - - - - - - - - - - -         .       .      .
  //                                                           .       .      v
  //  buffer --------------------------------------------------+-------+--------
  //                                                           .       .
  //                                                  safe_interval    .
  //                                                           v       .
  //   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -     .
  //                     ^v 1 unit                                     .
  //  boundary_low -------------------------                     unsafe_interval
  //                     ^v 1 unit                                     v
  //  too_low  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  //
  //
  // Note that the value of buffer could lie anywhere inside the range too_low
  // to too_high.
  //
  // boundary_low, boundary_high and w are approximations of the real boundaries
  // and v (the input number). They are guaranteed to be precise up to one unit.
  // In fact the error is guaranteed to be strictly less than one unit.
  //
  // Anything that lies outside the unsafe interval is guaranteed not to round
  // to v when read again.
  // Anything that lies inside the safe interval is guaranteed to round to v
  // when read again.
  // If the number inside the buffer lies inside the unsafe interval but not
  // inside the safe interval then we simply do not know and bail out (returning
  // false).
  //
  // Similarly we have to take into account the imprecision of 'w' when rounding
  // the buffer. If we have two potential representations we need to make sure
  // that the chosen one is closer to w_low and w_high since v can be anywhere
  // between them.
  //
  // By generating the digits of too_high we got the largest (closest to
  // too_high) buffer that is still in the unsafe interval. In the case where
  // w_high < buffer < too_high we try to decrement the buffer.
  // This way the buffer approaches (rounds towards) w.
  // There are 3 conditions that stop the decrementation process:
  //   1) the buffer is already below w_high
  //   2) decrementing the buffer would make it leave the unsafe interval
  //   3) decrementing the buffer would yield a number below w_high and farther
  //      away than the current number. In other words:
  //              (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high
  // Instead of using the buffer directly we use its distance to too_high.
  // Conceptually rest ~= too_high - buffer
  while (rest < small_distance &&  // Negated condition 1
         unsafe_interval - rest >= ten_kappa &&  // Negated condition 2
         (rest + ten_kappa < small_distance ||  // buffer{-1} > w_high
          small_distance - rest >= rest + ten_kappa - small_distance)) {
    buffer[length - 1]--;
    rest += ten_kappa;
  }

  // We have approached w+ as much as possible. We now test if approaching w-
  // would require changing the buffer. If yes, then we have two possible
  // representations close to w, but we cannot decide which one is closer.
  if (rest < big_distance &&
      unsafe_interval - rest >= ten_kappa &&
      (rest + ten_kappa < big_distance ||
       big_distance - rest > rest + ten_kappa - big_distance)) {
    return false;
  }

  // Weeding test.
  //   The safe interval is [too_low + 2 ulp; too_high - 2 ulp]
  //   Since too_low = too_high - unsafe_interval this is equivalent to
  //      [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp]
  //   Conceptually we have: rest ~= too_high - buffer
  return (2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit);
}



static const uint32_t kTen4 = 10000;
static const uint32_t kTen5 = 100000;
static const uint32_t kTen6 = 1000000;
static const uint32_t kTen7 = 10000000;
static const uint32_t kTen8 = 100000000;
static const uint32_t kTen9 = 1000000000;

// Returns the biggest power of ten that is less than or equal than the given
// number. We furthermore receive the maximum number of bits 'number' has.
// If number_bits == 0 then 0^-1 is returned
// The number of bits must be <= 32.
// Precondition: (1 << number_bits) <= number < (1 << (number_bits + 1)).
static void BiggestPowerTen(uint32_t number,
                            int number_bits,
                            uint32_t* power,
                            int* exponent) {
  switch (number_bits) {
    case 32:
    case 31:
    case 30:
      if (kTen9 <= number) {
        *power = kTen9;
        *exponent = 9;
        break;
      }  // else fallthrough
    case 29:
    case 28:
    case 27:
      if (kTen8 <= number) {
        *power = kTen8;
        *exponent = 8;
        break;
      }  // else fallthrough
    case 26:
    case 25:
    case 24:
      if (kTen7 <= number) {
        *power = kTen7;
        *exponent = 7;
        break;
      }  // else fallthrough
    case 23:
    case 22:
    case 21:
    case 20:
      if (kTen6 <= number) {
        *power = kTen6;
        *exponent = 6;
        break;
      }  // else fallthrough
    case 19:
    case 18:
    case 17:
      if (kTen5 <= number) {
        *power = kTen5;
        *exponent = 5;
        break;
      }  // else fallthrough
    case 16:
    case 15:
    case 14:
      if (kTen4 <= number) {
        *power = kTen4;
        *exponent = 4;
        break;
      }  // else fallthrough
    case 13:
    case 12:
    case 11:
    case 10:
      if (1000 <= number) {
        *power = 1000;
        *exponent = 3;
        break;
      }  // else fallthrough
    case 9:
    case 8:
    case 7:
      if (100 <= number) {
        *power = 100;
        *exponent = 2;
        break;
      }  // else fallthrough
    case 6:
    case 5:
    case 4:
      if (10 <= number) {
        *power = 10;
        *exponent = 1;
        break;
      }  // else fallthrough
    case 3:
    case 2:
    case 1:
      if (1 <= number) {
        *power = 1;
        *exponent = 0;
        break;
      }  // else fallthrough
    case 0:
      *power = 0;
      *exponent = -1;
      break;
    default:
      // Following assignments are here to silence compiler warnings.
      *power = 0;
      *exponent = 0;
      UNREACHABLE();
  }
}


// Generates the digits of input number w.
// w is a floating-point number (DiyFp), consisting of a significand and an
// exponent. Its exponent is bounded by minimal_target_exponent and
// maximal_target_exponent.
//       Hence -60 <= w.e() <= -32.
//
// Returns false if it fails, in which case the generated digits in the buffer
// should not be used.
// Preconditions:
//  * low, w and high are correct up to 1 ulp (unit in the last place). That
//    is, their error must be less that a unit of their last digits.
//  * low.e() == w.e() == high.e()
//  * low < w < high, and taking into account their error: low~ <= high~
//  * minimal_target_exponent <= w.e() <= maximal_target_exponent
// Postconditions: returns false if procedure fails.
//   otherwise:
//     * buffer is not null-terminated, but len contains the number of digits.
//     * buffer contains the shortest possible decimal digit-sequence
//       such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the
//       correct values of low and high (without their error).
//     * if more than one decimal representation gives the minimal number of
//       decimal digits then the one closest to W (where W is the correct value
//       of w) is chosen.
// Remark: this procedure takes into account the imprecision of its input
//   numbers. If the precision is not enough to guarantee all the postconditions
//   then false is returned. This usually happens rarely (~0.5%).
//
// Say, for the sake of example, that
//   w.e() == -48, and w.f() == 0x1234567890abcdef
// w's value can be computed by w.f() * 2^w.e()
// We can obtain w's integral digits by simply shifting w.f() by -w.e().
//  -> w's integral part is 0x1234
//  w's fractional part is therefore 0x567890abcdef.
// Printing w's integral part is easy (simply print 0x1234 in decimal).
// In order to print its fraction we repeatedly multiply the fraction by 10 and
// get each digit. Example the first digit after the point would be computed by
//   (0x567890abcdef * 10) >> 48. -> 3
// The whole thing becomes slightly more complicated because we want to stop
// once we have enough digits. That is, once the digits inside the buffer
// represent 'w' we can stop. Everything inside the interval low - high
// represents w. However we have to pay attention to low, high and w's
// imprecision.
bool DigitGen(DiyFp low,
              DiyFp w,
              DiyFp high,
              Vector<char> buffer,
              int* length,
              int* kappa) {
  ASSERT(low.e() == w.e() && w.e() == high.e());
  ASSERT(low.f() + 1 <= high.f() - 1);
  ASSERT(minimal_target_exponent <= w.e() && w.e() <= maximal_target_exponent);
  // low, w and high are imprecise, but by less than one ulp (unit in the last
  // place).
  // If we remove (resp. add) 1 ulp from low (resp. high) we are certain that
  // the new numbers are outside of the interval we want the final
  // representation to lie in.
  // Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield
  // numbers that are certain to lie in the interval. We will use this fact
  // later on.
  // We will now start by generating the digits within the uncertain
  // interval. Later we will weed out representations that lie outside the safe
  // interval and thus _might_ lie outside the correct interval.
  uint64_t unit = 1;
  DiyFp too_low = DiyFp(low.f() - unit, low.e());
  DiyFp too_high = DiyFp(high.f() + unit, high.e());
  // too_low and too_high are guaranteed to lie outside the interval we want the
  // generated number in.
  DiyFp unsafe_interval = DiyFp::Minus(too_high, too_low);
  // We now cut the input number into two parts: the integral digits and the
  // fractionals. We will not write any decimal separator though, but adapt
  // kappa instead.
  // Reminder: we are currently computing the digits (stored inside the buffer)
  // such that:   too_low < buffer * 10^kappa < too_high
  // We use too_high for the digit_generation and stop as soon as possible.
  // If we stop early we effectively round down.
  DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e());
  // Division by one is a shift.
  uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e());
  // Modulo by one is an and.
  uint64_t fractionals = too_high.f() & (one.f() - 1);
  uint32_t divider;
  int divider_exponent;
  BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()),
                  &divider, &divider_exponent);
  *kappa = divider_exponent + 1;
  *length = 0;
  // Loop invariant: buffer = too_high / 10^kappa  (integer division)
  // The invariant holds for the first iteration: kappa has been initialized
  // with the divider exponent + 1. And the divider is the biggest power of ten
  // that is smaller than integrals.
  while (*kappa > 0) {
    int digit = integrals / divider;
    buffer[*length] = '0' + digit;
    (*length)++;
    integrals %= divider;
    (*kappa)--;
    // Note that kappa now equals the exponent of the divider and that the
    // invariant thus holds again.
    uint64_t rest =
        (static_cast<uint64_t>(integrals) << -one.e()) + fractionals;
    // Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())
    // Reminder: unsafe_interval.e() == one.e()
    if (rest < unsafe_interval.f()) {
      // Rounding down (by not emitting the remaining digits) yields a number
      // that lies within the unsafe interval.
      return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(),
                       unsafe_interval.f(), rest,
                       static_cast<uint64_t>(divider) << -one.e(), unit);
    }
    divider /= 10;
  }

  // The integrals have been generated. We are at the point of the decimal
  // separator. In the following loop we simply multiply the remaining digits by
  // 10 and divide by one. We just need to pay attention to multiply associated
  // data (like the interval or 'unit'), too.
  // Instead of multiplying by 10 we multiply by 5 (cheaper operation) and
  // increase its (imaginary) exponent. At the same time we decrease the
  // divider's (one's) exponent and shift its significand.
  // Basically, if fractionals was a DiyFp (with fractionals.e == one.e):
  //      fractionals.f *= 10;
  //      fractionals.f >>= 1; fractionals.e++; // value remains unchanged.
  //      one.f >>= 1; one.e++;                 // value remains unchanged.
  //      and we have again fractionals.e == one.e which allows us to divide
  //           fractionals.f() by one.f()
  // We simply combine the *= 10 and the >>= 1.
  while (true) {
    fractionals *= 5;
    unit *= 5;
    unsafe_interval.set_f(unsafe_interval.f() * 5);
    unsafe_interval.set_e(unsafe_interval.e() + 1);  // Will be optimized out.
    one.set_f(one.f() >> 1);
    one.set_e(one.e() + 1);
    // Integer division by one.
    int digit = static_cast<int>(fractionals >> -one.e());
    buffer[*length] = '0' + digit;
    (*length)++;
    fractionals &= one.f() - 1;  // Modulo by one.
    (*kappa)--;
    if (fractionals < unsafe_interval.f()) {
      return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f() * unit,
                       unsafe_interval.f(), fractionals, one.f(), unit);
    }
  }
}


// Provides a decimal representation of v.
// Returns true if it succeeds, otherwise the result cannot be trusted.
// There will be *length digits inside the buffer (not null-terminated).
// If the function returns true then
//        v == (double) (buffer * 10^decimal_exponent).
// The digits in the buffer are the shortest representation possible: no
// 0.09999999999999999 instead of 0.1. The shorter representation will even be
// chosen even if the longer one would be closer to v.
// The last digit will be closest to the actual v. That is, even if several
// digits might correctly yield 'v' when read again, the closest will be
// computed.
bool grisu3(double v, Vector<char> buffer, int* length, int* decimal_exponent) {
  DiyFp w = Double(v).AsNormalizedDiyFp();
  // boundary_minus and boundary_plus are the boundaries between v and its
  // closest floating-point neighbors. Any number strictly between
  // boundary_minus and boundary_plus will round to v when convert to a double.
  // Grisu3 will never output representations that lie exactly on a boundary.
  DiyFp boundary_minus, boundary_plus;
  Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus);
  ASSERT(boundary_plus.e() == w.e());
  DiyFp ten_mk;  // Cached power of ten: 10^-k
  int mk;        // -k
  GetCachedPower(w.e() + DiyFp::kSignificandSize, minimal_target_exponent,
                 maximal_target_exponent, &mk, &ten_mk);
  ASSERT(minimal_target_exponent <= w.e() + ten_mk.e() +
         DiyFp::kSignificandSize &&
         maximal_target_exponent >= w.e() + ten_mk.e() +
         DiyFp::kSignificandSize);
  // Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
  // 64 bit significand and ten_mk is thus only precise up to 64 bits.

  // The DiyFp::Times procedure rounds its result, and ten_mk is approximated
  // too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
  // off by a small amount.
  // In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
  // In other words: let f = scaled_w.f() and e = scaled_w.e(), then
  //           (f-1) * 2^e < w*10^k < (f+1) * 2^e
  DiyFp scaled_w = DiyFp::Times(w, ten_mk);
  ASSERT(scaled_w.e() ==
         boundary_plus.e() + ten_mk.e() + DiyFp::kSignificandSize);
  // In theory it would be possible to avoid some recomputations by computing
  // the difference between w and boundary_minus/plus (a power of 2) and to
  // compute scaled_boundary_minus/plus by subtracting/adding from
  // scaled_w. However the code becomes much less readable and the speed
  // enhancements are not terriffic.
  DiyFp scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk);
  DiyFp scaled_boundary_plus  = DiyFp::Times(boundary_plus,  ten_mk);

  // DigitGen will generate the digits of scaled_w. Therefore we have
  // v == (double) (scaled_w * 10^-mk).
  // Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an
  // integer than it will be updated. For instance if scaled_w == 1.23 then
  // the buffer will be filled with "123" und the decimal_exponent will be
  // decreased by 2.
  int kappa;
  bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus,
                         buffer, length, &kappa);
  *decimal_exponent = -mk + kappa;
  return result;
}


bool FastDtoa(double v,
              Vector<char> buffer,
              int* length,
              int* point) {
  ASSERT(v > 0);
  ASSERT(!Double(v).IsSpecial());

  int decimal_exponent;
  bool result = grisu3(v, buffer, length, &decimal_exponent);
  *point = *length + decimal_exponent;
  buffer[*length] = '\0';
  return result;
}

} }  // namespace v8::internal