path: root/contrib/libs/double-conversion/bignum-dtoa.cc

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#include <cmath>
 
#include "bignum-dtoa.h" 
 
#include "bignum.h" 
#include "ieee.h" 
 
namespace double_conversion { 
 
static int NormalizedExponent(uint64_t significand, int exponent) { 
  ASSERT(significand != 0); 
  while ((significand & Double::kHiddenBit) == 0) { 
    significand = significand << 1; 
    exponent = exponent - 1; 
  } 
  return exponent; 
} 
 
 
// Forward declarations: 
// Returns an estimation of k such that 10^(k-1) <= v < 10^k. 
static int EstimatePower(int exponent); 
// Computes v / 10^estimated_power exactly, as a ratio of two bignums, numerator 
// and denominator. 
static void InitialScaledStartValues(uint64_t significand, 
                                     int exponent, 
                                     bool lower_boundary_is_closer, 
                                     int estimated_power, 
                                     bool need_boundary_deltas, 
                                     Bignum* numerator, 
                                     Bignum* denominator, 
                                     Bignum* delta_minus, 
                                     Bignum* delta_plus); 
// Multiplies numerator/denominator so that its values lies in the range 1-10. 
// Returns decimal_point s.t. 
//  v = numerator'/denominator' * 10^(decimal_point-1) 
//     where numerator' and denominator' are the values of numerator and 
//     denominator after the call to this function. 
static void FixupMultiply10(int estimated_power, bool is_even, 
                            int* decimal_point, 
                            Bignum* numerator, Bignum* denominator, 
                            Bignum* delta_minus, Bignum* delta_plus); 
// Generates digits from the left to the right and stops when the generated 
// digits yield the shortest decimal representation of v. 
static void GenerateShortestDigits(Bignum* numerator, Bignum* denominator, 
                                   Bignum* delta_minus, Bignum* delta_plus, 
                                   bool is_even, 
                                   Vector<char> buffer, int* length); 
// Generates 'requested_digits' after the decimal point. 
static void BignumToFixed(int requested_digits, int* decimal_point, 
                          Bignum* numerator, Bignum* denominator, 
                          Vector<char>(buffer), int* length); 
// Generates 'count' digits of numerator/denominator. 
// Once 'count' digits have been produced rounds the result depending on the 
// remainder (remainders of exactly .5 round upwards). Might update the 
// decimal_point when rounding up (for example for 0.9999). 
static void GenerateCountedDigits(int count, int* decimal_point, 
                                  Bignum* numerator, Bignum* denominator, 
                                  Vector<char>(buffer), int* length); 
 
 
void BignumDtoa(double v, BignumDtoaMode mode, int requested_digits, 
                Vector<char> buffer, int* length, int* decimal_point) { 
  ASSERT(v > 0); 
  ASSERT(!Double(v).IsSpecial()); 
  uint64_t significand; 
  int exponent; 
  bool lower_boundary_is_closer; 
  if (mode == BIGNUM_DTOA_SHORTEST_SINGLE) { 
    float f = static_cast<float>(v); 
    ASSERT(f == v); 
    significand = Single(f).Significand(); 
    exponent = Single(f).Exponent(); 
    lower_boundary_is_closer = Single(f).LowerBoundaryIsCloser(); 
  } else { 
    significand = Double(v).Significand(); 
    exponent = Double(v).Exponent(); 
    lower_boundary_is_closer = Double(v).LowerBoundaryIsCloser(); 
  } 
  bool need_boundary_deltas = 
      (mode == BIGNUM_DTOA_SHORTEST || mode == BIGNUM_DTOA_SHORTEST_SINGLE); 
 
  bool is_even = (significand & 1) == 0; 
  int normalized_exponent = NormalizedExponent(significand, exponent); 
  // estimated_power might be too low by 1. 
  int estimated_power = EstimatePower(normalized_exponent); 
 
  // Shortcut for Fixed. 
  // The requested digits correspond to the digits after the point. If the 
  // number is much too small, then there is no need in trying to get any 
  // digits. 
  if (mode == BIGNUM_DTOA_FIXED && -estimated_power - 1 > requested_digits) { 
    buffer[0] = '\0'; 
    *length = 0; 
    // Set decimal-point to -requested_digits. This is what Gay does. 
    // Note that it should not have any effect anyways since the string is 
    // empty. 
    *decimal_point = -requested_digits; 
    return; 
  } 
 
  Bignum numerator; 
  Bignum denominator; 
  Bignum delta_minus; 
  Bignum delta_plus; 
  // Make sure the bignum can grow large enough. The smallest double equals 
  // 4e-324. In this case the denominator needs fewer than 324*4 binary digits. 
  // The maximum double is 1.7976931348623157e308 which needs fewer than 
  // 308*4 binary digits. 
  ASSERT(Bignum::kMaxSignificantBits >= 324*4); 
  InitialScaledStartValues(significand, exponent, lower_boundary_is_closer, 
                           estimated_power, need_boundary_deltas, 
                           &numerator, &denominator, 
                           &delta_minus, &delta_plus); 
  // We now have v = (numerator / denominator) * 10^estimated_power. 
  FixupMultiply10(estimated_power, is_even, decimal_point, 
                  &numerator, &denominator, 
                  &delta_minus, &delta_plus); 
  // We now have v = (numerator / denominator) * 10^(decimal_point-1), and 
  //  1 <= (numerator + delta_plus) / denominator < 10 
  switch (mode) { 
    case BIGNUM_DTOA_SHORTEST: 
    case BIGNUM_DTOA_SHORTEST_SINGLE: 
      GenerateShortestDigits(&numerator, &denominator, 
                             &delta_minus, &delta_plus, 
                             is_even, buffer, length); 
      break; 
    case BIGNUM_DTOA_FIXED: 
      BignumToFixed(requested_digits, decimal_point, 
                    &numerator, &denominator, 
                    buffer, length); 
      break; 
    case BIGNUM_DTOA_PRECISION: 
      GenerateCountedDigits(requested_digits, decimal_point, 
                            &numerator, &denominator, 
                            buffer, length); 
      break; 
    default: 
      UNREACHABLE(); 
  } 
  buffer[*length] = '\0'; 
} 
 
 
// The procedure starts generating digits from the left to the right and stops 
// when the generated digits yield the shortest decimal representation of v. A 
// decimal representation of v is a number lying closer to v than to any other 
// double, so it converts to v when read. 
// 
// This is true if d, the decimal representation, is between m- and m+, the 
// upper and lower boundaries. d must be strictly between them if !is_even. 
//           m- := (numerator - delta_minus) / denominator 
//           m+ := (numerator + delta_plus) / denominator 
// 
// Precondition: 0 <= (numerator+delta_plus) / denominator < 10. 
//   If 1 <= (numerator+delta_plus) / denominator < 10 then no leading 0 digit 
//   will be produced. This should be the standard precondition. 
static void GenerateShortestDigits(Bignum* numerator, Bignum* denominator, 
                                   Bignum* delta_minus, Bignum* delta_plus, 
                                   bool is_even, 
                                   Vector<char> buffer, int* length) { 
  // Small optimization: if delta_minus and delta_plus are the same just reuse 
  // one of the two bignums. 
  if (Bignum::Equal(*delta_minus, *delta_plus)) { 
    delta_plus = delta_minus; 
  } 
  *length = 0; 
  for (;;) { 
    uint16_t digit; 
    digit = numerator->DivideModuloIntBignum(*denominator); 
    ASSERT(digit <= 9);  // digit is a uint16_t and therefore always positive. 
    // digit = numerator / denominator (integer division). 
    // numerator = numerator % denominator. 
    buffer[(*length)++] = static_cast<char>(digit + '0'); 
 
    // Can we stop already? 
    // If the remainder of the division is less than the distance to the lower 
    // boundary we can stop. In this case we simply round down (discarding the 
    // remainder). 
    // Similarly we test if we can round up (using the upper boundary). 
    bool in_delta_room_minus; 
    bool in_delta_room_plus; 
    if (is_even) { 
      in_delta_room_minus = Bignum::LessEqual(*numerator, *delta_minus); 
    } else { 
      in_delta_room_minus = Bignum::Less(*numerator, *delta_minus); 
    } 
    if (is_even) { 
      in_delta_room_plus = 
          Bignum::PlusCompare(*numerator, *delta_plus, *denominator) >= 0; 
    } else { 
      in_delta_room_plus = 
          Bignum::PlusCompare(*numerator, *delta_plus, *denominator) > 0; 
    } 
    if (!in_delta_room_minus && !in_delta_room_plus) { 
      // Prepare for next iteration. 
      numerator->Times10(); 
      delta_minus->Times10(); 
      // We optimized delta_plus to be equal to delta_minus (if they share the 
      // same value). So don't multiply delta_plus if they point to the same 
      // object. 
      if (delta_minus != delta_plus) { 
        delta_plus->Times10(); 
      } 
    } else if (in_delta_room_minus && in_delta_room_plus) { 
      // Let's see if 2*numerator < denominator. 
      // If yes, then the next digit would be < 5 and we can round down. 
      int compare = Bignum::PlusCompare(*numerator, *numerator, *denominator); 
      if (compare < 0) { 
        // Remaining digits are less than .5. -> Round down (== do nothing). 
      } else if (compare > 0) { 
        // Remaining digits are more than .5 of denominator. -> Round up. 
        // Note that the last digit could not be a '9' as otherwise the whole 
        // loop would have stopped earlier. 
        // We still have an assert here in case the preconditions were not 
        // satisfied. 
        ASSERT(buffer[(*length) - 1] != '9'); 
        buffer[(*length) - 1]++; 
      } else { 
        // Halfway case. 
        // TODO(floitsch): need a way to solve half-way cases. 
        //   For now let's round towards even (since this is what Gay seems to 
        //   do). 
 
        if ((buffer[(*length) - 1] - '0') % 2 == 0) { 
          // Round down => Do nothing. 
        } else { 
          ASSERT(buffer[(*length) - 1] != '9'); 
          buffer[(*length) - 1]++; 
        } 
      } 
      return; 
    } else if (in_delta_room_minus) { 
      // Round down (== do nothing). 
      return; 
    } else {  // in_delta_room_plus 
      // Round up. 
      // Note again that the last digit could not be '9' since this would have 
      // stopped the loop earlier. 
      // We still have an ASSERT here, in case the preconditions were not 
      // satisfied. 
      ASSERT(buffer[(*length) -1] != '9'); 
      buffer[(*length) - 1]++; 
      return; 
    } 
  } 
} 
 
 
// Let v = numerator / denominator < 10. 
// Then we generate 'count' digits of d = x.xxxxx... (without the decimal point) 
// from left to right. Once 'count' digits have been produced we decide wether 
// to round up or down. Remainders of exactly .5 round upwards. Numbers such 
// as 9.999999 propagate a carry all the way, and change the 
// exponent (decimal_point), when rounding upwards. 
static void GenerateCountedDigits(int count, int* decimal_point, 
                                  Bignum* numerator, Bignum* denominator, 
                                  Vector<char> buffer, int* length) { 
  ASSERT(count >= 0); 
  for (int i = 0; i < count - 1; ++i) { 
    uint16_t digit; 
    digit = numerator->DivideModuloIntBignum(*denominator); 
    ASSERT(digit <= 9);  // digit is a uint16_t and therefore always positive. 
    // digit = numerator / denominator (integer division). 
    // numerator = numerator % denominator. 
    buffer[i] = static_cast<char>(digit + '0'); 
    // Prepare for next iteration. 
    numerator->Times10(); 
  } 
  // Generate the last digit. 
  uint16_t digit; 
  digit = numerator->DivideModuloIntBignum(*denominator); 
  if (Bignum::PlusCompare(*numerator, *numerator, *denominator) >= 0) { 
    digit++; 
  } 
  ASSERT(digit <= 10); 
  buffer[count - 1] = static_cast<char>(digit + '0'); 
  // Correct bad digits (in case we had a sequence of '9's). Propagate the 
  // carry until we hat a non-'9' or til we reach the first digit. 
  for (int i = count - 1; i > 0; --i) { 
    if (buffer[i] != '0' + 10) break; 
    buffer[i] = '0'; 
    buffer[i - 1]++; 
  } 
  if (buffer[0] == '0' + 10) { 
    // Propagate a carry past the top place. 
    buffer[0] = '1'; 
    (*decimal_point)++; 
  } 
  *length = count; 
} 
 
 
// Generates 'requested_digits' after the decimal point. It might omit 
// trailing '0's. If the input number is too small then no digits at all are 
// generated (ex.: 2 fixed digits for 0.00001). 
// 
// Input verifies:  1 <= (numerator + delta) / denominator < 10. 
static void BignumToFixed(int requested_digits, int* decimal_point, 
                          Bignum* numerator, Bignum* denominator, 
                          Vector<char>(buffer), int* length) { 
  // Note that we have to look at more than just the requested_digits, since 
  // a number could be rounded up. Example: v=0.5 with requested_digits=0. 
  // Even though the power of v equals 0 we can't just stop here. 
  if (-(*decimal_point) > requested_digits) { 
    // The number is definitively too small. 
    // Ex: 0.001 with requested_digits == 1. 
    // Set decimal-point to -requested_digits. This is what Gay does. 
    // Note that it should not have any effect anyways since the string is 
    // empty. 
    *decimal_point = -requested_digits; 
    *length = 0; 
    return; 
  } else if (-(*decimal_point) == requested_digits) { 
    // We only need to verify if the number rounds down or up. 
    // Ex: 0.04 and 0.06 with requested_digits == 1. 
    ASSERT(*decimal_point == -requested_digits); 
    // Initially the fraction lies in range (1, 10]. Multiply the denominator 
    // by 10 so that we can compare more easily. 
    denominator->Times10(); 
    if (Bignum::PlusCompare(*numerator, *numerator, *denominator) >= 0) { 
      // If the fraction is >= 0.5 then we have to include the rounded 
      // digit. 
      buffer[0] = '1'; 
      *length = 1; 
      (*decimal_point)++; 
    } else { 
      // Note that we caught most of similar cases earlier. 
      *length = 0; 
    } 
    return; 
  } else { 
    // The requested digits correspond to the digits after the point. 
    // The variable 'needed_digits' includes the digits before the point. 
    int needed_digits = (*decimal_point) + requested_digits; 
    GenerateCountedDigits(needed_digits, decimal_point, 
                          numerator, denominator, 
                          buffer, length); 
  } 
} 
 
 
// Returns an estimation of k such that 10^(k-1) <= v < 10^k where 
// v = f * 2^exponent and 2^52 <= f < 2^53. 
// v is hence a normalized double with the given exponent. The output is an 
// approximation for the exponent of the decimal approimation .digits * 10^k. 
// 
// The result might undershoot by 1 in which case 10^k <= v < 10^k+1. 
// Note: this property holds for v's upper boundary m+ too. 
//    10^k <= m+ < 10^k+1. 
//   (see explanation below). 
// 
// Examples: 
//  EstimatePower(0)   => 16 
//  EstimatePower(-52) => 0 
// 
// Note: e >= 0 => EstimatedPower(e) > 0. No similar claim can be made for e<0. 
static int EstimatePower(int exponent) { 
  // This function estimates log10 of v where v = f*2^e (with e == exponent). 
  // Note that 10^floor(log10(v)) <= v, but v <= 10^ceil(log10(v)). 
  // Note that f is bounded by its container size. Let p = 53 (the double's 
  // significand size). Then 2^(p-1) <= f < 2^p. 
  // 
  // Given that log10(v) == log2(v)/log2(10) and e+(len(f)-1) is quite close 
  // to log2(v) the function is simplified to (e+(len(f)-1)/log2(10)). 
  // The computed number undershoots by less than 0.631 (when we compute log3 
  // and not log10). 
  // 
  // Optimization: since we only need an approximated result this computation 
  // can be performed on 64 bit integers. On x86/x64 architecture the speedup is 
  // not really measurable, though. 
  // 
  // Since we want to avoid overshooting we decrement by 1e10 so that 
  // floating-point imprecisions don't affect us. 
  // 
  // Explanation for v's boundary m+: the computation takes advantage of 
  // the fact that 2^(p-1) <= f < 2^p. Boundaries still satisfy this requirement 
  // (even for denormals where the delta can be much more important). 
 
  const double k1Log10 = 0.30102999566398114;  // 1/lg(10) 
 
  // For doubles len(f) == 53 (don't forget the hidden bit). 
  const int kSignificandSize = Double::kSignificandSize; 
  double estimate = ceil((exponent + kSignificandSize - 1) * k1Log10 - 1e-10); 
  return static_cast<int>(estimate); 
} 
 
 
// See comments for InitialScaledStartValues. 
static void InitialScaledStartValuesPositiveExponent( 
    uint64_t significand, int exponent, 
    int estimated_power, bool need_boundary_deltas, 
    Bignum* numerator, Bignum* denominator, 
    Bignum* delta_minus, Bignum* delta_plus) { 
  // A positive exponent implies a positive power. 
  ASSERT(estimated_power >= 0); 
  // Since the estimated_power is positive we simply multiply the denominator 
  // by 10^estimated_power. 
 
  // numerator = v. 
  numerator->AssignUInt64(significand); 
  numerator->ShiftLeft(exponent); 
  // denominator = 10^estimated_power. 
  denominator->AssignPowerUInt16(10, estimated_power); 
 
  if (need_boundary_deltas) { 
    // Introduce a common denominator so that the deltas to the boundaries are 
    // integers. 
    denominator->ShiftLeft(1); 
    numerator->ShiftLeft(1); 
    // Let v = f * 2^e, then m+ - v = 1/2 * 2^e; With the common 
    // denominator (of 2) delta_plus equals 2^e. 
    delta_plus->AssignUInt16(1); 
    delta_plus->ShiftLeft(exponent); 
    // Same for delta_minus. The adjustments if f == 2^p-1 are done later. 
    delta_minus->AssignUInt16(1); 
    delta_minus->ShiftLeft(exponent); 
  } 
} 
 
 
// See comments for InitialScaledStartValues 
static void InitialScaledStartValuesNegativeExponentPositivePower( 
    uint64_t significand, int exponent, 
    int estimated_power, bool need_boundary_deltas, 
    Bignum* numerator, Bignum* denominator, 
    Bignum* delta_minus, Bignum* delta_plus) { 
  // v = f * 2^e with e < 0, and with estimated_power >= 0. 
  // This means that e is close to 0 (have a look at how estimated_power is 
  // computed). 
 
  // numerator = significand 
  //  since v = significand * 2^exponent this is equivalent to 
  //  numerator = v * / 2^-exponent 
  numerator->AssignUInt64(significand); 
  // denominator = 10^estimated_power * 2^-exponent (with exponent < 0) 
  denominator->AssignPowerUInt16(10, estimated_power); 
  denominator->ShiftLeft(-exponent); 
 
  if (need_boundary_deltas) { 
    // Introduce a common denominator so that the deltas to the boundaries are 
    // integers. 
    denominator->ShiftLeft(1); 
    numerator->ShiftLeft(1); 
    // Let v = f * 2^e, then m+ - v = 1/2 * 2^e; With the common 
    // denominator (of 2) delta_plus equals 2^e. 
    // Given that the denominator already includes v's exponent the distance 
    // to the boundaries is simply 1. 
    delta_plus->AssignUInt16(1); 
    // Same for delta_minus. The adjustments if f == 2^p-1 are done later. 
    delta_minus->AssignUInt16(1); 
  } 
} 
 
 
// See comments for InitialScaledStartValues 
static void InitialScaledStartValuesNegativeExponentNegativePower( 
    uint64_t significand, int exponent, 
    int estimated_power, bool need_boundary_deltas, 
    Bignum* numerator, Bignum* denominator, 
    Bignum* delta_minus, Bignum* delta_plus) { 
  // Instead of multiplying the denominator with 10^estimated_power we 
  // multiply all values (numerator and deltas) by 10^-estimated_power. 
 
  // Use numerator as temporary container for power_ten. 
  Bignum* power_ten = numerator; 
  power_ten->AssignPowerUInt16(10, -estimated_power); 
 
  if (need_boundary_deltas) { 
    // Since power_ten == numerator we must make a copy of 10^estimated_power 
    // before we complete the computation of the numerator. 
    // delta_plus = delta_minus = 10^estimated_power 
    delta_plus->AssignBignum(*power_ten); 
    delta_minus->AssignBignum(*power_ten); 
  } 
 
  // numerator = significand * 2 * 10^-estimated_power 
  //  since v = significand * 2^exponent this is equivalent to 
  // numerator = v * 10^-estimated_power * 2 * 2^-exponent. 
  // Remember: numerator has been abused as power_ten. So no need to assign it 
  //  to itself. 
  ASSERT(numerator == power_ten); 
  numerator->MultiplyByUInt64(significand); 
 
  // denominator = 2 * 2^-exponent with exponent < 0. 
  denominator->AssignUInt16(1); 
  denominator->ShiftLeft(-exponent); 
 
  if (need_boundary_deltas) { 
    // Introduce a common denominator so that the deltas to the boundaries are 
    // integers. 
    numerator->ShiftLeft(1); 
    denominator->ShiftLeft(1); 
    // With this shift the boundaries have their correct value, since 
    // delta_plus = 10^-estimated_power, and 
    // delta_minus = 10^-estimated_power. 
    // These assignments have been done earlier. 
    // The adjustments if f == 2^p-1 (lower boundary is closer) are done later. 
  } 
} 
 
 
// Let v = significand * 2^exponent. 
// Computes v / 10^estimated_power exactly, as a ratio of two bignums, numerator 
// and denominator. The functions GenerateShortestDigits and 
// GenerateCountedDigits will then convert this ratio to its decimal 
// representation d, with the required accuracy. 
// Then d * 10^estimated_power is the representation of v. 
// (Note: the fraction and the estimated_power might get adjusted before 
// generating the decimal representation.) 
// 
// The initial start values consist of: 
//  - a scaled numerator: s.t. numerator/denominator == v / 10^estimated_power. 
//  - a scaled (common) denominator. 
//  optionally (used by GenerateShortestDigits to decide if it has the shortest 
//  decimal converting back to v): 
//  - v - m-: the distance to the lower boundary. 
//  - m+ - v: the distance to the upper boundary. 
// 
// v, m+, m-, and therefore v - m- and m+ - v all share the same denominator. 
// 
// Let ep == estimated_power, then the returned values will satisfy: 
//  v / 10^ep = numerator / denominator. 
//  v's boundarys m- and m+: 
//    m- / 10^ep == v / 10^ep - delta_minus / denominator 
//    m+ / 10^ep == v / 10^ep + delta_plus / denominator 
//  Or in other words: 
//    m- == v - delta_minus * 10^ep / denominator; 
//    m+ == v + delta_plus * 10^ep / denominator; 
// 
// Since 10^(k-1) <= v < 10^k    (with k == estimated_power) 
//  or       10^k <= v < 10^(k+1) 
//  we then have 0.1 <= numerator/denominator < 1 
//           or    1 <= numerator/denominator < 10 
// 
// It is then easy to kickstart the digit-generation routine. 
// 
// The boundary-deltas are only filled if the mode equals BIGNUM_DTOA_SHORTEST 
// or BIGNUM_DTOA_SHORTEST_SINGLE. 
 
static void InitialScaledStartValues(uint64_t significand, 
                                     int exponent, 
                                     bool lower_boundary_is_closer, 
                                     int estimated_power, 
                                     bool need_boundary_deltas, 
                                     Bignum* numerator, 
                                     Bignum* denominator, 
                                     Bignum* delta_minus, 
                                     Bignum* delta_plus) { 
  if (exponent >= 0) { 
    InitialScaledStartValuesPositiveExponent( 
        significand, exponent, estimated_power, need_boundary_deltas, 
        numerator, denominator, delta_minus, delta_plus); 
  } else if (estimated_power >= 0) { 
    InitialScaledStartValuesNegativeExponentPositivePower( 
        significand, exponent, estimated_power, need_boundary_deltas, 
        numerator, denominator, delta_minus, delta_plus); 
  } else { 
    InitialScaledStartValuesNegativeExponentNegativePower( 
        significand, exponent, estimated_power, need_boundary_deltas, 
        numerator, denominator, delta_minus, delta_plus); 
  } 
 
  if (need_boundary_deltas && lower_boundary_is_closer) { 
    // The lower boundary is closer at half the distance of "normal" numbers. 
    // Increase the common denominator and adapt all but the delta_minus. 
    denominator->ShiftLeft(1);  // *2 
    numerator->ShiftLeft(1);    // *2 
    delta_plus->ShiftLeft(1);   // *2 
  } 
} 
 
 
// This routine multiplies numerator/denominator so that its values lies in the 
// range 1-10. That is after a call to this function we have: 
//    1 <= (numerator + delta_plus) /denominator < 10. 
// Let numerator the input before modification and numerator' the argument 
// after modification, then the output-parameter decimal_point is such that 
//  numerator / denominator * 10^estimated_power == 
//    numerator' / denominator' * 10^(decimal_point - 1) 
// In some cases estimated_power was too low, and this is already the case. We 
// then simply adjust the power so that 10^(k-1) <= v < 10^k (with k == 
// estimated_power) but do not touch the numerator or denominator. 
// Otherwise the routine multiplies the numerator and the deltas by 10. 
static void FixupMultiply10(int estimated_power, bool is_even, 
                            int* decimal_point, 
                            Bignum* numerator, Bignum* denominator, 
                            Bignum* delta_minus, Bignum* delta_plus) { 
  bool in_range; 
  if (is_even) { 
    // For IEEE doubles half-way cases (in decimal system numbers ending with 5) 
    // are rounded to the closest floating-point number with even significand. 
    in_range = Bignum::PlusCompare(*numerator, *delta_plus, *denominator) >= 0; 
  } else { 
    in_range = Bignum::PlusCompare(*numerator, *delta_plus, *denominator) > 0; 
  } 
  if (in_range) { 
    // Since numerator + delta_plus >= denominator we already have 
    // 1 <= numerator/denominator < 10. Simply update the estimated_power. 
    *decimal_point = estimated_power + 1; 
  } else { 
    *decimal_point = estimated_power; 
    numerator->Times10(); 
    if (Bignum::Equal(*delta_minus, *delta_plus)) { 
      delta_minus->Times10(); 
      delta_plus->AssignBignum(*delta_minus); 
    } else { 
      delta_minus->Times10(); 
      delta_plus->Times10(); 
    } 
  } 
} 
 
}  // namespace double_conversion