michael@0: // Copyright 2010 the V8 project authors. All rights reserved.
michael@0: // Redistribution and use in source and binary forms, with or without
michael@0: // modification, are permitted provided that the following conditions are
michael@0: // met:
michael@0: //
michael@0: //     * Redistributions of source code must retain the above copyright
michael@0: //       notice, this list of conditions and the following disclaimer.
michael@0: //     * Redistributions in binary form must reproduce the above
michael@0: //       copyright notice, this list of conditions and the following
michael@0: //       disclaimer in the documentation and/or other materials provided
michael@0: //       with the distribution.
michael@0: //     * Neither the name of Google Inc. nor the names of its
michael@0: //       contributors may be used to endorse or promote products derived
michael@0: //       from this software without specific prior written permission.
michael@0: //
michael@0: // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
michael@0: // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
michael@0: // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
michael@0: // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
michael@0: // OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
michael@0: // SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
michael@0: // LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
michael@0: // DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
michael@0: // THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
michael@0: // (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
michael@0: // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
michael@0: 
michael@0: #include <math.h>
michael@0: 
michael@0: #include "bignum-dtoa.h"
michael@0: 
michael@0: #include "bignum.h"
michael@0: #include "ieee.h"
michael@0: 
michael@0: namespace double_conversion {
michael@0: 
michael@0: static int NormalizedExponent(uint64_t significand, int exponent) {
michael@0:   ASSERT(significand != 0);
michael@0:   while ((significand & Double::kHiddenBit) == 0) {
michael@0:     significand = significand << 1;
michael@0:     exponent = exponent - 1;
michael@0:   }
michael@0:   return exponent;
michael@0: }
michael@0: 
michael@0: 
michael@0: // Forward declarations:
michael@0: // Returns an estimation of k such that 10^(k-1) <= v < 10^k.
michael@0: static int EstimatePower(int exponent);
michael@0: // Computes v / 10^estimated_power exactly, as a ratio of two bignums, numerator
michael@0: // and denominator.
michael@0: static void InitialScaledStartValues(uint64_t significand,
michael@0:                                      int exponent,
michael@0:                                      bool lower_boundary_is_closer,
michael@0:                                      int estimated_power,
michael@0:                                      bool need_boundary_deltas,
michael@0:                                      Bignum* numerator,
michael@0:                                      Bignum* denominator,
michael@0:                                      Bignum* delta_minus,
michael@0:                                      Bignum* delta_plus);
michael@0: // Multiplies numerator/denominator so that its values lies in the range 1-10.
michael@0: // Returns decimal_point s.t.
michael@0: //  v = numerator'/denominator' * 10^(decimal_point-1)
michael@0: //     where numerator' and denominator' are the values of numerator and
michael@0: //     denominator after the call to this function.
michael@0: static void FixupMultiply10(int estimated_power, bool is_even,
michael@0:                             int* decimal_point,
michael@0:                             Bignum* numerator, Bignum* denominator,
michael@0:                             Bignum* delta_minus, Bignum* delta_plus);
michael@0: // Generates digits from the left to the right and stops when the generated
michael@0: // digits yield the shortest decimal representation of v.
michael@0: static void GenerateShortestDigits(Bignum* numerator, Bignum* denominator,
michael@0:                                    Bignum* delta_minus, Bignum* delta_plus,
michael@0:                                    bool is_even,
michael@0:                                    Vector<char> buffer, int* length);
michael@0: // Generates 'requested_digits' after the decimal point.
michael@0: static void BignumToFixed(int requested_digits, int* decimal_point,
michael@0:                           Bignum* numerator, Bignum* denominator,
michael@0:                           Vector<char>(buffer), int* length);
michael@0: // Generates 'count' digits of numerator/denominator.
michael@0: // Once 'count' digits have been produced rounds the result depending on the
michael@0: // remainder (remainders of exactly .5 round upwards). Might update the
michael@0: // decimal_point when rounding up (for example for 0.9999).
michael@0: static void GenerateCountedDigits(int count, int* decimal_point,
michael@0:                                   Bignum* numerator, Bignum* denominator,
michael@0:                                   Vector<char>(buffer), int* length);
michael@0: 
michael@0: 
michael@0: void BignumDtoa(double v, BignumDtoaMode mode, int requested_digits,
michael@0:                 Vector<char> buffer, int* length, int* decimal_point) {
michael@0:   ASSERT(v > 0);
michael@0:   ASSERT(!Double(v).IsSpecial());
michael@0:   uint64_t significand;
michael@0:   int exponent;
michael@0:   bool lower_boundary_is_closer;
michael@0:   if (mode == BIGNUM_DTOA_SHORTEST_SINGLE) {
michael@0:     float f = static_cast<float>(v);
michael@0:     ASSERT(f == v);
michael@0:     significand = Single(f).Significand();
michael@0:     exponent = Single(f).Exponent();
michael@0:     lower_boundary_is_closer = Single(f).LowerBoundaryIsCloser();
michael@0:   } else {
michael@0:     significand = Double(v).Significand();
michael@0:     exponent = Double(v).Exponent();
michael@0:     lower_boundary_is_closer = Double(v).LowerBoundaryIsCloser();
michael@0:   }
michael@0:   bool need_boundary_deltas =
michael@0:       (mode == BIGNUM_DTOA_SHORTEST || mode == BIGNUM_DTOA_SHORTEST_SINGLE);
michael@0: 
michael@0:   bool is_even = (significand & 1) == 0;
michael@0:   int normalized_exponent = NormalizedExponent(significand, exponent);
michael@0:   // estimated_power might be too low by 1.
michael@0:   int estimated_power = EstimatePower(normalized_exponent);
michael@0: 
michael@0:   // Shortcut for Fixed.
michael@0:   // The requested digits correspond to the digits after the point. If the
michael@0:   // number is much too small, then there is no need in trying to get any
michael@0:   // digits.
michael@0:   if (mode == BIGNUM_DTOA_FIXED && -estimated_power - 1 > requested_digits) {
michael@0:     buffer[0] = '\0';
michael@0:     *length = 0;
michael@0:     // Set decimal-point to -requested_digits. This is what Gay does.
michael@0:     // Note that it should not have any effect anyways since the string is
michael@0:     // empty.
michael@0:     *decimal_point = -requested_digits;
michael@0:     return;
michael@0:   }
michael@0: 
michael@0:   Bignum numerator;
michael@0:   Bignum denominator;
michael@0:   Bignum delta_minus;
michael@0:   Bignum delta_plus;
michael@0:   // Make sure the bignum can grow large enough. The smallest double equals
michael@0:   // 4e-324. In this case the denominator needs fewer than 324*4 binary digits.
michael@0:   // The maximum double is 1.7976931348623157e308 which needs fewer than
michael@0:   // 308*4 binary digits.
michael@0:   ASSERT(Bignum::kMaxSignificantBits >= 324*4);
michael@0:   InitialScaledStartValues(significand, exponent, lower_boundary_is_closer,
michael@0:                            estimated_power, need_boundary_deltas,
michael@0:                            &numerator, &denominator,
michael@0:                            &delta_minus, &delta_plus);
michael@0:   // We now have v = (numerator / denominator) * 10^estimated_power.
michael@0:   FixupMultiply10(estimated_power, is_even, decimal_point,
michael@0:                   &numerator, &denominator,
michael@0:                   &delta_minus, &delta_plus);
michael@0:   // We now have v = (numerator / denominator) * 10^(decimal_point-1), and
michael@0:   //  1 <= (numerator + delta_plus) / denominator < 10
michael@0:   switch (mode) {
michael@0:     case BIGNUM_DTOA_SHORTEST:
michael@0:     case BIGNUM_DTOA_SHORTEST_SINGLE:
michael@0:       GenerateShortestDigits(&numerator, &denominator,
michael@0:                              &delta_minus, &delta_plus,
michael@0:                              is_even, buffer, length);
michael@0:       break;
michael@0:     case BIGNUM_DTOA_FIXED:
michael@0:       BignumToFixed(requested_digits, decimal_point,
michael@0:                     &numerator, &denominator,
michael@0:                     buffer, length);
michael@0:       break;
michael@0:     case BIGNUM_DTOA_PRECISION:
michael@0:       GenerateCountedDigits(requested_digits, decimal_point,
michael@0:                             &numerator, &denominator,
michael@0:                             buffer, length);
michael@0:       break;
michael@0:     default:
michael@0:       UNREACHABLE();
michael@0:   }
michael@0:   buffer[*length] = '\0';
michael@0: }
michael@0: 
michael@0: 
michael@0: // The procedure starts generating digits from the left to the right and stops
michael@0: // when the generated digits yield the shortest decimal representation of v. A
michael@0: // decimal representation of v is a number lying closer to v than to any other
michael@0: // double, so it converts to v when read.
michael@0: //
michael@0: // This is true if d, the decimal representation, is between m- and m+, the
michael@0: // upper and lower boundaries. d must be strictly between them if !is_even.
michael@0: //           m- := (numerator - delta_minus) / denominator
michael@0: //           m+ := (numerator + delta_plus) / denominator
michael@0: //
michael@0: // Precondition: 0 <= (numerator+delta_plus) / denominator < 10.
michael@0: //   If 1 <= (numerator+delta_plus) / denominator < 10 then no leading 0 digit
michael@0: //   will be produced. This should be the standard precondition.
michael@0: static void GenerateShortestDigits(Bignum* numerator, Bignum* denominator,
michael@0:                                    Bignum* delta_minus, Bignum* delta_plus,
michael@0:                                    bool is_even,
michael@0:                                    Vector<char> buffer, int* length) {
michael@0:   // Small optimization: if delta_minus and delta_plus are the same just reuse
michael@0:   // one of the two bignums.
michael@0:   if (Bignum::Equal(*delta_minus, *delta_plus)) {
michael@0:     delta_plus = delta_minus;
michael@0:   }
michael@0:   *length = 0;
michael@0:   while (true) {
michael@0:     uint16_t digit;
michael@0:     digit = numerator->DivideModuloIntBignum(*denominator);
michael@0:     ASSERT(digit <= 9);  // digit is a uint16_t and therefore always positive.
michael@0:     // digit = numerator / denominator (integer division).
michael@0:     // numerator = numerator % denominator.
michael@0:     buffer[(*length)++] = digit + '0';
michael@0: 
michael@0:     // Can we stop already?
michael@0:     // If the remainder of the division is less than the distance to the lower
michael@0:     // boundary we can stop. In this case we simply round down (discarding the
michael@0:     // remainder).
michael@0:     // Similarly we test if we can round up (using the upper boundary).
michael@0:     bool in_delta_room_minus;
michael@0:     bool in_delta_room_plus;
michael@0:     if (is_even) {
michael@0:       in_delta_room_minus = Bignum::LessEqual(*numerator, *delta_minus);
michael@0:     } else {
michael@0:       in_delta_room_minus = Bignum::Less(*numerator, *delta_minus);
michael@0:     }
michael@0:     if (is_even) {
michael@0:       in_delta_room_plus =
michael@0:           Bignum::PlusCompare(*numerator, *delta_plus, *denominator) >= 0;
michael@0:     } else {
michael@0:       in_delta_room_plus =
michael@0:           Bignum::PlusCompare(*numerator, *delta_plus, *denominator) > 0;
michael@0:     }
michael@0:     if (!in_delta_room_minus && !in_delta_room_plus) {
michael@0:       // Prepare for next iteration.
michael@0:       numerator->Times10();
michael@0:       delta_minus->Times10();
michael@0:       // We optimized delta_plus to be equal to delta_minus (if they share the
michael@0:       // same value). So don't multiply delta_plus if they point to the same
michael@0:       // object.
michael@0:       if (delta_minus != delta_plus) {
michael@0:         delta_plus->Times10();
michael@0:       }
michael@0:     } else if (in_delta_room_minus && in_delta_room_plus) {
michael@0:       // Let's see if 2*numerator < denominator.
michael@0:       // If yes, then the next digit would be < 5 and we can round down.
michael@0:       int compare = Bignum::PlusCompare(*numerator, *numerator, *denominator);
michael@0:       if (compare < 0) {
michael@0:         // Remaining digits are less than .5. -> Round down (== do nothing).
michael@0:       } else if (compare > 0) {
michael@0:         // Remaining digits are more than .5 of denominator. -> Round up.
michael@0:         // Note that the last digit could not be a '9' as otherwise the whole
michael@0:         // loop would have stopped earlier.
michael@0:         // We still have an assert here in case the preconditions were not
michael@0:         // satisfied.
michael@0:         ASSERT(buffer[(*length) - 1] != '9');
michael@0:         buffer[(*length) - 1]++;
michael@0:       } else {
michael@0:         // Halfway case.
michael@0:         // TODO(floitsch): need a way to solve half-way cases.
michael@0:         //   For now let's round towards even (since this is what Gay seems to
michael@0:         //   do).
michael@0: 
michael@0:         if ((buffer[(*length) - 1] - '0') % 2 == 0) {
michael@0:           // Round down => Do nothing.
michael@0:         } else {
michael@0:           ASSERT(buffer[(*length) - 1] != '9');
michael@0:           buffer[(*length) - 1]++;
michael@0:         }
michael@0:       }
michael@0:       return;
michael@0:     } else if (in_delta_room_minus) {
michael@0:       // Round down (== do nothing).
michael@0:       return;
michael@0:     } else {  // in_delta_room_plus
michael@0:       // Round up.
michael@0:       // Note again that the last digit could not be '9' since this would have
michael@0:       // stopped the loop earlier.
michael@0:       // We still have an ASSERT here, in case the preconditions were not
michael@0:       // satisfied.
michael@0:       ASSERT(buffer[(*length) -1] != '9');
michael@0:       buffer[(*length) - 1]++;
michael@0:       return;
michael@0:     }
michael@0:   }
michael@0: }
michael@0: 
michael@0: 
michael@0: // Let v = numerator / denominator < 10.
michael@0: // Then we generate 'count' digits of d = x.xxxxx... (without the decimal point)
michael@0: // from left to right. Once 'count' digits have been produced we decide wether
michael@0: // to round up or down. Remainders of exactly .5 round upwards. Numbers such
michael@0: // as 9.999999 propagate a carry all the way, and change the
michael@0: // exponent (decimal_point), when rounding upwards.
michael@0: static void GenerateCountedDigits(int count, int* decimal_point,
michael@0:                                   Bignum* numerator, Bignum* denominator,
michael@0:                                   Vector<char>(buffer), int* length) {
michael@0:   ASSERT(count >= 0);
michael@0:   for (int i = 0; i < count - 1; ++i) {
michael@0:     uint16_t digit;
michael@0:     digit = numerator->DivideModuloIntBignum(*denominator);
michael@0:     ASSERT(digit <= 9);  // digit is a uint16_t and therefore always positive.
michael@0:     // digit = numerator / denominator (integer division).
michael@0:     // numerator = numerator % denominator.
michael@0:     buffer[i] = digit + '0';
michael@0:     // Prepare for next iteration.
michael@0:     numerator->Times10();
michael@0:   }
michael@0:   // Generate the last digit.
michael@0:   uint16_t digit;
michael@0:   digit = numerator->DivideModuloIntBignum(*denominator);
michael@0:   if (Bignum::PlusCompare(*numerator, *numerator, *denominator) >= 0) {
michael@0:     digit++;
michael@0:   }
michael@0:   buffer[count - 1] = digit + '0';
michael@0:   // Correct bad digits (in case we had a sequence of '9's). Propagate the
michael@0:   // carry until we hat a non-'9' or til we reach the first digit.
michael@0:   for (int i = count - 1; i > 0; --i) {
michael@0:     if (buffer[i] != '0' + 10) break;
michael@0:     buffer[i] = '0';
michael@0:     buffer[i - 1]++;
michael@0:   }
michael@0:   if (buffer[0] == '0' + 10) {
michael@0:     // Propagate a carry past the top place.
michael@0:     buffer[0] = '1';
michael@0:     (*decimal_point)++;
michael@0:   }
michael@0:   *length = count;
michael@0: }
michael@0: 
michael@0: 
michael@0: // Generates 'requested_digits' after the decimal point. It might omit
michael@0: // trailing '0's. If the input number is too small then no digits at all are
michael@0: // generated (ex.: 2 fixed digits for 0.00001).
michael@0: //
michael@0: // Input verifies:  1 <= (numerator + delta) / denominator < 10.
michael@0: static void BignumToFixed(int requested_digits, int* decimal_point,
michael@0:                           Bignum* numerator, Bignum* denominator,
michael@0:                           Vector<char>(buffer), int* length) {
michael@0:   // Note that we have to look at more than just the requested_digits, since
michael@0:   // a number could be rounded up. Example: v=0.5 with requested_digits=0.
michael@0:   // Even though the power of v equals 0 we can't just stop here.
michael@0:   if (-(*decimal_point) > requested_digits) {
michael@0:     // The number is definitively too small.
michael@0:     // Ex: 0.001 with requested_digits == 1.
michael@0:     // Set decimal-point to -requested_digits. This is what Gay does.
michael@0:     // Note that it should not have any effect anyways since the string is
michael@0:     // empty.
michael@0:     *decimal_point = -requested_digits;
michael@0:     *length = 0;
michael@0:     return;
michael@0:   } else if (-(*decimal_point) == requested_digits) {
michael@0:     // We only need to verify if the number rounds down or up.
michael@0:     // Ex: 0.04 and 0.06 with requested_digits == 1.
michael@0:     ASSERT(*decimal_point == -requested_digits);
michael@0:     // Initially the fraction lies in range (1, 10]. Multiply the denominator
michael@0:     // by 10 so that we can compare more easily.
michael@0:     denominator->Times10();
michael@0:     if (Bignum::PlusCompare(*numerator, *numerator, *denominator) >= 0) {
michael@0:       // If the fraction is >= 0.5 then we have to include the rounded
michael@0:       // digit.
michael@0:       buffer[0] = '1';
michael@0:       *length = 1;
michael@0:       (*decimal_point)++;
michael@0:     } else {
michael@0:       // Note that we caught most of similar cases earlier.
michael@0:       *length = 0;
michael@0:     }
michael@0:     return;
michael@0:   } else {
michael@0:     // The requested digits correspond to the digits after the point.
michael@0:     // The variable 'needed_digits' includes the digits before the point.
michael@0:     int needed_digits = (*decimal_point) + requested_digits;
michael@0:     GenerateCountedDigits(needed_digits, decimal_point,
michael@0:                           numerator, denominator,
michael@0:                           buffer, length);
michael@0:   }
michael@0: }
michael@0: 
michael@0: 
michael@0: // Returns an estimation of k such that 10^(k-1) <= v < 10^k where
michael@0: // v = f * 2^exponent and 2^52 <= f < 2^53.
michael@0: // v is hence a normalized double with the given exponent. The output is an
michael@0: // approximation for the exponent of the decimal approimation .digits * 10^k.
michael@0: //
michael@0: // The result might undershoot by 1 in which case 10^k <= v < 10^k+1.
michael@0: // Note: this property holds for v's upper boundary m+ too.
michael@0: //    10^k <= m+ < 10^k+1.
michael@0: //   (see explanation below).
michael@0: //
michael@0: // Examples:
michael@0: //  EstimatePower(0)   => 16
michael@0: //  EstimatePower(-52) => 0
michael@0: //
michael@0: // Note: e >= 0 => EstimatedPower(e) > 0. No similar claim can be made for e<0.
michael@0: static int EstimatePower(int exponent) {
michael@0:   // This function estimates log10 of v where v = f*2^e (with e == exponent).
michael@0:   // Note that 10^floor(log10(v)) <= v, but v <= 10^ceil(log10(v)).
michael@0:   // Note that f is bounded by its container size. Let p = 53 (the double's
michael@0:   // significand size). Then 2^(p-1) <= f < 2^p.
michael@0:   //
michael@0:   // Given that log10(v) == log2(v)/log2(10) and e+(len(f)-1) is quite close
michael@0:   // to log2(v) the function is simplified to (e+(len(f)-1)/log2(10)).
michael@0:   // The computed number undershoots by less than 0.631 (when we compute log3
michael@0:   // and not log10).
michael@0:   //
michael@0:   // Optimization: since we only need an approximated result this computation
michael@0:   // can be performed on 64 bit integers. On x86/x64 architecture the speedup is
michael@0:   // not really measurable, though.
michael@0:   //
michael@0:   // Since we want to avoid overshooting we decrement by 1e10 so that
michael@0:   // floating-point imprecisions don't affect us.
michael@0:   //
michael@0:   // Explanation for v's boundary m+: the computation takes advantage of
michael@0:   // the fact that 2^(p-1) <= f < 2^p. Boundaries still satisfy this requirement
michael@0:   // (even for denormals where the delta can be much more important).
michael@0: 
michael@0:   const double k1Log10 = 0.30102999566398114;  // 1/lg(10)
michael@0: 
michael@0:   // For doubles len(f) == 53 (don't forget the hidden bit).
michael@0:   const int kSignificandSize = Double::kSignificandSize;
michael@0:   double estimate = ceil((exponent + kSignificandSize - 1) * k1Log10 - 1e-10);
michael@0:   return static_cast<int>(estimate);
michael@0: }
michael@0: 
michael@0: 
michael@0: // See comments for InitialScaledStartValues.
michael@0: static void InitialScaledStartValuesPositiveExponent(
michael@0:     uint64_t significand, int exponent,
michael@0:     int estimated_power, bool need_boundary_deltas,
michael@0:     Bignum* numerator, Bignum* denominator,
michael@0:     Bignum* delta_minus, Bignum* delta_plus) {
michael@0:   // A positive exponent implies a positive power.
michael@0:   ASSERT(estimated_power >= 0);
michael@0:   // Since the estimated_power is positive we simply multiply the denominator
michael@0:   // by 10^estimated_power.
michael@0: 
michael@0:   // numerator = v.
michael@0:   numerator->AssignUInt64(significand);
michael@0:   numerator->ShiftLeft(exponent);
michael@0:   // denominator = 10^estimated_power.
michael@0:   denominator->AssignPowerUInt16(10, estimated_power);
michael@0: 
michael@0:   if (need_boundary_deltas) {
michael@0:     // Introduce a common denominator so that the deltas to the boundaries are
michael@0:     // integers.
michael@0:     denominator->ShiftLeft(1);
michael@0:     numerator->ShiftLeft(1);
michael@0:     // Let v = f * 2^e, then m+ - v = 1/2 * 2^e; With the common
michael@0:     // denominator (of 2) delta_plus equals 2^e.
michael@0:     delta_plus->AssignUInt16(1);
michael@0:     delta_plus->ShiftLeft(exponent);
michael@0:     // Same for delta_minus. The adjustments if f == 2^p-1 are done later.
michael@0:     delta_minus->AssignUInt16(1);
michael@0:     delta_minus->ShiftLeft(exponent);
michael@0:   }
michael@0: }
michael@0: 
michael@0: 
michael@0: // See comments for InitialScaledStartValues
michael@0: static void InitialScaledStartValuesNegativeExponentPositivePower(
michael@0:     uint64_t significand, int exponent,
michael@0:     int estimated_power, bool need_boundary_deltas,
michael@0:     Bignum* numerator, Bignum* denominator,
michael@0:     Bignum* delta_minus, Bignum* delta_plus) {
michael@0:   // v = f * 2^e with e < 0, and with estimated_power >= 0.
michael@0:   // This means that e is close to 0 (have a look at how estimated_power is
michael@0:   // computed).
michael@0: 
michael@0:   // numerator = significand
michael@0:   //  since v = significand * 2^exponent this is equivalent to
michael@0:   //  numerator = v * / 2^-exponent
michael@0:   numerator->AssignUInt64(significand);
michael@0:   // denominator = 10^estimated_power * 2^-exponent (with exponent < 0)
michael@0:   denominator->AssignPowerUInt16(10, estimated_power);
michael@0:   denominator->ShiftLeft(-exponent);
michael@0: 
michael@0:   if (need_boundary_deltas) {
michael@0:     // Introduce a common denominator so that the deltas to the boundaries are
michael@0:     // integers.
michael@0:     denominator->ShiftLeft(1);
michael@0:     numerator->ShiftLeft(1);
michael@0:     // Let v = f * 2^e, then m+ - v = 1/2 * 2^e; With the common
michael@0:     // denominator (of 2) delta_plus equals 2^e.
michael@0:     // Given that the denominator already includes v's exponent the distance
michael@0:     // to the boundaries is simply 1.
michael@0:     delta_plus->AssignUInt16(1);
michael@0:     // Same for delta_minus. The adjustments if f == 2^p-1 are done later.
michael@0:     delta_minus->AssignUInt16(1);
michael@0:   }
michael@0: }
michael@0: 
michael@0: 
michael@0: // See comments for InitialScaledStartValues
michael@0: static void InitialScaledStartValuesNegativeExponentNegativePower(
michael@0:     uint64_t significand, int exponent,
michael@0:     int estimated_power, bool need_boundary_deltas,
michael@0:     Bignum* numerator, Bignum* denominator,
michael@0:     Bignum* delta_minus, Bignum* delta_plus) {
michael@0:   // Instead of multiplying the denominator with 10^estimated_power we
michael@0:   // multiply all values (numerator and deltas) by 10^-estimated_power.
michael@0: 
michael@0:   // Use numerator as temporary container for power_ten.
michael@0:   Bignum* power_ten = numerator;
michael@0:   power_ten->AssignPowerUInt16(10, -estimated_power);
michael@0: 
michael@0:   if (need_boundary_deltas) {
michael@0:     // Since power_ten == numerator we must make a copy of 10^estimated_power
michael@0:     // before we complete the computation of the numerator.
michael@0:     // delta_plus = delta_minus = 10^estimated_power
michael@0:     delta_plus->AssignBignum(*power_ten);
michael@0:     delta_minus->AssignBignum(*power_ten);
michael@0:   }
michael@0: 
michael@0:   // numerator = significand * 2 * 10^-estimated_power
michael@0:   //  since v = significand * 2^exponent this is equivalent to
michael@0:   // numerator = v * 10^-estimated_power * 2 * 2^-exponent.
michael@0:   // Remember: numerator has been abused as power_ten. So no need to assign it
michael@0:   //  to itself.
michael@0:   ASSERT(numerator == power_ten);
michael@0:   numerator->MultiplyByUInt64(significand);
michael@0: 
michael@0:   // denominator = 2 * 2^-exponent with exponent < 0.
michael@0:   denominator->AssignUInt16(1);
michael@0:   denominator->ShiftLeft(-exponent);
michael@0: 
michael@0:   if (need_boundary_deltas) {
michael@0:     // Introduce a common denominator so that the deltas to the boundaries are
michael@0:     // integers.
michael@0:     numerator->ShiftLeft(1);
michael@0:     denominator->ShiftLeft(1);
michael@0:     // With this shift the boundaries have their correct value, since
michael@0:     // delta_plus = 10^-estimated_power, and
michael@0:     // delta_minus = 10^-estimated_power.
michael@0:     // These assignments have been done earlier.
michael@0:     // The adjustments if f == 2^p-1 (lower boundary is closer) are done later.
michael@0:   }
michael@0: }
michael@0: 
michael@0: 
michael@0: // Let v = significand * 2^exponent.
michael@0: // Computes v / 10^estimated_power exactly, as a ratio of two bignums, numerator
michael@0: // and denominator. The functions GenerateShortestDigits and
michael@0: // GenerateCountedDigits will then convert this ratio to its decimal
michael@0: // representation d, with the required accuracy.
michael@0: // Then d * 10^estimated_power is the representation of v.
michael@0: // (Note: the fraction and the estimated_power might get adjusted before
michael@0: // generating the decimal representation.)
michael@0: //
michael@0: // The initial start values consist of:
michael@0: //  - a scaled numerator: s.t. numerator/denominator == v / 10^estimated_power.
michael@0: //  - a scaled (common) denominator.
michael@0: //  optionally (used by GenerateShortestDigits to decide if it has the shortest
michael@0: //  decimal converting back to v):
michael@0: //  - v - m-: the distance to the lower boundary.
michael@0: //  - m+ - v: the distance to the upper boundary.
michael@0: //
michael@0: // v, m+, m-, and therefore v - m- and m+ - v all share the same denominator.
michael@0: //
michael@0: // Let ep == estimated_power, then the returned values will satisfy:
michael@0: //  v / 10^ep = numerator / denominator.
michael@0: //  v's boundarys m- and m+:
michael@0: //    m- / 10^ep == v / 10^ep - delta_minus / denominator
michael@0: //    m+ / 10^ep == v / 10^ep + delta_plus / denominator
michael@0: //  Or in other words:
michael@0: //    m- == v - delta_minus * 10^ep / denominator;
michael@0: //    m+ == v + delta_plus * 10^ep / denominator;
michael@0: //
michael@0: // Since 10^(k-1) <= v < 10^k    (with k == estimated_power)
michael@0: //  or       10^k <= v < 10^(k+1)
michael@0: //  we then have 0.1 <= numerator/denominator < 1
michael@0: //           or    1 <= numerator/denominator < 10
michael@0: //
michael@0: // It is then easy to kickstart the digit-generation routine.
michael@0: //
michael@0: // The boundary-deltas are only filled if the mode equals BIGNUM_DTOA_SHORTEST
michael@0: // or BIGNUM_DTOA_SHORTEST_SINGLE.
michael@0: 
michael@0: static void InitialScaledStartValues(uint64_t significand,
michael@0:                                      int exponent,
michael@0:                                      bool lower_boundary_is_closer,
michael@0:                                      int estimated_power,
michael@0:                                      bool need_boundary_deltas,
michael@0:                                      Bignum* numerator,
michael@0:                                      Bignum* denominator,
michael@0:                                      Bignum* delta_minus,
michael@0:                                      Bignum* delta_plus) {
michael@0:   if (exponent >= 0) {
michael@0:     InitialScaledStartValuesPositiveExponent(
michael@0:         significand, exponent, estimated_power, need_boundary_deltas,
michael@0:         numerator, denominator, delta_minus, delta_plus);
michael@0:   } else if (estimated_power >= 0) {
michael@0:     InitialScaledStartValuesNegativeExponentPositivePower(
michael@0:         significand, exponent, estimated_power, need_boundary_deltas,
michael@0:         numerator, denominator, delta_minus, delta_plus);
michael@0:   } else {
michael@0:     InitialScaledStartValuesNegativeExponentNegativePower(
michael@0:         significand, exponent, estimated_power, need_boundary_deltas,
michael@0:         numerator, denominator, delta_minus, delta_plus);
michael@0:   }
michael@0: 
michael@0:   if (need_boundary_deltas && lower_boundary_is_closer) {
michael@0:     // The lower boundary is closer at half the distance of "normal" numbers.
michael@0:     // Increase the common denominator and adapt all but the delta_minus.
michael@0:     denominator->ShiftLeft(1);  // *2
michael@0:     numerator->ShiftLeft(1);    // *2
michael@0:     delta_plus->ShiftLeft(1);   // *2
michael@0:   }
michael@0: }
michael@0: 
michael@0: 
michael@0: // This routine multiplies numerator/denominator so that its values lies in the
michael@0: // range 1-10. That is after a call to this function we have:
michael@0: //    1 <= (numerator + delta_plus) /denominator < 10.
michael@0: // Let numerator the input before modification and numerator' the argument
michael@0: // after modification, then the output-parameter decimal_point is such that
michael@0: //  numerator / denominator * 10^estimated_power ==
michael@0: //    numerator' / denominator' * 10^(decimal_point - 1)
michael@0: // In some cases estimated_power was too low, and this is already the case. We
michael@0: // then simply adjust the power so that 10^(k-1) <= v < 10^k (with k ==
michael@0: // estimated_power) but do not touch the numerator or denominator.
michael@0: // Otherwise the routine multiplies the numerator and the deltas by 10.
michael@0: static void FixupMultiply10(int estimated_power, bool is_even,
michael@0:                             int* decimal_point,
michael@0:                             Bignum* numerator, Bignum* denominator,
michael@0:                             Bignum* delta_minus, Bignum* delta_plus) {
michael@0:   bool in_range;
michael@0:   if (is_even) {
michael@0:     // For IEEE doubles half-way cases (in decimal system numbers ending with 5)
michael@0:     // are rounded to the closest floating-point number with even significand.
michael@0:     in_range = Bignum::PlusCompare(*numerator, *delta_plus, *denominator) >= 0;
michael@0:   } else {
michael@0:     in_range = Bignum::PlusCompare(*numerator, *delta_plus, *denominator) > 0;
michael@0:   }
michael@0:   if (in_range) {
michael@0:     // Since numerator + delta_plus >= denominator we already have
michael@0:     // 1 <= numerator/denominator < 10. Simply update the estimated_power.
michael@0:     *decimal_point = estimated_power + 1;
michael@0:   } else {
michael@0:     *decimal_point = estimated_power;
michael@0:     numerator->Times10();
michael@0:     if (Bignum::Equal(*delta_minus, *delta_plus)) {
michael@0:       delta_minus->Times10();
michael@0:       delta_plus->AssignBignum(*delta_minus);
michael@0:     } else {
michael@0:       delta_minus->Times10();
michael@0:       delta_plus->Times10();
michael@0:     }
michael@0:   }
michael@0: }
michael@0: 
michael@0: }  // namespace double_conversion