The Tor Browser: comparison mfbt/double-conversion/bignum-dtoa.cc

--1:000000000000
+:e70d15c0a449
+// Copyright 2010 the V8 project authors. All rights reserved.
+// Redistribution and use in source and binary forms, with or without
+// modification, are permitted provided that the following conditions are
+// met:
+//
+//     * Redistributions of source code must retain the above copyright
+//       notice, this list of conditions and the following disclaimer.
+//     * Redistributions in binary form must reproduce the above
+//       copyright notice, this list of conditions and the following
+//       disclaimer in the documentation and/or other materials provided
+//       with the distribution.
+//     * Neither the name of Google Inc. nor the names of its
+//       contributors may be used to endorse or promote products derived
+//       from this software without specific prior written permission.
+//
+// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
+// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
+// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
+// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
+// OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
+// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
+// LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
+// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
+// THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
+// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
+// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
+#include <math.h>
+#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;
+while (true) {
+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)++] = 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] = 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++;
+}
+buffer[count - 1] = 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

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