| // 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 |
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| // (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 "config.h" |
| |
| #include <math.h> |
| |
| #include "bignum-dtoa.h" |
| |
| #include "bignum.h" |
| #include "double.h" |
| |
| namespace WTF { |
| |
| 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(double v, |
| 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, |
| BufferReference<char> buffer, int* length); |
| // Generates 'requested_digits' after the decimal point. |
| static void BignumToFixed(int requested_digits, int* decimal_point, |
| Bignum* numerator, Bignum* denominator, |
| BufferReference<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, |
| BufferReference<char>(buffer), int* length); |
| |
| |
| void BignumDtoa(double v, BignumDtoaMode mode, int requested_digits, |
| BufferReference<char> buffer, int* length, int* decimal_point) { |
| ASSERT(v > 0); |
| ASSERT(!Double(v).IsSpecial()); |
| uint64_t significand = Double(v).Significand(); |
| bool is_even = (significand & 1) == 0; |
| int exponent = Double(v).Exponent(); |
| 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); |
| bool need_boundary_deltas = (mode == BIGNUM_DTOA_SHORTEST); |
| InitialScaledStartValues(v, 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: |
| 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, |
| BufferReference<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, |
| BufferReference<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, |
| BufferReference<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 = 53; |
| double estimate = ceil((exponent + kSignificandSize - 1) * k1Log10 - 1e-10); |
| return static_cast<int>(estimate); |
| } |
| |
| |
| // See comments for InitialScaledStartValues. |
| static void InitialScaledStartValuesPositiveExponent( |
| double v, 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(Double(v).Significand()); |
| numerator->ShiftLeft(Double(v).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(Double(v).Exponent()); |
| // Same for delta_minus (with adjustments below if f == 2^p-1). |
| delta_minus->AssignUInt16(1); |
| delta_minus->ShiftLeft(Double(v).Exponent()); |
| |
| // If the significand (without the hidden bit) is 0, then the lower |
| // boundary is closer than just half a ulp (unit in the last place). |
| // There is only one exception: if the next lower number is a denormal then |
| // the distance is 1 ulp. This cannot be the case for exponent >= 0 (but we |
| // have to test it in the other function where exponent < 0). |
| uint64_t v_bits = Double(v).AsUint64(); |
| if ((v_bits & Double::kSignificandMask) == 0) { |
| // 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 |
| } |
| } |
| } |
| |
| |
| // See comments for InitialScaledStartValues |
| static void InitialScaledStartValuesNegativeExponentPositivePower( |
| double v, int estimated_power, bool need_boundary_deltas, |
| Bignum* numerator, Bignum* denominator, |
| Bignum* delta_minus, Bignum* delta_plus) { |
| uint64_t significand = Double(v).Significand(); |
| int exponent = Double(v).Exponent(); |
| // 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 (with adjustments below if f == 2^p-1). |
| delta_minus->AssignUInt16(1); |
| |
| // If the significand (without the hidden bit) is 0, then the lower |
| // boundary is closer than just one ulp (unit in the last place). |
| // There is only one exception: if the next lower number is a denormal |
| // then the distance is 1 ulp. Since the exponent is close to zero |
| // (otherwise estimated_power would have been negative) this cannot happen |
| // here either. |
| uint64_t v_bits = Double(v).AsUint64(); |
| if ((v_bits & Double::kSignificandMask) == 0) { |
| // The lower boundary is closer at half the distance of "normal" numbers. |
| // Increase the denominator and adapt all but the delta_minus. |
| denominator->ShiftLeft(1); // *2 |
| numerator->ShiftLeft(1); // *2 |
| delta_plus->ShiftLeft(1); // *2 |
| } |
| } |
| } |
| |
| |
| // See comments for InitialScaledStartValues |
| static void InitialScaledStartValuesNegativeExponentNegativePower( |
| double v, int estimated_power, bool need_boundary_deltas, |
| Bignum* numerator, Bignum* denominator, |
| Bignum* delta_minus, Bignum* delta_plus) { |
| const uint64_t kMinimalNormalizedExponent = |
| UINT64_2PART_C(0x00100000, 00000000); |
| uint64_t significand = Double(v).Significand(); |
| int exponent = Double(v).Exponent(); |
| // 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 special case where the lower boundary is twice as close. |
| // This time we have to look out for the exception too. |
| uint64_t v_bits = Double(v).AsUint64(); |
| if ((v_bits & Double::kSignificandMask) == 0 && |
| // The only exception where a significand == 0 has its boundaries at |
| // "normal" distances: |
| (v_bits & Double::kExponentMask) != kMinimalNormalizedExponent) { |
| numerator->ShiftLeft(1); // *2 |
| denominator->ShiftLeft(1); // *2 |
| delta_plus->ShiftLeft(1); // *2 |
| } |
| } |
| } |
| |
| |
| // 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 need_boundary_deltas is set. |
| static void InitialScaledStartValues(double v, |
| int estimated_power, |
| bool need_boundary_deltas, |
| Bignum* numerator, |
| Bignum* denominator, |
| Bignum* delta_minus, |
| Bignum* delta_plus) { |
| if (Double(v).Exponent() >= 0) { |
| InitialScaledStartValuesPositiveExponent( |
| v, estimated_power, need_boundary_deltas, |
| numerator, denominator, delta_minus, delta_plus); |
| } else if (estimated_power >= 0) { |
| InitialScaledStartValuesNegativeExponentPositivePower( |
| v, estimated_power, need_boundary_deltas, |
| numerator, denominator, delta_minus, delta_plus); |
| } else { |
| InitialScaledStartValuesNegativeExponentNegativePower( |
| v, estimated_power, need_boundary_deltas, |
| numerator, denominator, delta_minus, delta_plus); |
| } |
| } |
| |
| |
| // 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 |
| |
| } // namespace WTF |
