void SubtractBignum(Bignum other) { Debug.Assert(IsClamped()); Debug.Assert(other.IsClamped()); // We require this to be bigger than other. Debug.Assert(LessEqual(other, this)); Align(other); int offset = other.exponent_ - exponent_; uint borrow = 0; int i; for (i = 0; i < other.used_digits_; ++i) { Debug.Assert((borrow == 0) || (borrow == 1)); uint difference = bigits_[i + offset] - other.bigits_[i] - borrow; bigits_[i + offset] = difference & kBigitMask; borrow = difference >> (kChunkSize - 1); } while (borrow != 0) { uint difference = bigits_[i + offset] - borrow; bigits_[i + offset] = difference & kBigitMask; borrow = difference >> (kChunkSize - 1); ++i; } Clamp(); }
void SubtractBignum(Bignum other) { Align(other); int offset = other.exponent_ - exponent_; uint borrow = 0; int i; for (i = 0; i < other.used_digits_; ++i) { uint difference = bigits_[i + offset] - other.bigits_[i] - borrow; bigits_[i + offset] = difference & kBigitMask; borrow = difference >> (kChunkSize - 1); } while (borrow != 0) { uint difference = bigits_[i + offset] - borrow; bigits_[i + offset] = difference & kBigitMask; borrow = difference >> (kChunkSize - 1); ++i; } Clamp(); }
static int Compare(Bignum a, Bignum b) { int bigit_length_a = a.BigitLength(); int bigit_length_b = b.BigitLength(); if (bigit_length_a < bigit_length_b) { return(-1); } if (bigit_length_a > bigit_length_b) { return(+1); } for (int i = bigit_length_a - 1; i >= System.Math.Min(a.exponent_, b.exponent_); --i) { uint bigit_a = a.BigitAt(i); uint bigit_b = b.BigitAt(i); if (bigit_a < bigit_b) { return(-1); } if (bigit_a > bigit_b) { return(+1); } // Otherwise they are equal up to this digit. Try the next digit. } return(0); }
void Align(Bignum other) { if (exponent_ > other.exponent_) { // If "X" represents a "hidden" digit (by the exponent) then we are in the // following case (a == this, b == other): // a: aaaaaaXXXX or a: aaaaaXXX // b: bbbbbbX b: bbbbbbbbXX // We replace some of the hidden digits (X) of a with 0 digits. // a: aaaaaa000X or a: aaaaa0XX int zero_digits = exponent_ - other.exponent_; EnsureCapacity(used_digits_ + zero_digits); for (int i = used_digits_ - 1; i >= 0; --i) { bigits_[i + zero_digits] = bigits_[i]; } for (int i = 0; i < zero_digits; ++i) { bigits_[i] = 0; } used_digits_ += zero_digits; exponent_ -= zero_digits; } }
internal static int PlusCompare(Bignum a, Bignum b, Bignum c) { Debug.Assert(a.IsClamped()); Debug.Assert(b.IsClamped()); Debug.Assert(c.IsClamped()); if (a.BigitLength() < b.BigitLength()) { return(PlusCompare(b, a, c)); } if (a.BigitLength() + 1 < c.BigitLength()) { return(-1); } if (a.BigitLength() > c.BigitLength()) { return(+1); } // The exponent encodes 0-bigits. So if there are more 0-digits in 'a' than // 'b' has digits, then the bigit-length of 'a'+'b' must be equal to the one // of 'a'. if (a.exponent_ >= b.BigitLength() && a.BigitLength() < c.BigitLength()) { return(-1); } uint borrow = 0; // Starting at min_exponent all digits are == 0. So no need to compare them. int min_exponent = System.Math.Min(System.Math.Min(a.exponent_, b.exponent_), c.exponent_); for (int i = c.BigitLength() - 1; i >= min_exponent; --i) { uint chunk_a = a.BigitAt(i); uint chunk_b = b.BigitAt(i); uint chunk_c = c.BigitAt(i); uint sum = chunk_a + chunk_b; if (sum > chunk_c + borrow) { return(+1); } else { borrow = chunk_c + borrow - sum; if (borrow > 1) { return(-1); } borrow <<= kBigitSize; } } if (borrow == 0) { return(0); } return(-1); }
// See comments for InitialScaledStartValues private static void InitialScaledStartValuesNegativeExponentPositivePower( double v, int estimated_power, bool need_boundary_deltas, Bignum numerator, Bignum denominator, Bignum delta_minus, Bignum delta_plus) { var bits = (ulong)BitConverter.DoubleToInt64Bits(v); ulong significand = DoubleHelper.Significand(bits); int exponent = DoubleHelper.Exponent(bits); // 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. ulong v_bits = bits; if ((v_bits & DoubleHelper.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. private 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. Debug.Assert(estimated_power >= 0); // Since the estimated_power is positive we simply multiply the denominator // by 10^estimated_power. // numerator = v. var bits = (ulong)BitConverter.DoubleToInt64Bits(v); numerator.AssignUInt64(DoubleHelper.Significand(bits)); numerator.ShiftLeft(DoubleHelper.Exponent(bits)); // 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(DoubleHelper.Exponent(bits)); // Same for delta_minus (with adjustments below if f == 2^p-1). delta_minus.AssignUInt16(1); delta_minus.ShiftLeft(DoubleHelper.Exponent(bits)); // 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). ulong v_bits = bits; if ((v_bits & DoubleHelper.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 } } }
// 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, ref int decimal_point, Bignum numerator, Bignum denominator, DtoaBuilder buffer) { // 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; buffer.Reset(); return; } 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. Debug.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'; decimal_point++; } else { // Note that we caught most of similar cases earlier. buffer.Reset(); } } 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, ref decimal_point, numerator, denominator, buffer); } }
void SubtractTimes(Bignum other, uint factor) { #if DEBUG var a = new Bignum(); var b = new Bignum(); a.AssignBignum(this); b.AssignBignum(other); b.MultiplyByUInt32(factor); a.SubtractBignum(b); #endif Debug.Assert(exponent_ <= other.exponent_); if (factor < 3) { for (int i = 0; i < factor; ++i) { SubtractBignum(other); } return; } uint borrow = 0; int exponent_diff = other.exponent_ - exponent_; for (int i = 0; i < other.used_digits_; ++i) { ulong product = factor * other.bigits_[i]; ulong remove = borrow + product; uint difference = bigits_[i + exponent_diff] - (uint)(remove & kBigitMask); bigits_[i + exponent_diff] = difference & kBigitMask; borrow = (uint)((difference >> (kChunkSize - 1)) + (remove >> kBigitSize)); } for (int i = other.used_digits_ + exponent_diff; i < used_digits_; ++i) { if (borrow == 0) { return; } uint difference = bigits_[i] - borrow; bigits_[i] = difference & kBigitMask; borrow = difference >> (kChunkSize - 1); } Clamp(); #if DEBUG Debug.Assert(Equal(a, this)); #endif }
internal void AssignBignum(Bignum other) { exponent_ = other.exponent_; for (int i = 0; i < other.used_digits_; ++i) { bigits_[i] = other.bigits_[i]; } // Clear the excess digits (if there were any). for (int i = other.used_digits_; i < used_digits_; ++i) { bigits_[i] = 0; } used_digits_ = other.used_digits_; }
// 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. private static void InitialScaledStartValues( double v, int estimated_power, bool need_boundary_deltas, Bignum numerator, Bignum denominator, Bignum delta_minus, Bignum delta_plus) { var bits = (ulong)BitConverter.DoubleToInt64Bits(v); if (DoubleHelper.Exponent(bits) >= 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); } }
// 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, ref int decimal_point, Bignum numerator, Bignum denominator, DtoaBuilder buffer) { Debug.Assert(count >= 0); for (int i = 0; i < count - 1; ++i) { uint d = numerator.DivideModuloIntBignum(denominator); Debug.Assert(d <= 9); // digit is a uint and therefore always positive. // digit = numerator / denominator (integer division). // numerator = numerator % denominator. buffer.Append((char)(d + '0')); // Prepare for next iteration. numerator.Times10(); } // Generate the last digit. uint digit = numerator.DivideModuloIntBignum(denominator); if (Bignum.PlusCompare(numerator, numerator, denominator) >= 0) { digit++; } buffer.Append((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++; } }
// 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. private static void FixupMultiply10( int estimated_power, bool is_even, out int decimal_point, Bignum numerator, Bignum denominator, Bignum delta_minus, Bignum delta_plus) { bool in_range; if (is_even) { 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(); } } }
public static void NumberToString( double v, DtoaMode mode, int requested_digits, DtoaBuilder builder, out int decimal_point) { var bits = (ulong)BitConverter.DoubleToInt64Bits(v); var significand = DoubleHelper.Significand(bits); var is_even = (significand & 1) == 0; var exponent = DoubleHelper.Exponent(bits); var normalized_exponent = DoubleHelper.NormalizedExponent(significand, exponent); // estimated_power might be too low by 1. var 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 == DtoaMode.Fixed && -estimated_power - 1 > requested_digits) { // 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 = new Bignum(); Bignum denominator = new Bignum(); Bignum delta_minus = new Bignum(); Bignum delta_plus = new Bignum(); // 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. var need_boundary_deltas = mode == DtoaMode.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, out 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 DtoaMode.Shortest: GenerateShortestDigits( numerator, denominator, delta_minus, delta_plus, is_even, builder); break; case DtoaMode.Fixed: BignumToFixed( requested_digits, ref decimal_point, numerator, denominator, builder); break; case DtoaMode.Precision: GenerateCountedDigits( requested_digits, ref decimal_point, numerator, denominator, builder); break; default: ExceptionHelper.ThrowArgumentOutOfRangeException(); break; } }
// Precondition: this/other < 16bit. public uint DivideModuloIntBignum(Bignum other) { // Easy case: if we have less digits than the divisor than the result is 0. // Note: this handles the case where this == 0, too. if (BigitLength() < other.BigitLength()) { return(0); } Align(other); uint result = 0; // Start by removing multiples of 'other' until both numbers have the same // number of digits. while (BigitLength() > other.BigitLength()) { // This naive approach is extremely inefficient if the this divided other // might be big. This function is implemented for doubleToString where // the result should be small (less than 10). // Remove the multiples of the first digit. // Example this = 23 and other equals 9. -> Remove 2 multiples. result += bigits_[used_digits_ - 1]; SubtractTimes(other, bigits_[used_digits_ - 1]); } // Both bignums are at the same length now. // Since other has more than 0 digits we know that the access to // bigits_[used_digits_ - 1] is safe. var this_bigit = bigits_[used_digits_ - 1]; var other_bigit = other.bigits_[other.used_digits_ - 1]; if (other.used_digits_ == 1) { // Shortcut for easy (and common) case. uint quotient = this_bigit / other_bigit; bigits_[used_digits_ - 1] = this_bigit - other_bigit * quotient; result += quotient; Clamp(); return(result); } uint division_estimate = this_bigit / (other_bigit + 1); result += division_estimate; SubtractTimes(other, division_estimate); if (other_bigit * (division_estimate + 1) > this_bigit) { return(result); } while (LessEqual(other, this)) { SubtractBignum(other); result++; } return(result); }
// 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. private static void GenerateShortestDigits( Bignum numerator, Bignum denominator, Bignum delta_minus, Bignum delta_plus, bool is_even, DtoaBuilder buffer) { // 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; } buffer.Reset(); while (true) { uint digit; digit = numerator.DivideModuloIntBignum(denominator); // digit = numerator / denominator (integer division). // numerator = numerator % denominator. buffer.Append((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. buffer[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[buffer.Length - 1] - '0') % 2 == 0) { // Round down => Do nothing. } else { buffer[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 DCHECK here, in case the preconditions were not // satisfied. buffer[buffer.Length - 1]++; return; } } }
// Returns a + b < c static bool PlusLess(Bignum a, Bignum b, Bignum c) { return(PlusCompare(a, b, c) < 0); }
// Returns a + b == c static bool PlusEqual(Bignum a, Bignum b, Bignum c) { return(PlusCompare(a, b, c) == 0); }
internal static bool Less(Bignum a, Bignum b) { return(Compare(a, b) < 0); }
internal static bool LessEqual(Bignum a, Bignum b) { return(Compare(a, b) <= 0); }
internal static bool Equal(Bignum a, Bignum b) { return(Compare(a, b) == 0); }
// See comments for InitialScaledStartValues private static void InitialScaledStartValuesNegativeExponentNegativePower( double v, int estimated_power, bool need_boundary_deltas, Bignum numerator, Bignum denominator, Bignum delta_minus, Bignum delta_plus) { const ulong kMinimalNormalizedExponent = 0x0010000000000000; var bits = (ulong)BitConverter.DoubleToInt64Bits(v); ulong significand = DoubleHelper.Significand(bits); int exponent = DoubleHelper.Exponent(bits); // 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. 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. ulong v_bits = bits; if ((v_bits & DoubleHelper.KSignificandMask) == 0 && // The only exception where a significand == 0 has its boundaries at // "normal" distances: (v_bits & DoubleHelper.KExponentMask) != kMinimalNormalizedExponent) { numerator.ShiftLeft(1); // *2 denominator.ShiftLeft(1); // *2 delta_plus.ShiftLeft(1); // *2 } } }