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
}
}
}