// 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++;
}
}
// 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;
}
}
}