// If the function returns true then the result is the correct double.
// Otherwise it is either the correct double or the double that is just below
// the correct double.
static bool DiyFpStrtod(Vector buffer,
int exponent,
out double result)
{
DiyFp input;
int remaining_decimals;
ReadDiyFp(buffer, out input, out remaining_decimals);
// Since we may have dropped some digits the input is not accurate.
// If remaining_decimals is different than 0 than the error is at most
// .5 ulp (unit in the last place).
// We don't want to deal with fractions and therefore keep a common
// denominator.
const int kDenominatorLog = 3;
const int kDenominator = 1 << kDenominatorLog;
// Move the remaining decimals into the exponent.
exponent += remaining_decimals;
uint64_t error = (ulong)(remaining_decimals == 0 ? 0 : kDenominator / 2);
int old_e = input.e;
input.Normalize();
error <<= old_e - input.e;
if (exponent < PowersOfTenCache.kMinDecimalExponent)
{
result = 0.0;
return(true);
}
DiyFp cached_power;
int cached_decimal_exponent;
PowersOfTenCache.GetCachedPowerForDecimalExponent(exponent,
out cached_power,
out cached_decimal_exponent);
if (cached_decimal_exponent != exponent)
{
int adjustment_exponent = exponent - cached_decimal_exponent;
DiyFp adjustment_power = AdjustmentPowerOfTen(adjustment_exponent);
input.Multiply(ref adjustment_power);
if (kMaxUint64DecimalDigits - buffer.length() >= adjustment_exponent)
{
// The product of input with the adjustment power fits into a 64 bit
// integer.
}
else
{
// The adjustment power is exact. There is hence only an error of 0.5.
error += kDenominator / 2;
}
}
input.Multiply(ref cached_power);
// The error introduced by a multiplication of a*b equals
// error_a + error_b + error_a*error_b/2^64 + 0.5
// Substituting a with 'input' and b with 'cached_power' we have
// error_b = 0.5 (all cached powers have an error of less than 0.5 ulp),
// error_ab = 0 or 1 / kDenominator > error_a*error_b/ 2^64
int error_b = kDenominator / 2;
int error_ab = (error == 0 ? 0 : 1); // We round up to 1.
int fixed_error = kDenominator / 2;
error += (ulong)(error_b + error_ab + fixed_error);
old_e = input.e;
input.Normalize();
error <<= old_e - input.e;
// See if the double's significand changes if we add/subtract the error.
int order_of_magnitude = DiyFp.kSignificandSize + input.e;
int effective_significand_size = Double.SignificandSizeForOrderOfMagnitude(order_of_magnitude);
int precision_digits_count = DiyFp.kSignificandSize - effective_significand_size;
if (precision_digits_count + kDenominatorLog >= DiyFp.kSignificandSize)
{
// This can only happen for very small denormals. In this case the
// half-way multiplied by the denominator exceeds the range of an uint64.
// Simply shift everything to the right.
int shift_amount = (precision_digits_count + kDenominatorLog) -
DiyFp.kSignificandSize + 1;
input.f = (input.f >> shift_amount);
input.e = (input.e + shift_amount);
// We add 1 for the lost precision of error, and kDenominator for
// the lost precision of input.f().
error = (error >> shift_amount) + 1 + kDenominator;
precision_digits_count -= shift_amount;
}
// We use uint64_ts now. This only works if the DiyFp uses uint64_ts too.
uint64_t one64 = 1;
uint64_t precision_bits_mask = (one64 << precision_digits_count) - 1;
uint64_t precision_bits = input.f & precision_bits_mask;
uint64_t half_way = one64 << (precision_digits_count - 1);
precision_bits *= kDenominator;
half_way *= kDenominator;
DiyFp rounded_input = new DiyFp(input.f >> precision_digits_count, input.e + precision_digits_count);
if (precision_bits >= half_way + error)
{
rounded_input.f = (rounded_input.f + 1);
}
// If the last_bits are too close to the half-way case than we are too
// inaccurate and round down. In this case we return false so that we can
// fall back to a more precise algorithm.
result = new Double(rounded_input).value();
if (half_way - error < precision_bits && precision_bits < half_way + error)
{
// Too imprecise. The caller will have to fall back to a slower version.
// However the returned number is guaranteed to be either the correct
// double, or the next-lower double.
return(false);
}
else
{
return(true);
}
}
// If the function returns true then the result is the correct double.
// Otherwise it is either the correct double or the double that is just below
// the correct double.
private static bool DiyFpStrToDouble(ReadOnlySpan <byte> buffer, int exponent, out double result)
{
ReadDiyFp(buffer, out var input, out var remainingDecimals);
// Since we may have dropped some digits the input is not accurate.
// If remaining_decimals is different than 0 than the error is at most
// .5 ulp (unit in the last place).
// We don't want to deal with fractions and therefore keep a common
// denominator.
const int kDenominatorLog = 3;
const int kDenominator = 1 << kDenominatorLog;
// Move the remaining decimals into the exponent.
exponent += remainingDecimals;
var error = (ulong)(remainingDecimals == 0 ? 0 : kDenominator / 2);
var oldE = input.e;
input.Normalize();
error <<= oldE - input.e;
if (exponent < PowersOfTenCache.kMinDecimalExponent)
{
result = 0.0;
return(true);
}
PowersOfTenCache.GetCachedPowerForDecimalExponent(exponent, out var cachedPower, out var cachedDecimalExponent);
if (cachedDecimalExponent != exponent)
{
var adjustmentExponent = exponent - cachedDecimalExponent;
var adjustmentPower = AdjustmentPowerOfTen(adjustmentExponent);
input.Multiply(ref adjustmentPower);
if (KMaxUint64DecimalDigits - buffer.Length >= adjustmentExponent)
{
// The product of input with the adjustment power fits into a 64 bit
// integer.
}
else
{
// The adjustment power is exact. There is hence only an error of 0.5.
error += kDenominator / 2;
}
}
input.Multiply(ref cachedPower);
// The error introduced by a multiplication of a*b equals
// error_a + error_b + error_a*error_b/2^64 + 0.5
// Substituting a with 'input' and b with 'cached_power' we have
// error_b = 0.5 (all cached powers have an error of less than 0.5 ulp),
// error_ab = 0 or 1 / kDenominator > error_a*error_b/ 2^64
const int errorB = kDenominator / 2;
var errorAb = error == 0 ? 0 : 1; // We round up to 1.
const int fixedError = kDenominator / 2;
error += (ulong)(errorB + errorAb + fixedError);
oldE = input.e;
input.Normalize();
error <<= oldE - input.e;
// See if the double's significand changes if we add/subtract the error.
var orderOfMagnitude = DiyFp.kSignificandSize + input.e;
var effectiveSignificandSize = IeeeDouble.SignificandSizeForOrderOfMagnitude(orderOfMagnitude);
var precisionDigitsCount = DiyFp.kSignificandSize - effectiveSignificandSize;
if (precisionDigitsCount + kDenominatorLog >= DiyFp.kSignificandSize)
{
// This can only happen for very small denormals. In this case the
// half-way multiplied by the denominator exceeds the range of an uint64.
// Simply shift everything to the right.
var shiftAmount = precisionDigitsCount + kDenominatorLog -
DiyFp.kSignificandSize + 1;
input.f >>= shiftAmount;
input.e += shiftAmount;
// We add 1 for the lost precision of error, and kDenominator for
// the lost precision of input.f().
error = (error >> shiftAmount) + 1 + kDenominator;
precisionDigitsCount -= shiftAmount;
}
// We use uint64_ts now. This only works if the DiyFp uses uint64_ts too.
const ulong one64 = 1;
var precisionBitsMask = (one64 << precisionDigitsCount) - 1;
var precisionBits = input.f & precisionBitsMask;
var halfWay = one64 << (precisionDigitsCount - 1);
precisionBits *= kDenominator;
halfWay *= kDenominator;
var roundedInput = new DiyFp(input.f >> precisionDigitsCount, input.e + precisionDigitsCount);
if (precisionBits >= halfWay + error)
{
roundedInput.f++;
}
// If the last_bits are too close to the half-way case than we are too
// inaccurate and round down. In this case we return false so that we can
// fall back to a more precise algorithm.
result = new IeeeDouble(roundedInput).Value();
// Too imprecise. The caller will have to fall back to a slower version.
// However the returned number is guaranteed to be either the correct
// double, or the next-lower double.
return(halfWay - error >= precisionBits || precisionBits >= halfWay + error);
}
