// Provides a decimal representation of v.
// Returns true if it succeeds, otherwise the result cannot be trusted.
// There will be *length digits inside the buffer (not null-terminated).
// If the function returns true then
// v == (double) (buffer * 10^decimal_exponent).
// The digits in the buffer are the shortest representation possible: no
// 0.09999999999999999 instead of 0.1. The shorter representation will even be
// chosen even if the longer one would be closer to v.
// The last digit will be closest to the actual v. That is, even if several
// digits might correctly yield 'v' when read again, the closest will be
// computed.
private static bool Grisu3(double v, FastDtoaMode mode, Span <byte> buffer, out int length, out int decimalExponent)
{
var w = new IeeeDouble(v).AsNormalizedDiyFp();
// boundary_minus and boundary_plus are the boundaries between v and its
// closest floating-point neighbors. Any number strictly between
// boundary_minus and boundary_plus will round to v when convert to a double.
// Grisu3 will never output representations that lie exactly on a boundary.
DiyFp boundaryMinus, boundaryPlus;
switch (mode)
{
case FastDtoaMode.FastDtoaShortest:
new IeeeDouble(v).NormalizedBoundaries(out boundaryMinus, out boundaryPlus);
break;
case FastDtoaMode.FastDtoaShortestSingle:
{
var singleV = (float)v;
new IeeeSingle(singleV).NormalizedBoundaries(out boundaryMinus, out boundaryPlus);
break;
}
default:
throw new Exception("Invalid Mode.");
}
var tenMkMinimalBinaryExponent = KMinimalTargetExponent - (w.e + DiyFp.kSignificandSize);
PowersOfTenCache.GetCachedPowerForBinaryExponentRange(tenMkMinimalBinaryExponent, out var tenMk, out var mk);
// Note that ten_mk is only an approximation of 10^-k. A DiyFp only contains a
// 64 bit significand and ten_mk is thus only precise up to 64 bits.
// The DiyFp::Times procedure rounds its result, and ten_mk is approximated
// too. The variable scaled_w (as well as scaled_boundary_minus/plus) are now
// off by a small amount.
// In fact: scaled_w - w*10^k < 1ulp (unit in the last place) of scaled_w.
// In other words: let f = scaled_w.f() and e = scaled_w.e(), then
// (f-1) * 2^e < w*10^k < (f+1) * 2^e
var scaledW = DiyFp.Times(ref w, ref tenMk);
// In theory it would be possible to avoid some recomputations by computing
// the difference between w and boundary_minus/plus (a power of 2) and to
// compute scaled_boundary_minus/plus by subtracting/adding from
// scaled_w. However the code becomes much less readable and the speed
// enhancements are not terrific.
var scaledBoundaryMinus = DiyFp.Times(ref boundaryMinus, ref tenMk);
var scaledBoundaryPlus = DiyFp.Times(ref boundaryPlus, ref tenMk);
// DigitGen will generate the digits of scaled_w. Therefore we have
// v == (double) (scaled_w * 10^-mk).
// Set decimal_exponent == -mk and pass it to DigitGen. If scaled_w is not an
// integer than it will be updated. For instance if scaled_w == 1.23 then
// the buffer will be filled with "123" und the decimal_exponent will be
// decreased by 2.
var result = DigitGen(scaledBoundaryMinus, scaledW, scaledBoundaryPlus,
buffer, out length, out var kappa);
decimalExponent = -mk + kappa;
return(result);
}
// 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);
}
