// returns a * b;
public static DiyFp Times(ref DiyFp a, ref DiyFp b)
{
DiyFp result = a;
result.Multiply(ref b);
return(result);
}
// Returns a - b.
// The exponents of both numbers must be the same and this must be bigger
// than other. The result will not be normalized.
public static DiyFp Minus(ref DiyFp a, ref DiyFp b)
{
DiyFp result = a;
result.Subtract(ref b);
return(result);
}
// this = this * other.
public void Multiply(ref DiyFp other)
{
// Simply "emulates" a 128 bit multiplication.
// However: the resulting number only contains 64 bits. The least
// significant 64 bits are only used for rounding the most significant 64
// bits.
const long kM32 = 0xFFFFFFFFU;
ulong a = f >> 32;
ulong b = f & kM32;
ulong c = other.f >> 32;
ulong d = other.f & kM32;
ulong ac = a * c;
ulong bc = b * c;
ulong ad = a * d;
ulong bd = b * d;
ulong tmp = (bd >> 32) + (ad & kM32) + (bc & kM32);
// By adding 1U << 31 to tmp we round the final result.
// Halfway cases will be round up.
tmp += 1U << 31;
ulong result_f = ac + (ad >> 32) + (bc >> 32) + (tmp >> 32);
e += other.e + 64;
f = result_f;
}
public static DiyFp Normalize(ref DiyFp a)
{
DiyFp result = a;
result.Normalize();
return(result);
}
// Reads a DiyFp from the buffer.
// The returned DiyFp is not necessarily normalized.
// If remaining_decimals is zero then the returned DiyFp is accurate.
// Otherwise it has been rounded and has error of at most 1/2 ulp.
static void ReadDiyFp(Vector buffer,
out DiyFp result,
out int remaining_decimals)
{
int read_digits;
uint64_t significand = ReadUint64(buffer, out read_digits);
if (buffer.length() == read_digits)
{
result = new DiyFp(significand, 0);
remaining_decimals = 0;
}
else
{
// Round the significand.
if (buffer[read_digits] >= '5')
{
significand++;
}
// Compute the binary exponent.
int exponent = 0;
result = new DiyFp(significand, exponent);
remaining_decimals = buffer.length() - read_digits;
}
}
public static void GetCachedPowerForDecimalExponent(int requested_exponent,
out DiyFp power,
out int found_exponent)
{
int index = (requested_exponent + kCachedPowersOffset) / kDecimalExponentDistance;
CachedPower cached_power = kCachedPowers[index];
power = new DiyFp(cached_power.significand, cached_power.binary_exponent);
found_exponent = cached_power.decimal_exponent;
}
public static void GetCachedPowerForBinaryExponentRange(int min_exponent,
out DiyFp power,
out int decimal_exponent)
{
const int kQ = DiyFp.kSignificandSize;
var k = Math.Ceiling((min_exponent + kQ - 1) * kD_1_LOG2_10);
const int foo = kCachedPowersOffset;
var index = (foo + (int)k - 1) / kDecimalExponentDistance + 1;
var cached_power = kCachedPowers[index];
// (void)max_exponent; // Mark variable as used.
decimal_exponent = cached_power.decimal_exponent;
power = new DiyFp(cached_power.significand, cached_power.binary_exponent);
}
// Computes the two boundaries of this.
// The bigger boundary (m_plus) is normalized. The lower boundary has the same
// exponent as m_plus.
// Precondition: the value encoded by this Single must be greater than 0.
public void NormalizedBoundaries(out DiyFp out_m_minus, out DiyFp out_m_plus)
{
DiyFp v = this.AsDiyFp();
var __ = new DiyFp((v.f << 1) + 1, v.e - 1);
DiyFp m_plus = DiyFp.Normalize(ref __);
DiyFp m_minus;
if (LowerBoundaryIsCloser())
{
m_minus = new DiyFp((v.f << 2) - 1, v.e - 2);
}
else
{
m_minus = new DiyFp((v.f << 1) - 1, v.e - 1);
}
m_minus.f = (m_minus.f << (m_minus.e - m_plus.e));
m_minus.e = (m_plus.e);
out_m_plus = m_plus;
out_m_minus = m_minus;
}
// Reads a DiyFp from the buffer.
// The returned DiyFp is not necessarily normalized.
// If remaining_decimals is zero then the returned DiyFp is accurate.
// Otherwise it has been rounded and has error of at most 1/2 ulp.
private static void ReadDiyFp(ReadOnlySpan <byte> buffer, out DiyFp result, out int remainingDecimals)
{
var significand = ReadUint64(buffer, out var readDigits);
if (buffer.Length == readDigits)
{
result = new DiyFp(significand, 0);
remainingDecimals = 0;
}
else
{
// Round the significand.
if (buffer[readDigits] >= '5')
{
significand++;
}
// Compute the binary exponent.
const int exponent = 0;
result = new DiyFp(significand, exponent);
remainingDecimals = buffer.Length - readDigits;
}
}
public static ulong DiyFpToUint64(DiyFp diy_fp)
{
ulong significand = diy_fp.f;
int exponent = diy_fp.e;
while (significand > kHiddenBit + kSignificandMask)
{
significand >>= 1;
exponent++;
}
if (exponent >= kMaxExponent)
{
return(kInfinity);
}
if (exponent < kDenormalExponent)
{
return(0);
}
while (exponent > kDenormalExponent && (significand & kHiddenBit) == 0)
{
significand <<= 1;
exponent--;
}
ulong biased_exponent;
if (exponent == kDenormalExponent && (significand & kHiddenBit) == 0)
{
biased_exponent = 0;
}
else
{
biased_exponent = (ulong)(exponent + kExponentBias);
}
return((significand & kSignificandMask) |
(biased_exponent << kPhysicalSignificandSize));
}
// 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);
}
}
// this = this - other.
// The exponents of both numbers must be the same and the significand of this
// must be bigger than the significand of other.
// The result will not be normalized.
public void Subtract(ref DiyFp other)
{
f -= other.f;
}
// 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);
}
// Generates the digits of input number w.
// w is a floating-point number (DiyFp), consisting of a significand and an
// exponent. Its exponent is bounded by kMinimalTargetExponent and
// kMaximalTargetExponent.
// Hence -60 <= w.e() <= -32.
//
// Returns false if it fails, in which case the generated digits in the buffer
// should not be used.
// Preconditions:
// * low, w and high are correct up to 1 ulp (unit in the last place). That
// is, their error must be less than a unit of their last digits.
// * low.e() == w.e() == high.e()
// * low < w < high, and taking into account their error: low~ <= high~
// * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
// Post conditions: returns false if procedure fails.
// otherwise:
// * buffer is not null-terminated, but len contains the number of digits.
// * buffer contains the shortest possible decimal digit-sequence
// such that LOW < buffer * 10^kappa < HIGH, where LOW and HIGH are the
// correct values of low and high (without their error).
// * if more than one decimal representation gives the minimal number of
// decimal digits then the one closest to W (where W is the correct value
// of w) is chosen.
// Remark: this procedure takes into account the imprecision of its input
// numbers. If the precision is not enough to guarantee all the post conditions
// then false is returned. This usually happens rarely (~0.5%).
//
// Say, for the sake of example, that
// w.e() == -48, and w.f() == 0x1234567890abcdef
// w's value can be computed by w.f() * 2^w.e()
// We can obtain w's integral digits by simply shifting w.f() by -w.e().
// -> w's integral part is 0x1234
// w's fractional part is therefore 0x567890abcdef.
// Printing w's integral part is easy (simply print 0x1234 in decimal).
// In order to print its fraction we repeatedly multiply the fraction by 10 and
// get each digit. Example the first digit after the point would be computed by
// (0x567890abcdef * 10) >> 48. -> 3
// The whole thing becomes slightly more complicated because we want to stop
// once we have enough digits. That is, once the digits inside the buffer
// represent 'w' we can stop. Everything inside the interval low - high
// represents w. However we have to pay attention to low, high and w's
// imprecision.
private static bool DigitGen(DiyFp low,
DiyFp w,
DiyFp high,
Span <byte> buffer,
out int length,
out int kappa)
{
// low, w and high are imprecise, but by less than one ulp (unit in the last
// place).
// If we remove (resp. add) 1 ulp from low (resp. high) we are certain that
// the new numbers are outside of the interval we want the final
// representation to lie in.
// Inversely adding (resp. removing) 1 ulp from low (resp. high) would yield
// numbers that are certain to lie in the interval. We will use this fact
// later on.
// We will now start by generating the digits within the uncertain
// interval. Later we will weed out representations that lie outside the safe
// interval and thus _might_ lie outside the correct interval.
ulong unit = 1;
var tooLow = new DiyFp(low.f - unit, low.e);
var tooHigh = new DiyFp(high.f + unit, high.e);
// too_low and too_high are guaranteed to lie outside the interval we want the
// generated number in.
var unsafeInterval = DiyFp.Minus(ref tooHigh, ref tooLow);
// We now cut the input number into two parts: the integral digits and the
// fractionals. We will not write any decimal separator though, but adapt
// kappa instead.
// Reminder: we are currently computing the digits (stored inside the buffer)
// such that: too_low < buffer * 10^kappa < too_high
// We use too_high for the digit_generation and stop as soon as possible.
// If we stop early we effectively round down.
var one = new DiyFp((ulong)1 << -w.e, w.e);
// Division by one is a shift.
var integrals = (uint)(tooHigh.f >> -one.e);
// Modulo by one is an and.
var fractionals = tooHigh.f & (one.f - 1);
BiggestPowerTen(integrals, DiyFp.kSignificandSize - -one.e,
out var divisor, out var divisorExponentPlusOne);
kappa = divisorExponentPlusOne;
length = 0;
// Loop invariant: buffer = too_high / 10^kappa (integer division)
// The invariant holds for the first iteration: kappa has been initialized
// with the divisor exponent + 1. And the divisor is the biggest power of ten
// that is smaller than integrals.
while (kappa > 0)
{
var digit = unchecked ((int)(integrals / divisor));
buffer[length] = (byte)((byte)'0' + digit);
length++;
integrals %= divisor;
kappa--;
// Note that kappa now equals the exponent of the divisor and that the
// invariant thus holds again.
var rest =
((ulong)integrals << -one.e) + fractionals;
// Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())
// Reminder: unsafe_interval.e() == one.e()
if (rest < unsafeInterval.f)
{
// Rounding down (by not emitting the remaining digits) yields a number
// that lies within the unsafe interval.
return(RoundWeed(buffer, length, DiyFp.Minus(ref tooHigh, ref w).f,
unsafeInterval.f, rest,
(ulong)divisor << -one.e, unit));
}
divisor /= 10;
}
// The integrals have been generated. We are at the point of the decimal
// separator. In the following loop we simply multiply the remaining digits by
// 10 and divide by one. We just need to pay attention to multiply associated
// data (like the interval or 'unit'), too.
// Note that the multiplication by 10 does not overflow, because w.e >= -60
// and thus one.e >= -60.
for (; ;)
{
fractionals *= 10;
unit *= 10;
unsafeInterval.f *= 10;
// Integer division by one.
var digit = (int)(fractionals >> -one.e);
buffer[length] = (byte)((byte)'0' + digit);
length++;
fractionals &= one.f - 1; // Modulo by one.
kappa--;
if (fractionals < unsafeInterval.f)
{
return(RoundWeed(buffer, length, DiyFp.Minus(ref tooHigh, ref w).f *unit,
unsafeInterval.f, fractionals, one.f, unit));
}
}
}
public Double(DiyFp d)
{
d64_ = DiyFpToUint64(d);
}
// 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);
}
