// 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.
private void Subtract(DiyFp other)
{
Debug.Assert(E == other.E);
Debug.Assert(Uint64Gte(F, other.F));
F -= other.F;
}
// Returns the two boundaries of first argument.
// The bigger boundary (m_plus) is normalized. The lower boundary has the same
// exponent as m_plus.
internal static void NormalizedBoundaries(long d64, DiyFp mMinus, DiyFp mPlus)
{
DiyFp v = AsDiyFp(d64);
bool significandIsZero = (v.F == KHiddenBit);
mPlus.F = (v.F << 1) + 1;
mPlus.E = v.E - 1;
mPlus.Normalize();
if (significandIsZero && v.E != KDenormalExponent)
{
// The boundary is closer. Think of v = 1000e10 and v- = 9999e9.
// Then the boundary (== (v - v-)/2) is not just at a distance of 1e9 but
// at a distance of 1e8.
// The only exception is for the smallest normal: the largest denormal is
// at the same distance as its successor.
// Note: denormals have the same exponent as the smallest normals.
mMinus.F = (v.F << 2) - 1;
mMinus.E = v.E - 2;
}
else
{
mMinus.F = (v.F << 1) - 1;
mMinus.E = v.E - 1;
}
mMinus.F = mMinus.F << (mMinus.E - mPlus.E);
mMinus.E = mPlus.E;
}
internal static int GetCachedPower(int e, int alpha, int gamma, DiyFp cMk)
{
const int kQ = DiyFp.KSignificandSize;
double k = System.Math.Ceiling((alpha - e + kQ - 1)*Kd1Log210);
int index = (GrisuCacheOffset + (int) k - 1)/CachedPowersSpacing + 1;
CachedPower cachedPower = CACHED_POWERS[index];
cMk.F = cachedPower.Significand;
cMk.E = cachedPower.BinaryExponent;
Debug.Assert((alpha <= cMk.E + e) && (cMk.E + e <= gamma));
return cachedPower.DecimalExponent;
}
internal static int GetCachedPower(int e, int alpha, int gamma, DiyFp cMk)
{
const int kQ = DiyFp.KSignificandSize;
double k = System.Math.Ceiling((alpha - e + kQ - 1) * Kd1Log210);
int index = (GrisuCacheOffset + (int)k - 1) / CachedPowersSpacing + 1;
CachedPower cachedPower = CACHED_POWERS[index];
cMk.F = cachedPower.Significand;
cMk.E = cachedPower.BinaryExponent;
Debug.Assert((alpha <= cMk.E + e) && (cMk.E + e <= gamma));
return(cachedPower.DecimalExponent);
}
// 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, FastDtoaBuilder buffer)
{
long bits = BitConverter.DoubleToInt64Bits(v);
DiyFp w = DoubleHelper.AsNormalizedDiyFp(bits);
// 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 = new DiyFp(), boundaryPlus = new DiyFp();
DoubleHelper.NormalizedBoundaries(bits, boundaryMinus, boundaryPlus);
Debug.Assert(boundaryPlus.E == w.E);
var tenMk = new DiyFp(); // Cached power of ten: 10^-k
int mk = CachedPowers.GetCachedPower(w.E + DiyFp.KSignificandSize,
MinimalTargetExponent, MaximalTargetExponent, tenMk);
Debug.Assert(MinimalTargetExponent <= w.E + tenMk.E +
DiyFp.KSignificandSize &&
MaximalTargetExponent >= w.E + tenMk.E +
DiyFp.KSignificandSize);
// 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
DiyFp scaledW = DiyFp.Times(w, tenMk);
Debug.Assert(scaledW.E ==
boundaryPlus.E + tenMk.E + DiyFp.KSignificandSize);
// 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 terriffic.
DiyFp scaledBoundaryMinus = DiyFp.Times(boundaryMinus, tenMk);
DiyFp scaledBoundaryPlus = DiyFp.Times(boundaryPlus, 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.
return(DigitGen(scaledBoundaryMinus, scaledW, scaledBoundaryPlus, buffer, mk));
}
// 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, FastDtoaBuilder buffer)
{
long bits = BitConverter.DoubleToInt64Bits(v);
DiyFp w = DoubleHelper.AsNormalizedDiyFp(bits);
// 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 = new DiyFp(), boundaryPlus = new DiyFp();
DoubleHelper.NormalizedBoundaries(bits, boundaryMinus, boundaryPlus);
Debug.Assert(boundaryPlus.E == w.E);
var tenMk = new DiyFp(); // Cached power of ten: 10^-k
int mk = CachedPowers.GetCachedPower(w.E + DiyFp.KSignificandSize,
MinimalTargetExponent, MaximalTargetExponent, tenMk);
Debug.Assert(MinimalTargetExponent <= w.E + tenMk.E +
DiyFp.KSignificandSize &&
MaximalTargetExponent >= w.E + tenMk.E +
DiyFp.KSignificandSize);
// 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
DiyFp scaledW = DiyFp.Times(w, tenMk);
Debug.Assert(scaledW.E ==
boundaryPlus.E + tenMk.E + DiyFp.KSignificandSize);
// 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 terriffic.
DiyFp scaledBoundaryMinus = DiyFp.Times(boundaryMinus, tenMk);
DiyFp scaledBoundaryPlus = DiyFp.Times(boundaryPlus, 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.
return DigitGen(scaledBoundaryMinus, scaledW, scaledBoundaryPlus, buffer, mk);
}
// 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 minimal_target_exponent and
// maximal_target_exponent.
// 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 that a unit of their last digits.
// * low.e() == w.e() == high.e()
// * low < w < high, and taking into account their error: low~ <= high~
// * minimal_target_exponent <= w.e() <= maximal_target_exponent
// Postconditions: 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 postconditions
// 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,
FastDtoaBuilder buffer,
int mk)
{
// 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.
long 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(tooHigh, 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(1L << -w.E, w.E);
// Division by one is a shift.
var integrals = (int) (tooHigh.F.UnsignedShift(-one.E) & 0xffffffffL);
// Modulo by one is an and.
long fractionals = tooHigh.F & (one.F - 1);
long result = BiggestPowerTen(integrals, DiyFp.KSignificandSize - (-one.E));
var divider = (int) (result.UnsignedShift(32) & 0xffffffffL);
var dividerExponent = (int) (result & 0xffffffffL);
var kappa = dividerExponent + 1;
// Loop invariant: buffer = too_high / 10^kappa (integer division)
// The invariant holds for the first iteration: kappa has been initialized
// with the divider exponent + 1. And the divider is the biggest power of ten
// that is smaller than integrals.
while (kappa > 0)
{
int digit = integrals/divider;
buffer.Append((char) ('0' + digit));
integrals %= divider;
kappa--;
// Note that kappa now equals the exponent of the divider and that the
// invariant thus holds again.
long rest =
((long) 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.
buffer.Point = buffer.End - mk + kappa;
return RoundWeed(buffer, DiyFp.Minus(tooHigh, w).F,
unsafeInterval.F, rest,
(long) divider << -one.E, unit);
}
divider /= 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.
// Instead of multiplying by 10 we multiply by 5 (cheaper operation) and
// increase its (imaginary) exponent. At the same time we decrease the
// divider's (one's) exponent and shift its significand.
// Basically, if fractionals was a DiyFp (with fractionals.e == one.e):
// fractionals.f *= 10;
// fractionals.f >>= 1; fractionals.e++; // value remains unchanged.
// one.f >>= 1; one.e++; // value remains unchanged.
// and we have again fractionals.e == one.e which allows us to divide
// fractionals.f() by one.f()
// We simply combine the *= 10 and the >>= 1.
while (true)
{
fractionals *= 5;
unit *= 5;
unsafeInterval.F = unsafeInterval.F*5;
unsafeInterval.E = unsafeInterval.E + 1; // Will be optimized out.
one.F = one.F.UnsignedShift(1);
one.E = one.E + 1;
// Integer division by one.
var digit = (int) ((fractionals.UnsignedShift(-one.E)) & 0xffffffffL);
buffer.Append((char) ('0' + digit));
fractionals &= one.F - 1; // Modulo by one.
kappa--;
if (fractionals < unsafeInterval.F)
{
buffer.Point = buffer.End - mk + kappa;
return RoundWeed(buffer, DiyFp.Minus(tooHigh, w).F*unit,
unsafeInterval.F, fractionals, one.F, unit);
}
}
}
// 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 minimal_target_exponent and
// maximal_target_exponent.
// 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 that a unit of their last digits.
// * low.e() == w.e() == high.e()
// * low < w < high, and taking into account their error: low~ <= high~
// * minimal_target_exponent <= w.e() <= maximal_target_exponent
// Postconditions: 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 postconditions
// 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,
FastDtoaBuilder buffer,
int mk)
{
// 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.
long 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(tooHigh, 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(1L << -w.E, w.E);
// Division by one is a shift.
var integrals = (int)(tooHigh.F.UnsignedShift(-one.E) & 0xffffffffL);
// Modulo by one is an and.
long fractionals = tooHigh.F & (one.F - 1);
long result = BiggestPowerTen(integrals, DiyFp.KSignificandSize - (-one.E));
var divider = (int)(result.UnsignedShift(32) & 0xffffffffL);
var dividerExponent = (int)(result & 0xffffffffL);
var kappa = dividerExponent + 1;
// Loop invariant: buffer = too_high / 10^kappa (integer division)
// The invariant holds for the first iteration: kappa has been initialized
// with the divider exponent + 1. And the divider is the biggest power of ten
// that is smaller than integrals.
while (kappa > 0)
{
int digit = integrals / divider;
buffer.Append((char)('0' + digit));
integrals %= divider;
kappa--;
// Note that kappa now equals the exponent of the divider and that the
// invariant thus holds again.
long rest =
((long)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.
buffer.Point = buffer.End - mk + kappa;
return(RoundWeed(buffer, DiyFp.Minus(tooHigh, w).F,
unsafeInterval.F, rest,
(long)divider << -one.E, unit));
}
divider /= 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.
// Instead of multiplying by 10 we multiply by 5 (cheaper operation) and
// increase its (imaginary) exponent. At the same time we decrease the
// divider's (one's) exponent and shift its significand.
// Basically, if fractionals was a DiyFp (with fractionals.e == one.e):
// fractionals.f *= 10;
// fractionals.f >>= 1; fractionals.e++; // value remains unchanged.
// one.f >>= 1; one.e++; // value remains unchanged.
// and we have again fractionals.e == one.e which allows us to divide
// fractionals.f() by one.f()
// We simply combine the *= 10 and the >>= 1.
while (true)
{
fractionals *= 5;
unit *= 5;
unsafeInterval.F = unsafeInterval.F * 5;
unsafeInterval.E = unsafeInterval.E + 1; // Will be optimized out.
one.F = one.F.UnsignedShift(1);
one.E = one.E + 1;
// Integer division by one.
var digit = (int)((fractionals.UnsignedShift(-one.E)) & 0xffffffffL);
buffer.Append((char)('0' + digit));
fractionals &= one.F - 1; // Modulo by one.
kappa--;
if (fractionals < unsafeInterval.F)
{
buffer.Point = buffer.End - mk + kappa;
return(RoundWeed(buffer, DiyFp.Minus(tooHigh, w).F *unit,
unsafeInterval.F, fractionals, one.F, unit));
}
}
}
// this = this * other.
private void Multiply(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 = 0xFFFFFFFFL;
long a = F.UnsignedShift(32);
long b = F & kM32;
long c = other.F.UnsignedShift(32);
long d = other.F & kM32;
long ac = a*c;
long bc = b*c;
long ad = a*d;
long bd = b*d;
long tmp = bd.UnsignedShift(32) + (ad & kM32) + (bc & kM32);
// By adding 1U << 31 to tmp we round the final result.
// Halfway cases will be round up.
tmp += 1L << 31;
long resultF = ac + ad.UnsignedShift(32) + bc.UnsignedShift(32) + tmp.UnsignedShift(32);
E += other.E + 64;
F = resultF;
}
// 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.
internal static DiyFp Minus(DiyFp a, DiyFp b)
{
DiyFp result = new DiyFp(a.F, a.E);
result.Subtract(b);
return result;
}
// returns a * b;
internal static DiyFp Times(DiyFp a, DiyFp b)
{
DiyFp result = new DiyFp(a.F, a.E);
result.Multiply(b);
return result;
}
// Returns the two boundaries of first argument.
// The bigger boundary (m_plus) is normalized. The lower boundary has the same
// exponent as m_plus.
internal static void NormalizedBoundaries(long d64, DiyFp mMinus, DiyFp mPlus)
{
DiyFp v = AsDiyFp(d64);
bool significandIsZero = (v.F == KHiddenBit);
mPlus.F = (v.F << 1) + 1;
mPlus.E = v.E - 1;
mPlus.Normalize();
if (significandIsZero && v.E != KDenormalExponent)
{
// The boundary is closer. Think of v = 1000e10 and v- = 9999e9.
// Then the boundary (== (v - v-)/2) is not just at a distance of 1e9 but
// at a distance of 1e8.
// The only exception is for the smallest normal: the largest denormal is
// at the same distance as its successor.
// Note: denormals have the same exponent as the smallest normals.
mMinus.F = (v.F << 2) - 1;
mMinus.E = v.E - 2;
}
else
{
mMinus.F = (v.F << 1) - 1;
mMinus.E = v.E - 1;
}
mMinus.F = mMinus.F << (mMinus.E - mPlus.E);
mMinus.E = mPlus.E;
}