// returns a * b;
public static DiyFp Times(ref DiyFp a, ref DiyFp b)
{
DiyFp result = a;
result.Multiply(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 ulong 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;
}
// 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);
}
public static DiyFp Normalize(ref DiyFp a)
{
DiyFp result = a;
result.Normalize();
return(result);
}
// Returns a cached power of ten x ~= 10^k such that
// k <= decimal_exponent < k + kCachedPowersDecimalDistance.
// The given decimal_exponent must satisfy
// kMinDecimalExponent <= requested_exponent, and
// requested_exponent < kMaxDecimalExponent + kDecimalExponentDistance.
public static void GetCachedPowerForDecimalExponent(int requested_exponent,
out DiyFp power,
out int found_exponent)
{
Debug.Assert(kMinDecimalExponent <= requested_exponent);
Debug.Assert(requested_exponent < kMaxDecimalExponent + kDecimalExponentDistance);
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;
Debug.Assert(found_exponent <= requested_exponent);
Debug.Assert(requested_exponent < found_exponent + kDecimalExponentDistance);
}
// Returns a cached power of ten x ~= 10^k such that
// k <= decimal_exponent < k + kCachedPowersDecimalDistance.
// The given decimal_exponent must satisfy
// kMinDecimalExponent <= requested_exponent, and
// requested_exponent < kMaxDecimalExponent + kDecimalExponentDistance.
public static void GetCachedPowerForDecimalExponent(int requested_exponent,
out DiyFp power,
out int found_exponent)
{
Debug.Assert(kMinDecimalExponent <= requested_exponent);
Debug.Assert(requested_exponent < kMaxDecimalExponent + kDecimalExponentDistance);
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;
Debug.Assert(found_exponent <= requested_exponent);
Debug.Assert(requested_exponent < found_exponent + kDecimalExponentDistance);
}
// Returns a cached power-of-ten with a binary exponent in the range
// [min_exponent; max_exponent] (boundaries included).
public static void GetCachedPowerForBinaryExponentRange(int min_exponent,
int max_exponent,
out DiyFp power,
out int decimal_exponent)
{
int kQ = DiyFp.kSignificandSize;
double k = Math.Ceiling((min_exponent + kQ - 1) * kD_1_LOG2_10);
int foo = kCachedPowersOffset;
int index =
(foo + (int)(k) - 1) / kDecimalExponentDistance + 1;
Debug.Assert(0 <= index && index < kCachedPowersLength);
CachedPower cached_power = kCachedPowers[index];
Debug.Assert(min_exponent <= cached_power.binary_exponent);
Debug.Assert(cached_power.binary_exponent <= max_exponent);
decimal_exponent = cached_power.decimal_exponent;
power = new DiyFp(cached_power.significand, cached_power.binary_exponent);
}
// Returns a cached power-of-ten with a binary exponent in the range
// [min_exponent; max_exponent] (boundaries included).
public static void GetCachedPowerForBinaryExponentRange(int min_exponent,
int max_exponent,
out DiyFp power,
out int decimal_exponent)
{
int kQ = DiyFp.kSignificandSize;
double k = Math.Ceiling((min_exponent + kQ - 1) * kD_1_LOG2_10);
int foo = kCachedPowersOffset;
int index =
(foo + (int)(k) - 1) / kDecimalExponentDistance + 1;
Debug.Assert(0 <= index && index < kCachedPowersLength);
CachedPower cached_power = kCachedPowers[index];
Debug.Assert(min_exponent <= cached_power.binary_exponent);
Debug.Assert(cached_power.binary_exponent <= max_exponent);
decimal_exponent = cached_power.decimal_exponent;
power = new DiyFp(cached_power.significand, cached_power.binary_exponent);
}
// 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 ulong 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;
}
private 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));
}
// 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 Double must be greater than 0.
public void NormalizedBoundaries(out DiyFp out_m_minus, out DiyFp out_m_plus)
{
Debug.Assert(Value > 0.0);
ulong d64 = d64_;
ulong vF;
int vE;
if (IsDenormal)
{
vF = d64 & kSignificandMask;
vE = kDenormalExponent;
}
else
{
vF = (d64 & kSignificandMask) + kHiddenBit;
vE = (int)((d64 & kExponentMask) >> kPhysicalSignificandSize) - kExponentBias;
}
ulong plusF = (vF << 1) + 1;
int plusE = vE - 1;
// This code is manually inlined from the GrisuDouble.Normalize() method,
// because the .NET JIT (at least the 64-bit one as of version 4) is too
// incompetent to do it itself.
const ulong k10MSBits = 0xFFC0000000000000;
const ulong kUint64MSB = 0x8000000000000000;
while ((plusF & k10MSBits) == 0)
{
plusF <<= 10;
plusE -= 10;
}
while ((plusF & kUint64MSB) == 0)
{
plusF <<= 1;
plusE--;
}
ulong minusF;
int minusE;
bool significand_is_zero = (vF == kHiddenBit);
if (significand_is_zero && vE != 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.
minusF = (vF << 2) - 1;
minusE = vE - 2;
}
else
{
minusF = (vF << 1) - 1;
minusE = vE - 1;
}
out_m_minus = new DiyFp(minusF << (minusE - plusE), plusE);
out_m_plus = new DiyFp(plusF, plusE);
}
public static DiyFp Normalize(ref DiyFp a)
{
DiyFp result = a;
result.Normalize();
return result;
}
// 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.
// 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)
{
Debug.Assert(e_ == other.e_);
Debug.Assert(f_ >= other.f_);
f_ -= other.f_;
}
public GrisuDouble(DiyFp diy_fp)
{
d64_ = DiyFpToUInt64(diy_fp);
value_ = BitConverter.Int64BitsToDouble((long)d64_);
}
private 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);
}
// 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)
{
Debug.Assert(e_ == other.e_);
Debug.Assert(f_ >= other.f_);
f_ -= other.f_;
}
public GrisuDouble(DiyFp diy_fp)
{
d64_ = DiyFpToUInt64(diy_fp);
value_ = BitConverter.Int64BitsToDouble((long)d64_);
}
// 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(ref GrisuDouble v,
char[] buffer,
out int length,
out int decimal_exponent)
{
DiyFp w = 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 boundary_minus, boundary_plus;
v.NormalizedBoundaries(out boundary_minus, out boundary_plus);
Debug.Assert(boundary_plus.E == w.E);
DiyFp ten_mk; // Cached power of ten: 10^-k
int mk; // -k
int ten_mk_minimal_binary_exponent =
kMinimalTargetExponent - (w.E + DiyFp.kSignificandSize);
int ten_mk_maximal_binary_exponent =
kMaximalTargetExponent - (w.E + DiyFp.kSignificandSize);
PowersOfTenCache.GetCachedPowerForBinaryExponentRange(
ten_mk_minimal_binary_exponent,
ten_mk_maximal_binary_exponent,
out ten_mk, out mk);
Debug.Assert((kMinimalTargetExponent <= w.E + ten_mk.E +
DiyFp.kSignificandSize) &&
(kMaximalTargetExponent >= w.E + ten_mk.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 scaled_w = DiyFp.Times(ref w, ref ten_mk);
w.Multiply(ref ten_mk);
Debug.Assert(w.E ==
boundary_plus.E + ten_mk.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 scaled_boundary_minus = DiyFp.Times(ref boundary_minus, ref ten_mk);
boundary_minus.Multiply(ref ten_mk);
//DiyFp scaled_boundary_plus = DiyFp.Times(ref boundary_plus, ref ten_mk);
boundary_plus.Multiply(ref ten_mk);
// 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.
int kappa;
bool result = DigitGen(ref boundary_minus, ref w, ref boundary_plus,
buffer, out length, out kappa);
decimal_exponent = -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
// 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(ref DiyFp low,
ref DiyFp w,
ref DiyFp high,
char[] buffer,
out int length,
out int kappa)
{
Debug.Assert(low.E == w.E && w.E == high.E);
Debug.Assert(low.F + 1 <= high.F - 1);
Debug.Assert(kMinimalTargetExponent <= w.E && w.E <= kMaximalTargetExponent);
// 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;
DiyFp too_low = new DiyFp(low.F - unit, low.E);
DiyFp too_high = 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.
DiyFp unsafe_interval = DiyFp.Minus(ref too_high, ref too_low);
// 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.
DiyFp one = new DiyFp((ulong)(1) << -w.E, w.E);
// Division by one is a shift.
uint integrals = (uint)(too_high.F >> -one.E);
// Modulo by one is an and.
ulong fractionals = too_high.F & (one.F - 1);
uint divisor;
int divisor_exponent_plus_one;
BiggestPowerTen(integrals, DiyFp.kSignificandSize - (-one.E),
out divisor, out divisor_exponent_plus_one);
kappa = divisor_exponent_plus_one;
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.
ulong unsafeIntervalF = unsafe_interval.F;
while (kappa > 0)
{
int digit = (int)(integrals / divisor);
buffer[length] = (char)('0' + digit);
++length;
integrals %= divisor;
kappa--;
// Note that kappa now equals the exponent of the divisor and that the
// invariant thus holds again.
ulong rest =
((ulong)(integrals) << -one.E) + fractionals;
// Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())
// Reminder: unsafe_interval.e() == one.e()
if (rest < unsafeIntervalF)
{
// Rounding down (by not emitting the remaining digits) yields a number
// that lies within the unsafe interval.
too_high.Subtract(ref w);
return(RoundWeed(buffer, length, too_high.F,
unsafeIntervalF, 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.
Debug.Assert(one.E >= -60);
Debug.Assert(fractionals < one.F);
Debug.Assert(0xFFFFFFFFFFFFFFFF / 10 >= one.F);
while (true)
{
fractionals *= 10;
unit *= 10;
unsafe_interval.F *= 10;
// Integer division by one.
int digit = (int)(fractionals >> -one.E);
buffer[length] = (char)('0' + digit);
++length;
fractionals &= one.F - 1; // Modulo by one.
kappa--;
if (fractionals < unsafe_interval.F)
{
too_high.Subtract(ref w);
return(RoundWeed(buffer, length, too_high.F * unit,
unsafe_interval.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 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
// 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(ref DiyFp low,
ref DiyFp w,
ref DiyFp high,
char[] buffer,
out int length,
out int kappa)
{
Debug.Assert(low.E == w.E && w.E == high.E);
Debug.Assert(low.F + 1 <= high.F - 1);
Debug.Assert(kMinimalTargetExponent <= w.E && w.E <= kMaximalTargetExponent);
// 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;
DiyFp too_low = new DiyFp(low.F - unit, low.E);
DiyFp too_high = 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.
DiyFp unsafe_interval = DiyFp.Minus(ref too_high, ref too_low);
// 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.
DiyFp one = new DiyFp((ulong)(1) << -w.E, w.E);
// Division by one is a shift.
uint integrals = (uint)(too_high.F >> -one.E);
// Modulo by one is an and.
ulong fractionals = too_high.F & (one.F - 1);
uint divisor;
int divisor_exponent_plus_one;
BiggestPowerTen(integrals, DiyFp.kSignificandSize - (-one.E),
out divisor, out divisor_exponent_plus_one);
kappa = divisor_exponent_plus_one;
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.
ulong unsafeIntervalF = unsafe_interval.F;
while (kappa > 0)
{
int digit = (int)(integrals / divisor);
buffer[length] = (char)('0' + digit);
++length;
integrals %= divisor;
kappa--;
// Note that kappa now equals the exponent of the divisor and that the
// invariant thus holds again.
ulong rest =
((ulong)(integrals) << -one.E) + fractionals;
// Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())
// Reminder: unsafe_interval.e() == one.e()
if (rest < unsafeIntervalF)
{
// Rounding down (by not emitting the remaining digits) yields a number
// that lies within the unsafe interval.
too_high.Subtract(ref w);
return RoundWeed(buffer, length, too_high.F,
unsafeIntervalF, 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.
Debug.Assert(one.E >= -60);
Debug.Assert(fractionals < one.F);
Debug.Assert(0xFFFFFFFFFFFFFFFF / 10 >= one.F);
while (true)
{
fractionals *= 10;
unit *= 10;
unsafe_interval.F *= 10;
// Integer division by one.
int digit = (int)(fractionals >> -one.E);
buffer[length] = (char)('0' + digit);
++length;
fractionals &= one.F - 1; // Modulo by one.
kappa--;
if (fractionals < unsafe_interval.F)
{
too_high.Subtract(ref w);
return RoundWeed(buffer, length, too_high.F * unit,
unsafe_interval.F, fractionals, one.F, unit);
}
}
}
// 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 Double must be greater than 0.
public void NormalizedBoundaries(out DiyFp out_m_minus, out DiyFp out_m_plus)
{
Debug.Assert(Value > 0.0);
ulong d64 = d64_;
ulong vF;
int vE;
if (IsDenormal)
{
vF = d64 & kSignificandMask;
vE = kDenormalExponent;
}
else
{
vF = (d64 & kSignificandMask) + kHiddenBit;
vE = (int)((d64 & kExponentMask) >> kPhysicalSignificandSize) - kExponentBias;
}
ulong plusF = (vF << 1) + 1;
int plusE = vE - 1;
// This code is manually inlined from the GrisuDouble.Normalize() method,
// because the .NET JIT (at least the 64-bit one as of version 4) is too
// incompetent to do it itself.
const ulong k10MSBits = 0xFFC0000000000000;
const ulong kUint64MSB = 0x8000000000000000;
while ((plusF & k10MSBits) == 0)
{
plusF <<= 10;
plusE -= 10;
}
while ((plusF & kUint64MSB) == 0)
{
plusF <<= 1;
plusE--;
}
ulong minusF;
int minusE;
bool significand_is_zero = (vF == kHiddenBit);
if (significand_is_zero && vE != 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.
minusF = (vF << 2) - 1;
minusE = vE - 2;
}
else
{
minusF = (vF << 1) - 1;
minusE = vE - 1;
}
out_m_minus = new DiyFp(minusF << (minusE - plusE), plusE);
out_m_plus = new DiyFp(plusF, plusE);
}