// 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);
}
// 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));
}
}
}
