public static bool NumberToString(double v, NumberFastDToABuilder buffer)
{
buffer.Reset();
if (v < 0)
{
buffer.Append('-');
v = -v;
}
return(Dtoa(v, buffer));
}
// 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(NumberDiyFp low,
NumberDiyFp w,
NumberDiyFp high,
NumberFastDToABuilder buffer,
int mk)
{
Debug.Assert(low.E() == w.E() && w.E() == high.E());
Debug.Assert(Uint64_lte(low.F() + 1, high.F() - 1));
Debug.Assert(minimal_target_exponent <= w.E() && w.E() <= maximal_target_exponent);
// 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;
NumberDiyFp too_low = new NumberDiyFp(low.F() - unit, low.E());
NumberDiyFp too_high = new NumberDiyFp(high.F() + unit, high.E());
// too_low and too_high are guaranteed to lie outside the interval we want the
// generated number in.
NumberDiyFp unsafe_interval = NumberDiyFp.Minus(too_high, 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.
NumberDiyFp one = new NumberDiyFp(1L << -w.E(), w.E());
// Division by one is a shift.
int integrals = (int)(((ulong)too_high.F() >> -one.E()) & 0xffffffffL);
// Modulo by one is an and.
long fractionals = too_high.F() & (one.F() - 1);
long result = BiggestPowerTen(integrals, NumberDiyFp.kSignificandSize - (-one.E()));
int divider = (int)(((ulong)result >> 32) & 0xffffffffL);
int divider_exponent = (int)(result & 0xffffffffL);
int kappa = divider_exponent + 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 < unsafe_interval.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, NumberDiyFp.Minus(too_high, w).F(),
unsafe_interval.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;
unsafe_interval.SetF(unsafe_interval.F() * 5);
unsafe_interval.SetE(unsafe_interval.E() + 1); // Will be optimized out.
one.SetF((long)((ulong)one.F() >> 1));
one.SetE(one.E() + 1);
// Integer division by one.
int digit = (int)(((ulong)fractionals >> -one.E()) & 0xffffffffL);
buffer.Append((char)('0' + digit));
fractionals &= one.F() - 1; // Modulo by one.
kappa--;
if (fractionals < unsafe_interval.F())
{
buffer.point = buffer.end - mk + kappa;
return(RoundWeed(buffer, NumberDiyFp.Minus(too_high, w).F() * unit,
unsafe_interval.F(), fractionals, one.F(), unit));
}
}
}
