private static bool Dtoa(double v, NumberFastDToABuilder buffer)
{
Debug.Assert(v > 0);
Debug.Assert(!Double.IsNaN(v));
Debug.Assert(!Double.IsInfinity(v));
return(Grisu3(v, buffer));
}
public static bool NumberToString(double v, NumberFastDToABuilder buffer)
{
buffer.Reset();
if (v < 0)
{
buffer.Append('-');
v = -v;
}
return(Dtoa(v, buffer));
}
// 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, NumberFastDToABuilder buffer)
{
long bits = BitConverter.DoubleToInt64Bits(v);
NumberDiyFp w = NumberDoubleHelper.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.
NumberDiyFp boundary_minus = new NumberDiyFp(), boundary_plus = new NumberDiyFp();
NumberDoubleHelper.NormalizedBoundaries(bits, boundary_minus, boundary_plus);
Debug.Assert(boundary_plus.E() == w.E());
NumberDiyFp ten_mk = new NumberDiyFp(); // Cached power of ten: 10^-k
int mk = NumberCachedPowers.GetCachedPower(w.E() + NumberDiyFp.kSignificandSize,
minimal_target_exponent, maximal_target_exponent, ten_mk);
Debug.Assert(minimal_target_exponent <= w.E() + ten_mk.E() +
NumberDiyFp.kSignificandSize &&
maximal_target_exponent >= w.E() + ten_mk.E() +
NumberDiyFp.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
NumberDiyFp scaled_w = NumberDiyFp.Times(w, ten_mk);
Debug.Assert(scaled_w.E() ==
boundary_plus.E() + ten_mk.E() + NumberDiyFp.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.
NumberDiyFp scaled_boundary_minus = NumberDiyFp.Times(boundary_minus, ten_mk);
NumberDiyFp scaled_boundary_plus = NumberDiyFp.Times(boundary_plus, 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.
return(DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus, buffer, mk));
}
// Adjusts the last digit of the generated number, and screens out generated
// solutions that may be inaccurate. A solution may be inaccurate if it is
// outside the safe interval, or if we ctannot prove that it is closer to the
// input than a neighboring representation of the same length.
//
// Input: * buffer containing the digits of too_high / 10^kappa
// * distance_too_high_w == (too_high - w).F() * unit
// * unsafe_interval == (too_high - too_low).F() * unit
// * rest = (too_high - buffer * 10^kappa).F() * unit
// * ten_kappa = 10^kappa * unit
// * unit = the common multiplier
// Output: returns true if the buffer is guaranteed to contain the closest
// representable number to the input.
// Modifies the generated digits in the buffer to approach (round towards) w.
private static bool RoundWeed(NumberFastDToABuilder buffer,
long distance_too_high_w,
long unsafe_interval,
long rest,
long ten_kappa,
long unit)
{
long small_distance = distance_too_high_w - unit;
long big_distance = distance_too_high_w + unit;
// Let w_low = too_high - big_distance, and
// w_high = too_high - small_distance.
// Note: w_low < w < w_high
//
// The real w (* unit) must lie somewhere inside the interval
// ]w_low; w_low[ (often written as "(w_low; w_low)")
// Basically the buffer currently contains a number in the unsafe interval
// ]too_low; too_high[ with too_low < w < too_high
//
// too_high - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
// ^v 1 unit ^ ^ ^ ^
// boundary_high --------------------- . . . .
// ^v 1 unit . . . .
// - - - - - - - - - - - - - - - - - - - + - - + - - - - - - . .
// . . ^ . .
// . big_distance . . .
// . . . . rest
// small_distance . . . .
// v . . . .
// w_high - - - - - - - - - - - - - - - - - - . . . .
// ^v 1 unit . . . .
// w ---------------------------------------- . . . .
// ^v 1 unit v . . .
// w_low - - - - - - - - - - - - - - - - - - - - - . . .
// . . v
// buffer --------------------------------------------------+-------+--------
// . .
// safe_interval .
// v .
// - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - .
// ^v 1 unit .
// boundary_low ------------------------- unsafe_interval
// ^v 1 unit v
// too_low - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
//
//
// Note that the value of buffer could lie anywhere inside the range too_low
// to too_high.
//
// boundary_low, boundary_high and w are approximations of the real boundaries
// and v (the input number). They are guaranteed to be precise up to one unit.
// In fact the error is guaranteed to be strictly less than one unit.
//
// Anything that lies outside the unsafe interval is guaranteed not to round
// to v when read again.
// Anything that lies inside the safe interval is guaranteed to round to v
// when read again.
// If the number inside the buffer lies inside the unsafe interval but not
// inside the safe interval then we simply do not know and bail out (returning
// false).
//
// Similarly we have to take into account the imprecision of 'w' when rounding
// the buffer. If we have two potential representations we need to make sure
// that the chosen one is closer to w_low and w_high since v can be anywhere
// between them.
//
// By generating the digits of too_high we got the largest (closest to
// too_high) buffer that is still in the unsafe interval. In the case where
// w_high < buffer < too_high we try to decrement the buffer.
// This way the buffer approaches (rounds towards) w.
// There are 3 conditions that stop the decrementation process:
// 1) the buffer is already below w_high
// 2) decrementing the buffer would make it leave the unsafe interval
// 3) decrementing the buffer would yield a number below w_high and farther
// away than the current number. In other words:
// (buffer{-1} < w_high) && w_high - buffer{-1} > buffer - w_high
// Instead of using the buffer directly we use its distance to too_high.
// Conceptually rest ~= too_high - buffer
while (rest < small_distance && // Negated condition 1
unsafe_interval - rest >= ten_kappa && // Negated condition 2
(rest + ten_kappa < small_distance || // buffer{-1} > w_high
small_distance - rest >= rest + ten_kappa - small_distance))
{
buffer.DecreaseLast();
rest += ten_kappa;
}
// We have approached w+ as much as possible. We now test if approaching w-
// would require changing the buffer. If yes, then we have two possible
// representations close to w, but we cannot decide which one is closer.
if (rest < big_distance &&
unsafe_interval - rest >= ten_kappa &&
(rest + ten_kappa < big_distance ||
big_distance - rest > rest + ten_kappa - big_distance))
{
return(false);
}
// Weeding test.
// The safe interval is [too_low + 2 ulp; too_high - 2 ulp]
// Since too_low = too_high - unsafe_interval this is equivalent to
// [too_high - unsafe_interval + 4 ulp; too_high - 2 ulp]
// Conceptually we have: rest ~= too_high - buffer
return((2 * unit <= rest) && (rest <= unsafe_interval - 4 * unit));
}
public static String NumberToString(double v)
{
NumberFastDToABuilder buffer = new NumberFastDToABuilder();
return(NumberToString(v, buffer) ? buffer.Format() : null);
}
// 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));
}
}
}
