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doubleconv /
FastDtoa.java
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// file:
//
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package jdk.nashorn.internal.runtime.doubleconv;
// Fast Dtoa implementation supporting shortest and precision modes. Does not
// work for all numbers so BugnumDtoa is used as fallback.
class FastDtoa {
// FastDtoa will produce at most kFastDtoaMaximalLength digits. This does not
// include the terminating '\0' character.
static final int kFastDtoaMaximalLength = 17;
// The minimal and maximal target exponent define the range of w's binary
// exponent, where 'w' is the result of multiplying the input by a cached power
// of ten.
//
// A different range might be chosen on a different platform, to optimize digit
// generation, but a smaller range requires more powers of ten to be cached.
static final int kMinimalTargetExponent = -60;
static final int kMaximalTargetExponent = -32;
// 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 cannot 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.
static boolean roundWeed(final DtoaBuffer buffer,
final long distance_too_high_w,
final long unsafe_interval,
long rest,
final long ten_kappa,
final long unit) {
final long small_distance = distance_too_high_w - unit;
final 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_high[ (often written as "(w_low; w_high)")
// 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 finding
// the closest representation of 'w'. If we have two potential
// representations, and one is closer to both w_low and w_high, then we know
// it is closer to the actual value v.
//
// 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
// We need to do the following tests in this order to avoid over- and
// underflows.
assert (Long.compareUnsigned(rest, unsafe_interval) <= 0);
while (Long.compareUnsigned(rest, small_distance) < 0 && // Negated condition 1
Long.compareUnsigned(unsafe_interval - rest, ten_kappa) >= 0 && // Negated condition 2
(Long.compareUnsigned(rest + ten_kappa, small_distance) < 0 || // buffer{-1} > w_high
Long.compareUnsigned(small_distance - rest, rest + ten_kappa - small_distance) >= 0)) {
buffer.chars[buffer.length - 1]--;
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 (Long.compareUnsigned(rest, big_distance) < 0 &&
Long.compareUnsigned(unsafe_interval - rest, ten_kappa) >= 0 &&
(Long.compareUnsigned(rest + ten_kappa, big_distance) < 0 ||
Long.compareUnsigned(big_distance - rest, rest + ten_kappa - big_distance) > 0)) {
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 Long.compareUnsigned(2 * unit, rest) <= 0 && Long.compareUnsigned(rest, unsafe_interval - 4 * unit) <= 0;
}
// Rounds the buffer upwards if the result is closer to v by possibly adding
// 1 to the buffer. If the precision of the calculation is not sufficient to
// round correctly, return false.
// The rounding might shift the whole buffer in which case the kappa is
// adjusted. For example "99", kappa = 3 might become "10", kappa = 4.
//
// If 2*rest > ten_kappa then the buffer needs to be round up.
// rest can have an error of +/- 1 unit. This function accounts for the
// imprecision and returns false, if the rounding direction cannot be
// unambiguously determined.
//
// Precondition: rest < ten_kappa.
// Changed return type to int to let caller know they should increase kappa (return value 2)
static int roundWeedCounted(final char[] buffer,
final int length,
final long rest,
final long ten_kappa,
final long unit) {
assert(Long.compareUnsigned(rest, ten_kappa) < 0);
// The following tests are done in a specific order to avoid overflows. They
// will work correctly with any uint64 values of rest < ten_kappa and unit.
//
// If the unit is too big, then we don't know which way to round. For example
// a unit of 50 means that the real number lies within rest +/- 50. If
// 10^kappa == 40 then there is no way to tell which way to round.
if (Long.compareUnsigned(unit, ten_kappa) >= 0) return 0;
// Even if unit is just half the size of 10^kappa we are already completely
// lost. (And after the previous test we know that the expression will not
// over/underflow.)
if (Long.compareUnsigned(ten_kappa - unit, unit) <= 0) return 0;
// If 2 * (rest + unit) <= 10^kappa we can safely round down.
if (Long.compareUnsigned(ten_kappa - rest, rest) > 0 && Long.compareUnsigned(ten_kappa - 2 * rest, 2 * unit) >= 0) {
return 1;
}
// If 2 * (rest - unit) >= 10^kappa, then we can safely round up.
if (Long.compareUnsigned(rest, unit) > 0 && Long.compareUnsigned(ten_kappa - (rest - unit), (rest - unit)) <= 0) {
// Increment the last digit recursively until we find a non '9' digit.
buffer[length - 1]++;
for (int i = length - 1; i > 0; --i) {
if (buffer[i] != '0' + 10) break;
buffer[i] = '0';
buffer[i - 1]++;
}
// If the first digit is now '0'+ 10 we had a buffer with all '9's. With the
// exception of the first digit all digits are now '0'. Simply switch the
// first digit to '1' and adjust the kappa. Example: "99" becomes "10" and
// the power (the kappa) is increased.
if (buffer[0] == '0' + 10) {
buffer[0] = '1';
// Return value of 2 tells caller to increase (*kappa) += 1
return 2;
}
return 1;
}
return 0;
}
// Returns the biggest power of ten that is less than or equal to the given
// number. We furthermore receive the maximum number of bits 'number' has.
//
// Returns power == 10^(exponent_plus_one-1) such that
// power <= number < power * 10.
// If number_bits == 0 then 0^(0-1) is returned.
// The number of bits must be <= 32.
// Precondition: number < (1 << (number_bits + 1)).
// Inspired by the method for finding an integer log base 10 from here:
// http://graphics.stanford.edu/~seander/bithacks.html#IntegerLog10
static final int kSmallPowersOfTen[] =
{0, 1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000,
1000000000};
// Returns the biggest power of ten that is less than or equal than the given
// number. We furthermore receive the maximum number of bits 'number' has.
// If number_bits == 0 then 0^-1 is returned
// The number of bits must be <= 32.
// Precondition: (1 << number_bits) <= number < (1 << (number_bits + 1)).
static long biggestPowerTen(final int number,
final int number_bits) {
final int power, exponent_plus_one;
assert ((number & 0xFFFFFFFFL) < (1l << (number_bits + 1)));
// 1233/4096 is approximately 1/lg(10).
int exponent_plus_one_guess = ((number_bits + 1) * 1233 >>> 12);
// We increment to skip over the first entry in the kPowersOf10 table.
// Note: kPowersOf10[i] == 10^(i-1).
exponent_plus_one_guess++;
// We don't have any guarantees that 2^number_bits <= number.
if (number < kSmallPowersOfTen[exponent_plus_one_guess]) {
exponent_plus_one_guess--;
}
power = kSmallPowersOfTen[exponent_plus_one_guess];
exponent_plus_one = exponent_plus_one_guess;
return ((long) power << 32) | (long) exponent_plus_one;
}
// 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.
static boolean digitGen(final DiyFp low,
final DiyFp w,
final DiyFp high,
final DtoaBuffer buffer,
final int mk) {
assert(low.e() == w.e() && w.e() == high.e());
assert Long.compareUnsigned(low.f() + 1, high.f() - 1) <= 0;
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.
long unit = 1;
final DiyFp too_low = new DiyFp(low.f() - unit, low.e());
final 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.
final DiyFp unsafe_interval = DiyFp.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.
final DiyFp one = new DiyFp(1l << -w.e(), w.e());
// Division by one is a shift.
int integrals = (int)(too_high.f() >>> -one.e());
// Modulo by one is an and.
long fractionals = too_high.f() & (one.f() - 1);
int divisor;
final int divisor_exponent_plus_one;
final long result = biggestPowerTen(integrals, DiyFp.kSignificandSize - (-one.e()));
divisor = (int) (result >>> 32);
divisor_exponent_plus_one = (int) result;
int kappa = divisor_exponent_plus_one;
// 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.
while (kappa > 0) {
final int digit = integrals / divisor;
assert (digit <= 9);
buffer.append((char) ('0' + digit));
integrals %= divisor;
kappa--;
// Note that kappa now equals the exponent of the divisor and that the
// invariant thus holds again.
final long rest =
((long) integrals << -one.e()) + fractionals;
// Invariant: too_high = buffer * 10^kappa + DiyFp(rest, one.e())
// Reminder: unsafe_interval.e() == one.e()
if (Long.compareUnsigned(rest, unsafe_interval.f()) < 0) {
// Rounding down (by not emitting the remaining digits) yields a number
// that lies within the unsafe interval.
buffer.decimalPoint = buffer.length - mk + kappa;
return roundWeed(buffer, DiyFp.minus(too_high, w).f(),
unsafe_interval.f(), rest,
(long) 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.
assert (one.e() >= -60);
assert (fractionals < one.f());
assert (Long.compareUnsigned(Long.divideUnsigned(0xFFFFFFFFFFFFFFFFL, 10), one.f()) >= 0);
for (;;) {
fractionals *= 10;
unit *= 10;
unsafe_interval.setF(unsafe_interval.f() * 10);
// Integer division by one.
final int digit = (int) (fractionals >>> -one.e());
assert (digit <= 9);
buffer.append((char) ('0' + digit));
fractionals &= one.f() - 1; // Modulo by one.
kappa--;
if (Long.compareUnsigned(fractionals, unsafe_interval.f()) < 0) {
buffer.decimalPoint = buffer.length - mk + kappa;
return roundWeed(buffer, DiyFp.minus(too_high, w).f() * unit,
unsafe_interval.f(), fractionals, one.f(), unit);
}
}
}
// Generates (at most) requested_digits 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:
// * w is correct up to 1 ulp (unit in the last place). That
// is, its error must be strictly less than a unit of its last digit.
// * kMinimalTargetExponent <= w.e() <= kMaximalTargetExponent
//
// Postconditions: returns false if procedure fails.
// otherwise:
// * buffer is not null-terminated, but length contains the number of
// digits.
// * the representation in buffer is the most precise representation of
// requested_digits digits.
// * buffer contains at most requested_digits digits of w. If there are less
// than requested_digits digits then some trailing '0's have been removed.
// * kappa is such that
// w = buffer * 10^kappa + eps with |eps| < 10^kappa / 2.
//
// 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, but the failure-rate
// increases with higher requested_digits.
static boolean digitGenCounted(final DiyFp w,
int requested_digits,
final DtoaBuffer buffer,
final int mk) {
assert (kMinimalTargetExponent <= w.e() && w.e() <= kMaximalTargetExponent);
assert (kMinimalTargetExponent >= -60);
assert (kMaximalTargetExponent <= -32);
// w is assumed to have an error less than 1 unit. Whenever w is scaled we
// also scale its error.
long w_error = 1;
// We cut the input number into two parts: the integral digits and the
// fractional digits. We don't emit any decimal separator, but adapt kappa
// instead. Example: instead of writing "1.2" we put "12" into the buffer and
// increase kappa by 1.
final DiyFp one = new DiyFp(1l << -w.e(), w.e());
// Division by one is a shift.
int integrals = (int) (w.f() >>> -one.e());
// Modulo by one is an and.
long fractionals = w.f() & (one.f() - 1);
int divisor;
final int divisor_exponent_plus_one;
final long biggestPower = biggestPowerTen(integrals, DiyFp.kSignificandSize - (-one.e()));
divisor = (int) (biggestPower >>> 32);
divisor_exponent_plus_one = (int) biggestPower;
int kappa = divisor_exponent_plus_one;
// Loop invariant: buffer = w / 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'.
while (kappa > 0) {
final int digit = integrals / divisor;
assert (digit <= 9);
buffer.append((char) ('0' + digit));
requested_digits--;
integrals %= divisor;
kappa--;
// Note that kappa now equals the exponent of the divisor and that the
// invariant thus holds again.
if (requested_digits == 0) break;
divisor /= 10;
}
if (requested_digits == 0) {
final long rest =
((long) (integrals) << -one.e()) + fractionals;
final int result = roundWeedCounted(buffer.chars, buffer.length, rest,
(long) divisor << -one.e(), w_error);
buffer.decimalPoint = buffer.length - mk + kappa + (result == 2 ? 1 : 0);
return result > 0;
}
// The integrals have been generated. We are at the decimalPoint 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 (the 'unit'), too.
// Note that the multiplication by 10 does not overflow, because w.e >= -60
// and thus one.e >= -60.
assert (one.e() >= -60);
assert (fractionals < one.f());
assert (Long.compareUnsigned(Long.divideUnsigned(0xFFFFFFFFFFFFFFFFL, 10), one.f()) >= 0);
while (requested_digits > 0 && fractionals > w_error) {
fractionals *= 10;
w_error *= 10;
// Integer division by one.
final int digit = (int) (fractionals >>> -one.e());
assert (digit <= 9);
buffer.append((char) ('0' + digit));
requested_digits--;
fractionals &= one.f() - 1; // Modulo by one.
kappa--;
}
if (requested_digits != 0) return false;
final int result = roundWeedCounted(buffer.chars, buffer.length, fractionals, one.f(), w_error);
buffer.decimalPoint = buffer.length - mk + kappa + (result == 2 ? 1 : 0);
return result > 0;
}
// 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.
static boolean grisu3(final double v, final DtoaBuffer buffer) {
final long d64 = IeeeDouble.doubleToLong(v);
final DiyFp w = IeeeDouble.asNormalizedDiyFp(d64);
// 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.
final DiyFp boundary_minus = new DiyFp(), boundary_plus = new DiyFp();
IeeeDouble.normalizedBoundaries(d64, boundary_minus, boundary_plus);
assert(boundary_plus.e() == w.e());
final DiyFp ten_mk = new DiyFp(); // Cached power of ten: 10^-k
final int mk; // -k
final int ten_mk_minimal_binary_exponent =
kMinimalTargetExponent - (w.e() + DiyFp.kSignificandSize);
final int ten_mk_maximal_binary_exponent =
kMaximalTargetExponent - (w.e() + DiyFp.kSignificandSize);
mk = CachedPowers.getCachedPowerForBinaryExponentRange(
ten_mk_minimal_binary_exponent,
ten_mk_maximal_binary_exponent,
ten_mk);
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
final DiyFp scaled_w = DiyFp.times(w, ten_mk);
assert(scaled_w.e() ==
boundary_plus.e() + ten_mk.e() + DiyFp.kSignificandSize);
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