sebastienros.jint/Jint/Native/Number/Dtoa/FastDtoa.cs

// Copyright 2010 the V8 project authors. All rights reserved.
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// Ported to Java from Mozilla's version of V8-dtoa by Hannes Wallnoefer.
// The original revision was 67d1049b0bf9 from the mozilla-central tree.

using System;
using System.Diagnostics;

namespace Jint.Native.Number.Dtoa
{
    public class FastDtoa
    {

        // FastDtoa will produce at most kFastDtoaMaximalLength digits.
        public const 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.
        private const int MinimalTargetExponent = -60;
        private const int MaximalTargetExponent = -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 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(FastDtoaBuilder buffer,
            long distanceTooHighW,
            long unsafeInterval,
            long rest,
            long tenKappa,
            long unit)
        {
            long smallDistance = distanceTooHighW - unit;
            long bigDistance = distanceTooHighW + 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 < smallDistance && // Negated condition 1
                   unsafeInterval - rest >= tenKappa && // Negated condition 2
                   (rest + tenKappa < smallDistance || // buffer{-1} > w_high
                    smallDistance - rest >= rest + tenKappa - smallDistance))
            {
                buffer.DecreaseLast();
                rest += tenKappa;
            }

            // 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 < bigDistance &&
                unsafeInterval - rest >= tenKappa &&
                (rest + tenKappa < bigDistance ||
                 bigDistance - rest > rest + tenKappa - bigDistance))
            {
                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 <= unsafeInterval - 4*unit);
        }
        
        private const int KTen4 = 10000;
        private const int KTen5 = 100000;
        private const int KTen6 = 1000000;
        private const int KTen7 = 10000000;
        private const int KTen8 = 100000000;
        private const int KTen9 = 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)).
        private static long BiggestPowerTen(int number,
            int numberBits)
        {
            int power, exponent;
            switch (numberBits)
            {
                case 32:
                case 31:
                case 30:
                    if (KTen9 <= number)
                    {
                        power = KTen9;
                        exponent = 9;
                        break;
                    } // else fallthrough

                    goto case 29;
                case 29:
                case 28:
                case 27:
                    if (KTen8 <= number)
                    {
                        power = KTen8;
                        exponent = 8;
                        break;
                    } // else fallthrough
                    goto case 26;
                case 26:
                case 25:
                case 24:
                    if (KTen7 <= number)
                    {
                        power = KTen7;
                        exponent = 7;
                        break;
                    } // else fallthrough
                    goto case 23;
                case 23:
                case 22:
                case 21:
                case 20:
                    if (KTen6 <= number)
                    {
                        power = KTen6;
                        exponent = 6;
                        break;
                    } // else fallthrough
                    goto case 19;
                case 19:
                case 18:
                case 17:
                    if (KTen5 <= number)
                    {
                        power = KTen5;
                        exponent = 5;
                        break;
                    } // else fallthrough
                    goto case 16;
                case 16:
                case 15:
                case 14:
                    if (KTen4 <= number)
                    {
                        power = KTen4;
                        exponent = 4;
                        break;
                    } // else fallthrough
                    goto case 13;
                case 13:
                case 12:
                case 11:
                case 10:
                    if (1000 <= number)
                    {
                        power = 1000;
                        exponent = 3;
                        break;
                    } // else fallthrough
                    goto case 9;
                case 9:
                case 8:
                case 7:
                    if (100 <= number)
                    {
                        power = 100;
                        exponent = 2;
                        break;
                    } // else fallthrough
                    goto case 6;
                case 6:
                case 5:
                case 4:
                    if (10 <= number)
                    {
                        power = 10;
                        exponent = 1;
                        break;
                    } // else fallthrough
                    goto case 3;
                case 3:
                case 2:
                case 1:
                    if (1 <= number)
                    {
                        power = 1;
                        exponent = 0;
                        break;
                    } // else fallthrough
                    goto case 0;
                case 0:
                    power = 0;
                    exponent = -1;
                    break;
                default:
                    // Following assignments are here to silence compiler warnings.
                    power = 0;
                    exponent = 0;
                    // UNREACHABLE();
                    break;
            }
            return ((long) power << 32) | (0xffffffffL & exponent);
        }

        // 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(DiyFp low,
            DiyFp w,
            DiyFp high,
            FastDtoaBuilder buffer,
            int mk)
        {
            // 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;
            var tooLow = new DiyFp(low.F - unit, low.E);
            var tooHigh = 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.
            var unsafeInterval = DiyFp.Minus(tooHigh, tooLow);
            // 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.
            var one = new DiyFp(1L << -w.E, w.E);
            // Division by one is a shift.
            var integrals = (int) (tooHigh.F.UnsignedShift(-one.E) & 0xffffffffL);
            // Modulo by one is an and.
            long fractionals = tooHigh.F & (one.F - 1);
            long result = BiggestPowerTen(integrals, DiyFp.KSignificandSize - (-one.E));
            var divider = (int) (result.UnsignedShift(32) & 0xffffffffL);
            var dividerExponent = (int) (result & 0xffffffffL);
            var kappa = dividerExponent + 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 < unsafeInterval.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, DiyFp.Minus(tooHigh, w).F,
                        unsafeInterval.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;
                unsafeInterval.F = unsafeInterval.F*5;
                unsafeInterval.E = unsafeInterval.E + 1; // Will be optimized out.
                one.F = one.F.UnsignedShift(1);
                one.E = one.E + 1;
                // Integer division by one.
                var digit = (int) ((fractionals.UnsignedShift(-one.E)) & 0xffffffffL);
                buffer.Append((char) ('0' + digit));
                fractionals &= one.F - 1; // Modulo by one.
                kappa--;
                if (fractionals < unsafeInterval.F)
                {
                    buffer.Point = buffer.End - mk + kappa;
                    return RoundWeed(buffer, DiyFp.Minus(tooHigh, w).F*unit,
                        unsafeInterval.F, fractionals, one.F, unit);
                }
            }
        }

        // 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, FastDtoaBuilder buffer)
        {
            long bits = BitConverter.DoubleToInt64Bits(v);
            DiyFp w = DoubleHelper.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.
            DiyFp boundaryMinus = new DiyFp(), boundaryPlus = new DiyFp();
            DoubleHelper.NormalizedBoundaries(bits, boundaryMinus, boundaryPlus);
            Debug.Assert(boundaryPlus.E == w.E);
            var tenMk = new DiyFp(); // Cached power of ten: 10^-k
            int mk = CachedPowers.GetCachedPower(w.E + DiyFp.KSignificandSize,
                MinimalTargetExponent, MaximalTargetExponent, tenMk);
            Debug.Assert(MinimalTargetExponent <= w.E + tenMk.E +
                         DiyFp.KSignificandSize &&
                         MaximalTargetExponent >= w.E + tenMk.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 scaledW = DiyFp.Times(w, tenMk);
            Debug.Assert(scaledW.E ==
                         boundaryPlus.E + tenMk.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 scaledBoundaryMinus = DiyFp.Times(boundaryMinus, tenMk);
            DiyFp scaledBoundaryPlus = DiyFp.Times(boundaryPlus, tenMk);

            // 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(scaledBoundaryMinus, scaledW, scaledBoundaryPlus, buffer, mk);
        }

        public static bool Dtoa(double v, FastDtoaBuilder buffer)
        {
            Debug.Assert(v > 0);
            Debug.Assert(!Double.IsNaN(v));
            Debug.Assert(!Double.IsInfinity(v));

            return Grisu3(v, buffer);
        }

        public static string NumberToString(double v)
        {
            var buffer = new FastDtoaBuilder();
            return NumberToString(v, buffer) ? buffer.Format() : null;
        }

        public static bool NumberToString(double v, FastDtoaBuilder buffer)
        {
            buffer.Reset();
            if (v < 0)
            {
                buffer.Append('-');
                v = -v;
            }
            return Dtoa(v, buffer);
        }
    }
}