// Returns the two boundaries of first argument.
        // The bigger boundary (m_plus) is normalized. The lower boundary has the same
        // exponent as m_plus.
        internal static void NormalizedBoundaries(long d64, DiyFp mMinus, DiyFp mPlus)
        {
            DiyFp v = AsDiyFp(d64);
            bool  significandIsZero = (v.F == KHiddenBit);

            mPlus.F = (v.F << 1) + 1;
            mPlus.E = v.E - 1;
            mPlus.Normalize();
            if (significandIsZero && v.E != KDenormalExponent)
            {
                // The boundary is closer. Think of v = 1000e10 and v- = 9999e9.
                // Then the boundary (== (v - v-)/2) is not just at a distance of 1e9 but
                // at a distance of 1e8.
                // The only exception is for the smallest normal: the largest denormal is
                // at the same distance as its successor.
                // Note: denormals have the same exponent as the smallest normals.
                mMinus.F = (v.F << 2) - 1;
                mMinus.E = v.E - 2;
            }
            else
            {
                mMinus.F = (v.F << 1) - 1;
                mMinus.E = v.E - 1;
            }
            mMinus.F = mMinus.F << (mMinus.E - mPlus.E);
            mMinus.E = mPlus.E;
        }
Ejemplo n.º 2
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        // this = this * other.
        private void Multiply(DiyFp other)
        {
            // Simply "emulates" a 128 bit multiplication.
            // However: the resulting number only contains 64 bits. The least
            // significant 64 bits are only used for rounding the most significant 64
            // bits.
            const long kM32 = 0xFFFFFFFFL;
            long       a    = F.UnsignedShift(32);
            long       b    = F & kM32;
            long       c    = other.F.UnsignedShift(32);
            long       d    = other.F & kM32;
            long       ac   = a * c;
            long       bc   = b * c;
            long       ad   = a * d;
            long       bd   = b * d;
            long       tmp  = bd.UnsignedShift(32) + (ad & kM32) + (bc & kM32);

            // By adding 1U << 31 to tmp we round the final result.
            // Halfway cases will be round up.
            tmp += 1L << 31;
            long resultF = ac + ad.UnsignedShift(32) + bc.UnsignedShift(32) + tmp.UnsignedShift(32);

            E += other.E + 64;
            F  = resultF;
        }
Ejemplo n.º 3
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        // 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.
        internal static DiyFp Minus(DiyFp a, DiyFp b)
        {
            DiyFp result = new DiyFp(a.F, a.E);

            result.Subtract(b);
            return(result);
        }
Ejemplo n.º 4
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        // this = this - other.
        // The exponents of both numbers must be the same and the significand of this
        // must be bigger than the significand of other.
        // The result will not be normalized.
        private void Subtract(DiyFp other)
        {
            Debug.Assert(E == other.E);
            Debug.Assert(Uint64Gte(F, other.F));

            F -= other.F;
        }
Ejemplo n.º 5
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        // returns a * b;
        internal static DiyFp Times(DiyFp a, DiyFp b)
        {
            DiyFp result = new DiyFp(a.F, a.E);

            result.Multiply(b);
            return(result);
        }
        internal static int GetCachedPower(int e, int alpha, int gamma, DiyFp cMk)
        {
            const int   kQ          = DiyFp.KSignificandSize;
            double      k           = System.Math.Ceiling((alpha - e + kQ - 1) * Kd1Log210);
            int         index       = (GrisuCacheOffset + (int)k - 1) / CachedPowersSpacing + 1;
            CachedPower cachedPower = CACHED_POWERS[index];

            cMk.F = cachedPower.Significand;
            cMk.E = cachedPower.BinaryExponent;
            Debug.Assert((alpha <= cMk.E + e) && (cMk.E + e <= gamma));
            return(cachedPower.DecimalExponent);
        }
        // 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));
        }
        // 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));
                }
            }
        }