// Returns the two boundaries of first argument. // The bigger boundary (m_plus) is normalized. The lower boundary has the same // exponent as m_plus. public static void NormalizedBoundaries(long d64, NumberDiyFp m_minus, NumberDiyFp m_plus) { NumberDiyFp v = AsDiyFp(d64); bool significand_is_zero = (v.F() == kHiddenBit); m_plus.SetF((v.F() << 1) + 1); m_plus.SetE(v.E() - 1); m_plus.Normalize(); if (significand_is_zero && 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. m_minus.SetF((v.F() << 2) - 1); m_minus.SetE(v.E() - 2); } else { m_minus.SetF((v.F() << 1) - 1); m_minus.SetE(v.E() - 1); } m_minus.SetF(m_minus.F() << (m_minus.E() - m_plus.E())); m_minus.SetE(m_plus.E()); }
public static int GetCachedPower(int e, int alpha, int gamma, NumberDiyFp c_mk) { int kQ = NumberDiyFp.kSignificandSize; double k = Math.Ceiling((alpha - e + kQ - 1) * kD_1_LOG2_10); int index = (GRISU_CACHE_OFFSET + (int)k - 1) / CACHED_POWERS_SPACING + 1; CachedPower cachedPower = CACHED_POWERS[index]; c_mk.SetF(cachedPower.significand); c_mk.SetE(cachedPower.binaryExponent); Debug.Assert((alpha <= c_mk.E() + e) && (c_mk.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, 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)); }
// 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)); } } }