// Copyright 2010 the V8 project authors. All rights reserved. // Redistribution and use in source and binary forms, with or without // modification, are permitted provided that the following conditions are // met: // // * Redistributions of source code must retain the above copyright // notice, this list of conditions and the following disclaimer. // * Redistributions in binary form must reproduce the above // copyright notice, this list of conditions and the following // disclaimer in the documentation and/or other materials provided // with the distribution. // * Neither the name of Google Inc. nor the names of its // contributors may be used to endorse or promote products derived // from this software without specific prior written permission. // // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT // OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, // SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT // LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, // DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY // THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT // (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. #include "config.h" #include "fast-dtoa.h" #include "cached-powers.h" #include "diy-fp.h" #include "double.h" namespace WTF { namespace double_conversion { // 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 const int kMinimalTargetExponent = -60; static const 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 // * the buffer's length // * 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 bool RoundWeed(Vector<char> buffer, int length, uint64_t distance_too_high_w, uint64_t unsafe_interval, uint64_t rest, uint64_t ten_kappa, uint64_t unit) { uint64_t small_distance = distance_too_high_w - unit; uint64_t 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(rest <= unsafe_interval); 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[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 (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); } // 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. static bool RoundWeedCounted(Vector<char> buffer, int length, uint64_t rest, uint64_t ten_kappa, uint64_t unit, int* kappa) { ASSERT(rest < ten_kappa); // 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 (unit >= ten_kappa) return false; // 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 (ten_kappa - unit <= unit) return false; // If 2 * (rest + unit) <= 10^kappa we can safely round down. if ((ten_kappa - rest > rest) && (ten_kappa - 2 * rest >= 2 * unit)) { return true; } // If 2 * (rest - unit) >= 10^kappa, then we can safely round up. if ((rest > unit) && (ten_kappa - (rest - unit) <= (rest - unit))) { // 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'; (*kappa) += 1; } return true; } return false; } static const uint32_t kTen4 = 10000; static const uint32_t kTen5 = 100000; static const uint32_t kTen6 = 1000000; static const uint32_t kTen7 = 10000000; static const uint32_t kTen8 = 100000000; static const uint32_t kTen9 = 1000000000; // 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. // If number_bits == 0 then 0^-1 is returned // The number of bits must be <= 32. // Precondition: number < (1 << (number_bits + 1)). static void BiggestPowerTen(uint32_t number, int number_bits, uint32_t* power, int* exponent) { ASSERT(number < (uint32_t)(1 << (number_bits + 1))); switch (number_bits) { case 32: case 31: case 30: if (kTen9 <= number) { *power = kTen9; *exponent = 9; break; } // else fallthrough case 29: case 28: case 27: if (kTen8 <= number) { *power = kTen8; *exponent = 8; break; } // else fallthrough case 26: case 25: case 24: if (kTen7 <= number) { *power = kTen7; *exponent = 7; break; } // else fallthrough case 23: case 22: case 21: case 20: if (kTen6 <= number) { *power = kTen6; *exponent = 6; break; } // else fallthrough case 19: case 18: case 17: if (kTen5 <= number) { *power = kTen5; *exponent = 5; break; } // else fallthrough case 16: case 15: case 14: if (kTen4 <= number) { *power = kTen4; *exponent = 4; break; } // else fallthrough case 13: case 12: case 11: case 10: if (1000 <= number) { *power = 1000; *exponent = 3; break; } // else fallthrough case 9: case 8: case 7: if (100 <= number) { *power = 100; *exponent = 2; break; } // else fallthrough case 6: case 5: case 4: if (10 <= number) { *power = 10; *exponent = 1; break; } // else fallthrough case 3: case 2: case 1: if (1 <= number) { *power = 1; *exponent = 0; break; } // else fallthrough case 0: *power = 0; *exponent = -1; break; default: // Following assignments are here to silence compiler warnings. *power = 0; *exponent = 0; UNREACHABLE(); } } // 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 bool DigitGen(DiyFp low, DiyFp w, DiyFp high, Vector<char> buffer, int* length, int* kappa) { ASSERT(low.e() == w.e() && w.e() == high.e()); ASSERT(low.f() + 1 <= high.f() - 1); 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. uint64_t unit = 1; DiyFp too_low = DiyFp(low.f() - unit, low.e()); DiyFp too_high = DiyFp(high.f() + unit, high.e()); // too_low and too_high are guaranteed to lie outside the interval we want the // generated number in. 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. DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e()); // Division by one is a shift. uint32_t integrals = static_cast<uint32_t>(too_high.f() >> -one.e()); // Modulo by one is an and. uint64_t fractionals = too_high.f() & (one.f() - 1); uint32_t divisor; int divisor_exponent; BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()), &divisor, &divisor_exponent); *kappa = divisor_exponent + 1; *length = 0; // 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) { int digit = integrals / divisor; buffer[*length] = '0' + digit; (*length)++; integrals %= divisor; (*kappa)--; // Note that kappa now equals the exponent of the divisor and that the // invariant thus holds again. uint64_t rest = (static_cast<uint64_t>(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. return RoundWeed(buffer, *length, DiyFp::Minus(too_high, w).f(), unsafe_interval.f(), rest, static_cast<uint64_t>(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(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f()); while (true) { fractionals *= 10; unit *= 10; unsafe_interval.set_f(unsafe_interval.f() * 10); // Integer division by one. int digit = static_cast<int>(fractionals >> -one.e()); buffer[*length] = '0' + digit; (*length)++; fractionals &= one.f() - 1; // Modulo by one. (*kappa)--; if (fractionals < unsafe_interval.f()) { return RoundWeed(buffer, *length, 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 bool DigitGenCounted(DiyFp w, int requested_digits, Vector<char> buffer, int* length, int* kappa) { 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. uint64_t 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. DiyFp one = DiyFp(static_cast<uint64_t>(1) << -w.e(), w.e()); // Division by one is a shift. uint32_t integrals = static_cast<uint32_t>(w.f() >> -one.e()); // Modulo by one is an and. uint64_t fractionals = w.f() & (one.f() - 1); uint32_t divisor; int divisor_exponent; BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()), &divisor, &divisor_exponent); *kappa = divisor_exponent + 1; *length = 0; // 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) { int digit = integrals / divisor; buffer[*length] = '0' + digit; (*length)++; 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) { uint64_t rest = (static_cast<uint64_t>(integrals) << -one.e()) + fractionals; return RoundWeedCounted(buffer, *length, rest, static_cast<uint64_t>(divisor) << -one.e(), w_error, kappa); } // 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 (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(UINT64_2PART_C(0xFFFFFFFF, FFFFFFFF) / 10 >= one.f()); while (requested_digits > 0 && fractionals > w_error) { fractionals *= 10; w_error *= 10; // Integer division by one. int digit = static_cast<int>(fractionals >> -one.e()); buffer[*length] = '0' + digit; (*length)++; requested_digits--; fractionals &= one.f() - 1; // Modulo by one. (*kappa)--; } if (requested_digits != 0) return false; return RoundWeedCounted(buffer, *length, fractionals, one.f(), w_error, kappa); } // 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 bool Grisu3(double v, Vector<char> buffer, int* length, int* decimal_exponent) { DiyFp w = Double(v).AsNormalizedDiyFp(); // 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 boundary_minus, boundary_plus; Double(v).NormalizedBoundaries(&boundary_minus, &boundary_plus); ASSERT(boundary_plus.e() == w.e()); DiyFp ten_mk; // Cached power of ten: 10^-k int mk; // -k int ten_mk_minimal_binary_exponent = kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize); int ten_mk_maximal_binary_exponent = kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize); PowersOfTenCache::GetCachedPowerForBinaryExponentRange( ten_mk_minimal_binary_exponent, ten_mk_maximal_binary_exponent, &ten_mk, &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 DiyFp scaled_w = DiyFp::Times(w, ten_mk); ASSERT(scaled_w.e() == boundary_plus.e() + ten_mk.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 scaled_boundary_minus = DiyFp::Times(boundary_minus, ten_mk); DiyFp scaled_boundary_plus = DiyFp::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. int kappa; bool result = DigitGen(scaled_boundary_minus, scaled_w, scaled_boundary_plus, buffer, length, &kappa); *decimal_exponent = -mk + kappa; return result; } // The "counted" version of grisu3 (see above) only generates requested_digits // number of digits. This version does not generate the shortest representation, // and with enough requested digits 0.1 will at some point print as 0.9999999... // Grisu3 is too imprecise for real halfway cases (1.5 will not work) and // therefore the rounding strategy for halfway cases is irrelevant. static bool Grisu3Counted(double v, int requested_digits, Vector<char> buffer, int* length, int* decimal_exponent) { DiyFp w = Double(v).AsNormalizedDiyFp(); DiyFp ten_mk; // Cached power of ten: 10^-k int mk; // -k int ten_mk_minimal_binary_exponent = kMinimalTargetExponent - (w.e() + DiyFp::kSignificandSize); int ten_mk_maximal_binary_exponent = kMaximalTargetExponent - (w.e() + DiyFp::kSignificandSize); PowersOfTenCache::GetCachedPowerForBinaryExponentRange( ten_mk_minimal_binary_exponent, ten_mk_maximal_binary_exponent, &ten_mk, &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 DiyFp scaled_w = DiyFp::Times(w, ten_mk); // We now have (double) (scaled_w * 10^-mk). // DigitGen will generate the first requested_digits digits of scaled_w and // return together with a kappa such that scaled_w ~= buffer * 10^kappa. (It // will not always be exactly the same since DigitGenCounted only produces a // limited number of digits.) int kappa; bool result = DigitGenCounted(scaled_w, requested_digits, buffer, length, &kappa); *decimal_exponent = -mk + kappa; return result; } bool FastDtoa(double v, FastDtoaMode mode, int requested_digits, Vector<char> buffer, int* length, int* decimal_point) { ASSERT(v > 0); ASSERT(!Double(v).IsSpecial()); bool result = false; int decimal_exponent = 0; switch (mode) { case FAST_DTOA_SHORTEST: result = Grisu3(v, buffer, length, &decimal_exponent); break; case FAST_DTOA_PRECISION: result = Grisu3Counted(v, requested_digits, buffer, length, &decimal_exponent); break; default: UNREACHABLE(); } if (result) { *decimal_point = *length + decimal_exponent; buffer[*length] = '\0'; } return result; } } // namespace double_conversion } // namespace WTF