| // 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 "v8.h" |
| |
| #include "fast-dtoa.h" |
| |
| #include "cached-powers.h" |
| #include "diy-fp.h" |
| #include "double.h" |
| |
| namespace v8 { |
| namespace internal { |
| |
| // 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 minimal_target_exponent = -60; |
| static const int maximal_target_exponent = -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 |
| // * 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. |
| 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_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 < 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); |
| } |
| |
| |
| |
| 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 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 void BiggestPowerTen(uint32_t number, |
| int number_bits, |
| uint32_t* power, |
| int* exponent) { |
| 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 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. |
| 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(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. |
| 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 divider; |
| int divider_exponent; |
| BiggestPowerTen(integrals, DiyFp::kSignificandSize - (-one.e()), |
| ÷r, ÷r_exponent); |
| *kappa = divider_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 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[*length] = '0' + digit; |
| (*length)++; |
| integrals %= divider; |
| (*kappa)--; |
| // Note that kappa now equals the exponent of the divider 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>(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.set_f(unsafe_interval.f() * 5); |
| unsafe_interval.set_e(unsafe_interval.e() + 1); // Will be optimized out. |
| one.set_f(one.f() >> 1); |
| one.set_e(one.e() + 1); |
| // 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); |
| } |
| } |
| } |
| |
| |
| // 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. |
| 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 |
| GetCachedPower(w.e() + DiyFp::kSignificandSize, minimal_target_exponent, |
| maximal_target_exponent, &mk, &ten_mk); |
| ASSERT(minimal_target_exponent <= w.e() + ten_mk.e() + |
| DiyFp::kSignificandSize && |
| maximal_target_exponent >= 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; |
| } |
| |
| |
| bool FastDtoa(double v, |
| Vector<char> buffer, |
| int* length, |
| int* point) { |
| ASSERT(v > 0); |
| ASSERT(!Double(v).IsSpecial()); |
| |
| int decimal_exponent; |
| bool result = grisu3(v, buffer, length, &decimal_exponent); |
| *point = *length + decimal_exponent; |
| buffer[*length] = '\0'; |
| return result; |
| } |
| |
| } } // namespace v8::internal |