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 // Copyright 2011 the V8 project authors. All rights reserved. // Use of this source code is governed by a BSD-style license that can be // found in the LICENSE file. #include "src/bignum-dtoa.h" #include #include "src/base/logging.h" #include "src/bignum.h" #include "src/double.h" #include "src/utils.h" namespace v8 { namespace internal { static int NormalizedExponent(uint64_t significand, int exponent) { DCHECK(significand != 0); while ((significand & Double::kHiddenBit) == 0) { significand = significand << 1; exponent = exponent - 1; } return exponent; } // Forward declarations: // Returns an estimation of k such that 10^(k-1) <= v < 10^k. static int EstimatePower(int exponent); // Computes v / 10^estimated_power exactly, as a ratio of two bignums, numerator // and denominator. static void InitialScaledStartValues(double v, int estimated_power, bool need_boundary_deltas, Bignum* numerator, Bignum* denominator, Bignum* delta_minus, Bignum* delta_plus); // Multiplies numerator/denominator so that its values lies in the range 1-10. // Returns decimal_point s.t. // v = numerator'/denominator' * 10^(decimal_point-1) // where numerator' and denominator' are the values of numerator and // denominator after the call to this function. static void FixupMultiply10(int estimated_power, bool is_even, int* decimal_point, Bignum* numerator, Bignum* denominator, Bignum* delta_minus, Bignum* delta_plus); // Generates digits from the left to the right and stops when the generated // digits yield the shortest decimal representation of v. static void GenerateShortestDigits(Bignum* numerator, Bignum* denominator, Bignum* delta_minus, Bignum* delta_plus, bool is_even, Vector buffer, int* length); // Generates 'requested_digits' after the decimal point. static void BignumToFixed(int requested_digits, int* decimal_point, Bignum* numerator, Bignum* denominator, Vector(buffer), int* length); // Generates 'count' digits of numerator/denominator. // Once 'count' digits have been produced rounds the result depending on the // remainder (remainders of exactly .5 round upwards). Might update the // decimal_point when rounding up (for example for 0.9999). static void GenerateCountedDigits(int count, int* decimal_point, Bignum* numerator, Bignum* denominator, Vector(buffer), int* length); void BignumDtoa(double v, BignumDtoaMode mode, int requested_digits, Vector buffer, int* length, int* decimal_point) { DCHECK(v > 0); DCHECK(!Double(v).IsSpecial()); uint64_t significand = Double(v).Significand(); bool is_even = (significand & 1) == 0; int exponent = Double(v).Exponent(); int normalized_exponent = NormalizedExponent(significand, exponent); // estimated_power might be too low by 1. int estimated_power = EstimatePower(normalized_exponent); // Shortcut for Fixed. // The requested digits correspond to the digits after the point. If the // number is much too small, then there is no need in trying to get any // digits. if (mode == BIGNUM_DTOA_FIXED && -estimated_power - 1 > requested_digits) { buffer[0] = '\0'; *length = 0; // Set decimal-point to -requested_digits. This is what Gay does. // Note that it should not have any effect anyways since the string is // empty. *decimal_point = -requested_digits; return; } Bignum numerator; Bignum denominator; Bignum delta_minus; Bignum delta_plus; // Make sure the bignum can grow large enough. The smallest double equals // 4e-324. In this case the denominator needs fewer than 324*4 binary digits. // The maximum double is 1.7976931348623157e308 which needs fewer than // 308*4 binary digits. DCHECK(Bignum::kMaxSignificantBits >= 324*4); bool need_boundary_deltas = (mode == BIGNUM_DTOA_SHORTEST); InitialScaledStartValues(v, estimated_power, need_boundary_deltas, &numerator, &denominator, &delta_minus, &delta_plus); // We now have v = (numerator / denominator) * 10^estimated_power. FixupMultiply10(estimated_power, is_even, decimal_point, &numerator, &denominator, &delta_minus, &delta_plus); // We now have v = (numerator / denominator) * 10^(decimal_point-1), and // 1 <= (numerator + delta_plus) / denominator < 10 switch (mode) { case BIGNUM_DTOA_SHORTEST: GenerateShortestDigits(&numerator, &denominator, &delta_minus, &delta_plus, is_even, buffer, length); break; case BIGNUM_DTOA_FIXED: BignumToFixed(requested_digits, decimal_point, &numerator, &denominator, buffer, length); break; case BIGNUM_DTOA_PRECISION: GenerateCountedDigits(requested_digits, decimal_point, &numerator, &denominator, buffer, length); break; default: UNREACHABLE(); } buffer[*length] = '\0'; } // The procedure starts generating digits from the left to the right and stops // when the generated digits yield the shortest decimal representation of v. A // decimal representation of v is a number lying closer to v than to any other // double, so it converts to v when read. // // This is true if d, the decimal representation, is between m- and m+, the // upper and lower boundaries. d must be strictly between them if !is_even. // m- := (numerator - delta_minus) / denominator // m+ := (numerator + delta_plus) / denominator // // Precondition: 0 <= (numerator+delta_plus) / denominator < 10. // If 1 <= (numerator+delta_plus) / denominator < 10 then no leading 0 digit // will be produced. This should be the standard precondition. static void GenerateShortestDigits(Bignum* numerator, Bignum* denominator, Bignum* delta_minus, Bignum* delta_plus, bool is_even, Vector buffer, int* length) { // Small optimization: if delta_minus and delta_plus are the same just reuse // one of the two bignums. if (Bignum::Equal(*delta_minus, *delta_plus)) { delta_plus = delta_minus; } *length = 0; while (true) { uint16_t digit; digit = numerator->DivideModuloIntBignum(*denominator); DCHECK(digit <= 9); // digit is a uint16_t and therefore always positive. // digit = numerator / denominator (integer division). // numerator = numerator % denominator. buffer[(*length)++] = digit + '0'; // Can we stop already? // If the remainder of the division is less than the distance to the lower // boundary we can stop. In this case we simply round down (discarding the // remainder). // Similarly we test if we can round up (using the upper boundary). bool in_delta_room_minus; bool in_delta_room_plus; if (is_even) { in_delta_room_minus = Bignum::LessEqual(*numerator, *delta_minus); } else { in_delta_room_minus = Bignum::Less(*numerator, *delta_minus); } if (is_even) { in_delta_room_plus = Bignum::PlusCompare(*numerator, *delta_plus, *denominator) >= 0; } else { in_delta_room_plus = Bignum::PlusCompare(*numerator, *delta_plus, *denominator) > 0; } if (!in_delta_room_minus && !in_delta_room_plus) { // Prepare for next iteration. numerator->Times10(); delta_minus->Times10(); // We optimized delta_plus to be equal to delta_minus (if they share the // same value). So don't multiply delta_plus if they point to the same // object. if (delta_minus != delta_plus) { delta_plus->Times10(); } } else if (in_delta_room_minus && in_delta_room_plus) { // Let's see if 2*numerator < denominator. // If yes, then the next digit would be < 5 and we can round down. int compare = Bignum::PlusCompare(*numerator, *numerator, *denominator); if (compare < 0) { // Remaining digits are less than .5. -> Round down (== do nothing). } else if (compare > 0) { // Remaining digits are more than .5 of denominator. -> Round up. // Note that the last digit could not be a '9' as otherwise the whole // loop would have stopped earlier. // We still have an assert here in case the preconditions were not // satisfied. DCHECK(buffer[(*length) - 1] != '9'); buffer[(*length) - 1]++; } else { // Halfway case. // TODO(floitsch): need a way to solve half-way cases. // For now let's round towards even (since this is what Gay seems to // do). if ((buffer[(*length) - 1] - '0') % 2 == 0) { // Round down => Do nothing. } else { DCHECK(buffer[(*length) - 1] != '9'); buffer[(*length) - 1]++; } } return; } else if (in_delta_room_minus) { // Round down (== do nothing). return; } else { // in_delta_room_plus // Round up. // Note again that the last digit could not be '9' since this would have // stopped the loop earlier. // We still have an DCHECK here, in case the preconditions were not // satisfied. DCHECK(buffer[(*length) -1] != '9'); buffer[(*length) - 1]++; return; } } } // Let v = numerator / denominator < 10. // Then we generate 'count' digits of d = x.xxxxx... (without the decimal point) // from left to right. Once 'count' digits have been produced we decide wether // to round up or down. Remainders of exactly .5 round upwards. Numbers such // as 9.999999 propagate a carry all the way, and change the // exponent (decimal_point), when rounding upwards. static void GenerateCountedDigits(int count, int* decimal_point, Bignum* numerator, Bignum* denominator, Vector(buffer), int* length) { DCHECK(count >= 0); for (int i = 0; i < count - 1; ++i) { uint16_t digit; digit = numerator->DivideModuloIntBignum(*denominator); DCHECK(digit <= 9); // digit is a uint16_t and therefore always positive. // digit = numerator / denominator (integer division). // numerator = numerator % denominator. buffer[i] = digit + '0'; // Prepare for next iteration. numerator->Times10(); } // Generate the last digit. uint16_t digit; digit = numerator->DivideModuloIntBignum(*denominator); if (Bignum::PlusCompare(*numerator, *numerator, *denominator) >= 0) { digit++; } buffer[count - 1] = digit + '0'; // Correct bad digits (in case we had a sequence of '9's). Propagate the // carry until we hat a non-'9' or til we reach the first digit. for (int i = count - 1; i > 0; --i) { if (buffer[i] != '0' + 10) break; buffer[i] = '0'; buffer[i - 1]++; } if (buffer[0] == '0' + 10) { // Propagate a carry past the top place. buffer[0] = '1'; (*decimal_point)++; } *length = count; } // Generates 'requested_digits' after the decimal point. It might omit // trailing '0's. If the input number is too small then no digits at all are // generated (ex.: 2 fixed digits for 0.00001). // // Input verifies: 1 <= (numerator + delta) / denominator < 10. static void BignumToFixed(int requested_digits, int* decimal_point, Bignum* numerator, Bignum* denominator, Vector(buffer), int* length) { // Note that we have to look at more than just the requested_digits, since // a number could be rounded up. Example: v=0.5 with requested_digits=0. // Even though the power of v equals 0 we can't just stop here. if (-(*decimal_point) > requested_digits) { // The number is definitively too small. // Ex: 0.001 with requested_digits == 1. // Set decimal-point to -requested_digits. This is what Gay does. // Note that it should not have any effect anyways since the string is // empty. *decimal_point = -requested_digits; *length = 0; return; } else if (-(*decimal_point) == requested_digits) { // We only need to verify if the number rounds down or up. // Ex: 0.04 and 0.06 with requested_digits == 1. DCHECK(*decimal_point == -requested_digits); // Initially the fraction lies in range (1, 10]. Multiply the denominator // by 10 so that we can compare more easily. denominator->Times10(); if (Bignum::PlusCompare(*numerator, *numerator, *denominator) >= 0) { // If the fraction is >= 0.5 then we have to include the rounded // digit. buffer[0] = '1'; *length = 1; (*decimal_point)++; } else { // Note that we caught most of similar cases earlier. *length = 0; } return; } else { // The requested digits correspond to the digits after the point. // The variable 'needed_digits' includes the digits before the point. int needed_digits = (*decimal_point) + requested_digits; GenerateCountedDigits(needed_digits, decimal_point, numerator, denominator, buffer, length); } } // Returns an estimation of k such that 10^(k-1) <= v < 10^k where // v = f * 2^exponent and 2^52 <= f < 2^53. // v is hence a normalized double with the given exponent. The output is an // approximation for the exponent of the decimal approimation .digits * 10^k. // // The result might undershoot by 1 in which case 10^k <= v < 10^k+1. // Note: this property holds for v's upper boundary m+ too. // 10^k <= m+ < 10^k+1. // (see explanation below). // // Examples: // EstimatePower(0) => 16 // EstimatePower(-52) => 0 // // Note: e >= 0 => EstimatedPower(e) > 0. No similar claim can be made for e<0. static int EstimatePower(int exponent) { // This function estimates log10 of v where v = f*2^e (with e == exponent). // Note that 10^floor(log10(v)) <= v, but v <= 10^ceil(log10(v)). // Note that f is bounded by its container size. Let p = 53 (the double's // significand size). Then 2^(p-1) <= f < 2^p. // // Given that log10(v) == log2(v)/log2(10) and e+(len(f)-1) is quite close // to log2(v) the function is simplified to (e+(len(f)-1)/log2(10)). // The computed number undershoots by less than 0.631 (when we compute log3 // and not log10). // // Optimization: since we only need an approximated result this computation // can be performed on 64 bit integers. On x86/x64 architecture the speedup is // not really measurable, though. // // Since we want to avoid overshooting we decrement by 1e10 so that // floating-point imprecisions don't affect us. // // Explanation for v's boundary m+: the computation takes advantage of // the fact that 2^(p-1) <= f < 2^p. Boundaries still satisfy this requirement // (even for denormals where the delta can be much more important). const double k1Log10 = 0.30102999566398114; // 1/lg(10) // For doubles len(f) == 53 (don't forget the hidden bit). const int kSignificandSize = 53; double estimate = std::ceil((exponent + kSignificandSize - 1) * k1Log10 - 1e-10); return static_cast(estimate); } // See comments for InitialScaledStartValues. static void InitialScaledStartValuesPositiveExponent( double v, int estimated_power, bool need_boundary_deltas, Bignum* numerator, Bignum* denominator, Bignum* delta_minus, Bignum* delta_plus) { // A positive exponent implies a positive power. DCHECK(estimated_power >= 0); // Since the estimated_power is positive we simply multiply the denominator // by 10^estimated_power. // numerator = v. numerator->AssignUInt64(Double(v).Significand()); numerator->ShiftLeft(Double(v).Exponent()); // denominator = 10^estimated_power. denominator->AssignPowerUInt16(10, estimated_power); if (need_boundary_deltas) { // Introduce a common denominator so that the deltas to the boundaries are // integers. denominator->ShiftLeft(1); numerator->ShiftLeft(1); // Let v = f * 2^e, then m+ - v = 1/2 * 2^e; With the common // denominator (of 2) delta_plus equals 2^e. delta_plus->AssignUInt16(1); delta_plus->ShiftLeft(Double(v).Exponent()); // Same for delta_minus (with adjustments below if f == 2^p-1). delta_minus->AssignUInt16(1); delta_minus->ShiftLeft(Double(v).Exponent()); // If the significand (without the hidden bit) is 0, then the lower // boundary is closer than just half a ulp (unit in the last place). // There is only one exception: if the next lower number is a denormal then // the distance is 1 ulp. This cannot be the case for exponent >= 0 (but we // have to test it in the other function where exponent < 0). uint64_t v_bits = Double(v).AsUint64(); if ((v_bits & Double::kSignificandMask) == 0) { // The lower boundary is closer at half the distance of "normal" numbers. // Increase the common denominator and adapt all but the delta_minus. denominator->ShiftLeft(1); // *2 numerator->ShiftLeft(1); // *2 delta_plus->ShiftLeft(1); // *2 } } } // See comments for InitialScaledStartValues static void InitialScaledStartValuesNegativeExponentPositivePower( double v, int estimated_power, bool need_boundary_deltas, Bignum* numerator, Bignum* denominator, Bignum* delta_minus, Bignum* delta_plus) { uint64_t significand = Double(v).Significand(); int exponent = Double(v).Exponent(); // v = f * 2^e with e < 0, and with estimated_power >= 0. // This means that e is close to 0 (have a look at how estimated_power is // computed). // numerator = significand // since v = significand * 2^exponent this is equivalent to // numerator = v * / 2^-exponent numerator->AssignUInt64(significand); // denominator = 10^estimated_power * 2^-exponent (with exponent < 0) denominator->AssignPowerUInt16(10, estimated_power); denominator->ShiftLeft(-exponent); if (need_boundary_deltas) { // Introduce a common denominator so that the deltas to the boundaries are // integers. denominator->ShiftLeft(1); numerator->ShiftLeft(1); // Let v = f * 2^e, then m+ - v = 1/2 * 2^e; With the common // denominator (of 2) delta_plus equals 2^e. // Given that the denominator already includes v's exponent the distance // to the boundaries is simply 1. delta_plus->AssignUInt16(1); // Same for delta_minus (with adjustments below if f == 2^p-1). delta_minus->AssignUInt16(1); // If the significand (without the hidden bit) is 0, then the lower // boundary is closer than just one ulp (unit in the last place). // There is only one exception: if the next lower number is a denormal // then the distance is 1 ulp. Since the exponent is close to zero // (otherwise estimated_power would have been negative) this cannot happen // here either. uint64_t v_bits = Double(v).AsUint64(); if ((v_bits & Double::kSignificandMask) == 0) { // The lower boundary is closer at half the distance of "normal" numbers. // Increase the denominator and adapt all but the delta_minus. denominator->ShiftLeft(1); // *2 numerator->ShiftLeft(1); // *2 delta_plus->ShiftLeft(1); // *2 } } } // See comments for InitialScaledStartValues static void InitialScaledStartValuesNegativeExponentNegativePower( double v, int estimated_power, bool need_boundary_deltas, Bignum* numerator, Bignum* denominator, Bignum* delta_minus, Bignum* delta_plus) { const uint64_t kMinimalNormalizedExponent = V8_2PART_UINT64_C(0x00100000, 00000000); uint64_t significand = Double(v).Significand(); int exponent = Double(v).Exponent(); // Instead of multiplying the denominator with 10^estimated_power we // multiply all values (numerator and deltas) by 10^-estimated_power. // Use numerator as temporary container for power_ten. Bignum* power_ten = numerator; power_ten->AssignPowerUInt16(10, -estimated_power); if (need_boundary_deltas) { // Since power_ten == numerator we must make a copy of 10^estimated_power // before we complete the computation of the numerator. // delta_plus = delta_minus = 10^estimated_power delta_plus->AssignBignum(*power_ten); delta_minus->AssignBignum(*power_ten); } // numerator = significand * 2 * 10^-estimated_power // since v = significand * 2^exponent this is equivalent to // numerator = v * 10^-estimated_power * 2 * 2^-exponent. // Remember: numerator has been abused as power_ten. So no need to assign it // to itself. DCHECK(numerator == power_ten); numerator->MultiplyByUInt64(significand); // denominator = 2 * 2^-exponent with exponent < 0. denominator->AssignUInt16(1); denominator->ShiftLeft(-exponent); if (need_boundary_deltas) { // Introduce a common denominator so that the deltas to the boundaries are // integers. numerator->ShiftLeft(1); denominator->ShiftLeft(1); // With this shift the boundaries have their correct value, since // delta_plus = 10^-estimated_power, and // delta_minus = 10^-estimated_power. // These assignments have been done earlier. // The special case where the lower boundary is twice as close. // This time we have to look out for the exception too. uint64_t v_bits = Double(v).AsUint64(); if ((v_bits & Double::kSignificandMask) == 0 && // The only exception where a significand == 0 has its boundaries at // "normal" distances: (v_bits & Double::kExponentMask) != kMinimalNormalizedExponent) { numerator->ShiftLeft(1); // *2 denominator->ShiftLeft(1); // *2 delta_plus->ShiftLeft(1); // *2 } } } // Let v = significand * 2^exponent. // Computes v / 10^estimated_power exactly, as a ratio of two bignums, numerator // and denominator. The functions GenerateShortestDigits and // GenerateCountedDigits will then convert this ratio to its decimal // representation d, with the required accuracy. // Then d * 10^estimated_power is the representation of v. // (Note: the fraction and the estimated_power might get adjusted before // generating the decimal representation.) // // The initial start values consist of: // - a scaled numerator: s.t. numerator/denominator == v / 10^estimated_power. // - a scaled (common) denominator. // optionally (used by GenerateShortestDigits to decide if it has the shortest // decimal converting back to v): // - v - m-: the distance to the lower boundary. // - m+ - v: the distance to the upper boundary. // // v, m+, m-, and therefore v - m- and m+ - v all share the same denominator. // // Let ep == estimated_power, then the returned values will satisfy: // v / 10^ep = numerator / denominator. // v's boundarys m- and m+: // m- / 10^ep == v / 10^ep - delta_minus / denominator // m+ / 10^ep == v / 10^ep + delta_plus / denominator // Or in other words: // m- == v - delta_minus * 10^ep / denominator; // m+ == v + delta_plus * 10^ep / denominator; // // Since 10^(k-1) <= v < 10^k (with k == estimated_power) // or 10^k <= v < 10^(k+1) // we then have 0.1 <= numerator/denominator < 1 // or 1 <= numerator/denominator < 10 // // It is then easy to kickstart the digit-generation routine. // // The boundary-deltas are only filled if need_boundary_deltas is set. static void InitialScaledStartValues(double v, int estimated_power, bool need_boundary_deltas, Bignum* numerator, Bignum* denominator, Bignum* delta_minus, Bignum* delta_plus) { if (Double(v).Exponent() >= 0) { InitialScaledStartValuesPositiveExponent( v, estimated_power, need_boundary_deltas, numerator, denominator, delta_minus, delta_plus); } else if (estimated_power >= 0) { InitialScaledStartValuesNegativeExponentPositivePower( v, estimated_power, need_boundary_deltas, numerator, denominator, delta_minus, delta_plus); } else { InitialScaledStartValuesNegativeExponentNegativePower( v, estimated_power, need_boundary_deltas, numerator, denominator, delta_minus, delta_plus); } } // This routine multiplies numerator/denominator so that its values lies in the // range 1-10. That is after a call to this function we have: // 1 <= (numerator + delta_plus) /denominator < 10. // Let numerator the input before modification and numerator' the argument // after modification, then the output-parameter decimal_point is such that // numerator / denominator * 10^estimated_power == // numerator' / denominator' * 10^(decimal_point - 1) // In some cases estimated_power was too low, and this is already the case. We // then simply adjust the power so that 10^(k-1) <= v < 10^k (with k == // estimated_power) but do not touch the numerator or denominator. // Otherwise the routine multiplies the numerator and the deltas by 10. static void FixupMultiply10(int estimated_power, bool is_even, int* decimal_point, Bignum* numerator, Bignum* denominator, Bignum* delta_minus, Bignum* delta_plus) { bool in_range; if (is_even) { // For IEEE doubles half-way cases (in decimal system numbers ending with 5) // are rounded to the closest floating-point number with even significand. in_range = Bignum::PlusCompare(*numerator, *delta_plus, *denominator) >= 0; } else { in_range = Bignum::PlusCompare(*numerator, *delta_plus, *denominator) > 0; } if (in_range) { // Since numerator + delta_plus >= denominator we already have // 1 <= numerator/denominator < 10. Simply update the estimated_power. *decimal_point = estimated_power + 1; } else { *decimal_point = estimated_power; numerator->Times10(); if (Bignum::Equal(*delta_minus, *delta_plus)) { delta_minus->Times10(); delta_plus->AssignBignum(*delta_minus); } else { delta_minus->Times10(); delta_plus->Times10(); } } } } // namespace internal } // namespace v8