// 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 <cmath>
#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<char> buffer, int* length);
// Generates 'requested_digits' after the decimal point.
static void BignumToFixed(int requested_digits, int* decimal_point,
Bignum* numerator, Bignum* denominator,
Vector<char>(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<char>(buffer), int* length);
void BignumDtoa(double v, BignumDtoaMode mode, int requested_digits,
Vector<char> 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<char> 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<char>(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<char>(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<int>(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