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ladybird/AK/FloatingPointStringConversions.cpp

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AK: Add an exact and fast floating point parsing algorithm This is based on the paper by Daniel Lemire called "Number parsing at a Gigabyte per second", currently available at https://arxiv.org/abs/2101.11408 An implementation can be found at https://github.com/fastfloat/fast_float To support both strtod like methods and String::to_double we have two different APIs. The parse_first_floating_point gives back both the result, next character to read and the error/out of range status. Out of range here means we rounded to infinity 0. The other API, parse_floating_point_completely, will return a floating point only if the given character range contains just the floating point and nothing else. This can be much faster as we can skip actually computing the value if we notice we did not parse the whole range. Both of these APIs support a very lenient format to be usable in as many places as possible. Also it does not check for "named" values like "nan", "inf", "NAN" etc. Because this can be different for every usage. For integers and small values this new method is not faster and often even a tiny bit slower than the current strtod implementation. However the strtod implementation is wrong for a lot of values and has a much less predictable running time. For correctness this method was tested against known string -> double datasets from https://github.com/nigeltao/parse-number-fxx-test-data This method gives 100% accuracy. The old strtod gave an incorrect value in over 50% of the numbers tested.
2022-10-13 00:18:56 +00:00
/*
* Copyright (c) 2022, David Tuin <davidot@serenityos.org>
*
* SPDX-License-Identifier: BSD-2-Clause
*/
#include <AK/CharacterTypes.h>
#include <AK/FloatingPointStringConversions.h>
#include <AK/Format.h>
#include <AK/ScopeGuard.h>
#include <AK/StringView.h>
#include <AK/UFixedBigInt.h>
AK: Implement Knuth's algorithm D for dividing UFixedBigInt's
2023-02-04 15:57:10 +00:00
#include <AK/UFixedBigIntDivision.h>
AK: Add an exact and fast floating point parsing algorithm This is based on the paper by Daniel Lemire called "Number parsing at a Gigabyte per second", currently available at https://arxiv.org/abs/2101.11408 An implementation can be found at https://github.com/fastfloat/fast_float To support both strtod like methods and String::to_double we have two different APIs. The parse_first_floating_point gives back both the result, next character to read and the error/out of range status. Out of range here means we rounded to infinity 0. The other API, parse_floating_point_completely, will return a floating point only if the given character range contains just the floating point and nothing else. This can be much faster as we can skip actually computing the value if we notice we did not parse the whole range. Both of these APIs support a very lenient format to be usable in as many places as possible. Also it does not check for "named" values like "nan", "inf", "NAN" etc. Because this can be different for every usage. For integers and small values this new method is not faster and often even a tiny bit slower than the current strtod implementation. However the strtod implementation is wrong for a lot of values and has a much less predictable running time. For correctness this method was tested against known string -> double datasets from https://github.com/nigeltao/parse-number-fxx-test-data This method gives 100% accuracy. The old strtod gave an incorrect value in over 50% of the numbers tested.
2022-10-13 00:18:56 +00:00
namespace AK {
// This entire algorithm is an implementation of the paper: Number Parsing at a Gigabyte per Second
// by Daniel Lemire, available at https://arxiv.org/abs/2101.11408 and an implementation
// at https://github.com/fastfloat/fast_float
// There is also a perhaps more easily understandable explanation
// at https://nigeltao.github.io/blog/2020/eisel-lemire.html
template<typename T>
Everywhere: Run clang-format
2022-10-16 22:06:11 +00:00
concept ParseableFloatingPoint = IsFloatingPoint<T> && (sizeof(T) == sizeof(u32) || sizeof(T) == sizeof(u64));
AK: Add an exact and fast floating point parsing algorithm This is based on the paper by Daniel Lemire called "Number parsing at a Gigabyte per second", currently available at https://arxiv.org/abs/2101.11408 An implementation can be found at https://github.com/fastfloat/fast_float To support both strtod like methods and String::to_double we have two different APIs. The parse_first_floating_point gives back both the result, next character to read and the error/out of range status. Out of range here means we rounded to infinity 0. The other API, parse_floating_point_completely, will return a floating point only if the given character range contains just the floating point and nothing else. This can be much faster as we can skip actually computing the value if we notice we did not parse the whole range. Both of these APIs support a very lenient format to be usable in as many places as possible. Also it does not check for "named" values like "nan", "inf", "NAN" etc. Because this can be different for every usage. For integers and small values this new method is not faster and often even a tiny bit slower than the current strtod implementation. However the strtod implementation is wrong for a lot of values and has a much less predictable running time. For correctness this method was tested against known string -> double datasets from https://github.com/nigeltao/parse-number-fxx-test-data This method gives 100% accuracy. The old strtod gave an incorrect value in over 50% of the numbers tested.
2022-10-13 00:18:56 +00:00
template<ParseableFloatingPoint T>
struct FloatingPointInfo {
static_assert(sizeof(T) == sizeof(u64) || sizeof(T) == sizeof(u32));
using SameSizeUnsigned = Conditional<sizeof(T) == sizeof(u64), u64, u32>;
// Implementing just this gives all the other bit sizes and mask immediately.
static constexpr inline i32 mantissa_bits()
{
if constexpr (sizeof(T) == sizeof(u64))
return 52;
return 23;
}
static constexpr inline i32 exponent_bits()
{
return sizeof(T) * 8u - 1u - mantissa_bits();
}
static constexpr inline i32 exponent_bias()
{
return (1 << (exponent_bits() - 1)) - 1;
}
static constexpr inline i32 minimum_exponent()
{
return -exponent_bias();
}
static constexpr inline i32 infinity_exponent()
{
static_assert(exponent_bits() < 31);
return (1 << exponent_bits()) - 1;
}
static constexpr inline i32 sign_bit_index()
{
return sizeof(T) * 8 - 1;
}
static constexpr inline SameSizeUnsigned sign_mask()
{
return SameSizeUnsigned { 1 } << sign_bit_index();
}
static constexpr inline SameSizeUnsigned mantissa_mask()
{
return (SameSizeUnsigned { 1 } << mantissa_bits()) - 1;
}
static constexpr inline SameSizeUnsigned exponent_mask()
{
return SameSizeUnsigned { infinity_exponent() } << mantissa_bits();
}
static constexpr inline i32 max_exponent_round_to_even()
{
if constexpr (sizeof(T) == sizeof(u64))
return 23;
return 10;
}
static constexpr inline i32 min_exponent_round_to_even()
{
if constexpr (sizeof(T) == sizeof(u64))
return -4;
return -17;
}
static constexpr inline size_t max_possible_digits_needed_for_parsing()
{
if constexpr (sizeof(T) == sizeof(u64))
return 769;
return 114;
}
static constexpr inline i32 max_power_of_10()
{
if constexpr (sizeof(T) == sizeof(u64))
return 308;
return 38;
}
static constexpr inline i32 min_power_of_10()
{
// Closest double value to zero is xe-324 and since we have at most 19 digits
// we know that -324 -19 = -343 so exponent below that must be zero (for double)
if constexpr (sizeof(T) == sizeof(u64))
return -342;
return -65;
}
static constexpr inline i32 max_exact_power_of_10()
{
// These are the largest power of 10 representable in T
// So all powers of 10*i less than or equal to this should be the exact
// values, be careful as they can be above "safe integer" limits.
if constexpr (sizeof(T) == sizeof(u64))
return 22;
return 10;
}
static constexpr inline T power_of_ten(i32 exponent)
{
VERIFY(exponent <= max_exact_power_of_10());
VERIFY(exponent >= 0);
return m_powers_of_ten_stored[exponent];
}
template<u32 MaxPower>
static constexpr inline Array<T, MaxPower + 1> compute_powers_of_ten()
{
// All these values are guaranteed to be exact all powers of MaxPower is the
Array<T, MaxPower + 1> values {};
values[0] = T(1.0);
T ten = T(10.);
for (u32 i = 1; i <= MaxPower; ++i)
values[i] = values[i - 1] * ten;
return values;
}
static constexpr auto m_powers_of_ten_stored = compute_powers_of_ten<max_exact_power_of_10()>();
};
template<typename T>
using BitSizedUnsignedForFloatingPoint = typename FloatingPointInfo<T>::SameSizeUnsigned;
struct BasicParseResult {
u64 mantissa = 0;
i64 exponent = 0;
bool valid = false;
bool negative = false;
bool more_than_19_digits_with_overflow = false;
char const* last_parsed { nullptr };
StringView whole_part;
StringView fractional_part;
};
static constexpr auto max_representable_power_of_ten_in_u64 = 19;
static_assert(1e19 <= static_cast<double>(NumericLimits<u64>::max()));
static_assert(1e20 >= static_cast<double>(NumericLimits<u64>::max()));
#if __BYTE_ORDER__ == __ORDER_BIG_ENDIAN__
# error Float parsing currently assumes little endian, this fact is only used in fast parsing of 8 digits at a time \
you _should_ only need to change read eight_digits to make this big endian compatible.
#endif
constexpr u64 read_eight_digits(char const* string)
{
u64 val;
__builtin_memcpy(&val, string, sizeof(val));
return val;
}
constexpr static bool has_eight_digits(u64 value)
{
// The ascii digits 0-9 are hex 0x30 - 0x39
// If x is within that range then y := x + 0x46 is 0x76 to 0x7f
// z := x - 0x30 is 0x00 - 0x09
// y | z = 0x7t where t is in the range 0 - f so doing & 0x80 gives 0
// However if a character x is below 0x30 then x - 0x30 underflows setting
// the 0x80 bit of the next digit meaning & 0x80 will never be 0.
// Similarly if a character x is above 0x39 then x + 0x46 gives at least
// 0x80 thus & 0x80 will not be zero.
return (((value + 0x4646464646464646) | (value - 0x3030303030303030)) & 0x8080808080808080) == 0;
}
constexpr static u32 eight_digits_to_value(u64 value)
{
// THIS DOES ABSOLUTELY ASSUME has_eight_digits is true
// This trick is based on https://johnnylee-sde.github.io/Fast-numeric-string-to-int/
// FIXME: fast_float uses a slightly different version, but that is far harder
// to understand and does not seem to improve performance substantially.
// See https://github.com/fastfloat/fast_float/pull/28
// First convert the digits to their respectively numbers (0x30 -> 0x00 etc.)
value -= 0x3030303030303030;
// Because of little endian the first number will in fact be the least significant
// bits of value i.e. "12345678" -> 0x0807060504030201
// This means that we need to shift/multiply each digit with 8 - the byte it is in
// So the eight need to go down, and the 01 need to be multiplied with 10000000
// We effectively multiply by 10 and then shift those values to the right (2^8 = 256)
// We then shift the values back down, this leads to 4 digits pairs in the 2 byte parts
// The values between are "garbage" which we will ignore
value = (value * (256 * 10 + 1)) >> 8;
// So with our example this gives 0x$$4e$$38$$22$$0c, where $$ is garbage/ignored
// In decimal this gives 78 56 34 12
// Now we keep performing the same trick twice more
// First * 100 and shift of 16 (2^16 = 65536) and then shift back
value = ((value & 0x00FF00FF00FF00FF) * (65536 * 100 + 1)) >> 16;
// Again with our example this gives 0x$$$$162e$$$$04d2
// 5678 1234
// And finally with * 10000 and shift of 32 (2^32 = 4294967296)
value = ((value & 0x0000FFFF0000FFFF) * (4294967296 * 10000 + 1)) >> 32;
// With the example this gives 0x$$$$$$$$00bc614e
// 12345678
// Now we just truncate to the lower part
return u32(value);
}
template<typename IsDoneCallback, typename Has8CharsLeftCallback>
static BasicParseResult parse_numbers(char const* start, IsDoneCallback is_done, Has8CharsLeftCallback has_eight_chars_to_read)
{
char const* ptr = start;
BasicParseResult result {};
if (start == nullptr || is_done(ptr))
return result;
if (*ptr == '-' || *ptr == '+') {
result.negative = *ptr == '-';
++ptr;
if (is_done(ptr) || (!is_ascii_digit(*ptr) && *ptr != '.'))
return result;
}
auto const fast_parse_decimal = [&](auto& value) {
while (has_eight_chars_to_read(ptr) && has_eight_digits(read_eight_digits(ptr))) {
value = 100'000'000 * value + eight_digits_to_value(read_eight_digits(ptr));
ptr += 8;
}
while (!is_done(ptr) && is_ascii_digit(*ptr)) {
value = 10 * value + (*ptr - '0');
++ptr;
}
};
u64 mantissa = 0;
auto const* whole_part_start = ptr;
fast_parse_decimal(mantissa);
auto const* whole_part_end = ptr;
auto digits_found = whole_part_end - whole_part_start;
result.whole_part = StringView(whole_part_start, digits_found);
i64 exponent = 0;
auto const* start_of_fractional_part = ptr;
if (!is_done(ptr) && *ptr == '.') {
++ptr;
++start_of_fractional_part;
fast_parse_decimal(mantissa);
// We parsed x digits after the dot so need to multiply with 10^-x
exponent = -(ptr - start_of_fractional_part);
}
result.fractional_part = StringView(start_of_fractional_part, ptr - start_of_fractional_part);
digits_found += -exponent;
// If both the part
if (digits_found == 0)
return result;
i64 explicit_exponent = 0;
// We do this in a lambda to easily be able to get out of parsing the exponent
// and resetting the final character read to before the 'e'.
[&] {
if (is_done(ptr))
return;
if (*ptr != 'e' && *ptr != 'E')
return;
auto* pointer_before_e = ptr;
ArmedScopeGuard reset_ptr { [&] { ptr = pointer_before_e; } };
++ptr;
if (is_done(ptr))
return;
bool negative_exponent = false;
if (*ptr == '-' || *ptr == '+') {
negative_exponent = *ptr == '-';
++ptr;
if (is_done(ptr))
return;
}
if (!is_ascii_digit(*ptr))
return;
// Now we must have an optional sign and at least one digit so we
// will not reset
reset_ptr.disarm();
while (!is_done(ptr) && is_ascii_digit(*ptr)) {
// A massive exponent is not really a problem as this would
// require a lot of characters so we would fallback on precise
// parsing anyway (this is already 268435456 digits or 10 megabytes of digits)
if (explicit_exponent < 0x10'000'000)
explicit_exponent = 10 * explicit_exponent + (*ptr - '0');
++ptr;
}
explicit_exponent = negative_exponent ? -explicit_exponent : explicit_exponent;
exponent += explicit_exponent;
}();
result.valid = true;
result.last_parsed = ptr;
if (digits_found > max_representable_power_of_ten_in_u64) {
// There could be overflow but because we just count the digits it could be leading zeros
auto const* leading_digit = whole_part_start;
while (!is_done(leading_digit) && (*leading_digit == '0' || *leading_digit == '.')) {
if (*leading_digit == '0')
--digits_found;
++leading_digit;
}
if (digits_found > max_representable_power_of_ten_in_u64) {
// FIXME: We just removed leading zeros, we might be able to skip these easily again.
// If removing the leading zeros does not help we reparse and keep just the significant digits
result.more_than_19_digits_with_overflow = true;
mantissa = 0;
constexpr i64 smallest_nineteen_digit_number = { 1000000000000000000 };
char const* reparse_ptr = whole_part_start;
constexpr i64 smallest_eleven_digit_number = { 10000000000 };
while (mantissa < smallest_eleven_digit_number && (whole_part_end - reparse_ptr) >= 8) {
mantissa = 100'000'000 * mantissa + eight_digits_to_value(read_eight_digits(reparse_ptr));
reparse_ptr += 8;
}
while (mantissa < smallest_nineteen_digit_number && reparse_ptr != whole_part_end) {
mantissa = 10 * mantissa + (*reparse_ptr - '0');
++reparse_ptr;
}
if (mantissa >= smallest_nineteen_digit_number) {
// We still needed to parse (whole_part_end - reparse_ptr) digits so scale the exponent
exponent = explicit_exponent + (whole_part_end - reparse_ptr);
} else {
reparse_ptr = start_of_fractional_part;
char const* fractional_end = result.fractional_part.characters_without_null_termination() + result.fractional_part.length();
while (mantissa < smallest_eleven_digit_number && (fractional_end - reparse_ptr) >= 8) {
mantissa = 100'000'000 * mantissa + eight_digits_to_value(read_eight_digits(reparse_ptr));
reparse_ptr += 8;
}
while (mantissa < smallest_nineteen_digit_number && reparse_ptr != fractional_end) {
mantissa = 10 * mantissa + (*reparse_ptr - '0');
++reparse_ptr;
}
// Again we might be truncating fractional number so scale the exponent with that
// However here need to subtract 1 from the exponent for every fractional digit
exponent = explicit_exponent - (reparse_ptr - start_of_fractional_part);
}
}
}
result.mantissa = mantissa;
result.exponent = exponent;
return result;
}
constexpr static u128 compute_power_of_five(i64 exponent)
{
constexpr u4096 bit128 = u4096 { 1u } << 127u;
constexpr u4096 bit129 = u4096 { 1u } << 128u;
VERIFY(exponent <= 308);
VERIFY(exponent >= -342);
if (exponent >= 0) {
u4096 base { 1u };
for (auto i = 0u; i < exponent; ++i) {
base *= 5u;
}
while (base < bit128)
base <<= 1u;
while (base >= bit129)
base >>= 1u;
return u128 { base };
}
exponent *= -1;
if (exponent <= 27) {
u4096 base { 1u };
for (auto i = 0u; i < exponent; ++i) {
base *= 5u;
}
AK+LibAudio: Remove UFixedBigInt::my_size
2023-02-09 13:46:07 +00:00
auto z = 4096 - base.clz();
AK: Add an exact and fast floating point parsing algorithm This is based on the paper by Daniel Lemire called "Number parsing at a Gigabyte per second", currently available at https://arxiv.org/abs/2101.11408 An implementation can be found at https://github.com/fastfloat/fast_float To support both strtod like methods and String::to_double we have two different APIs. The parse_first_floating_point gives back both the result, next character to read and the error/out of range status. Out of range here means we rounded to infinity 0. The other API, parse_floating_point_completely, will return a floating point only if the given character range contains just the floating point and nothing else. This can be much faster as we can skip actually computing the value if we notice we did not parse the whole range. Both of these APIs support a very lenient format to be usable in as many places as possible. Also it does not check for "named" values like "nan", "inf", "NAN" etc. Because this can be different for every usage. For integers and small values this new method is not faster and often even a tiny bit slower than the current strtod implementation. However the strtod implementation is wrong for a lot of values and has a much less predictable running time. For correctness this method was tested against known string -> double datasets from https://github.com/nigeltao/parse-number-fxx-test-data This method gives 100% accuracy. The old strtod gave an incorrect value in over 50% of the numbers tested.
2022-10-13 00:18:56 +00:00
auto b = z + 127;
u4096 base2 { 1u };
for (auto i = 0u; i < b; ++i) {
base2 *= 2u;
}
base2 /= base;
base2 += 1u;
return u128 { base2 };
}
VERIFY(exponent <= 342);
VERIFY(exponent >= 28);
u4096 base { 1u };
for (auto i = 0u; i < exponent; ++i) {
base *= 5u;
}
AK+LibAudio: Remove UFixedBigInt::my_size
2023-02-09 13:46:07 +00:00
auto z = 4096 - base.clz();
AK: Add an exact and fast floating point parsing algorithm This is based on the paper by Daniel Lemire called "Number parsing at a Gigabyte per second", currently available at https://arxiv.org/abs/2101.11408 An implementation can be found at https://github.com/fastfloat/fast_float To support both strtod like methods and String::to_double we have two different APIs. The parse_first_floating_point gives back both the result, next character to read and the error/out of range status. Out of range here means we rounded to infinity 0. The other API, parse_floating_point_completely, will return a floating point only if the given character range contains just the floating point and nothing else. This can be much faster as we can skip actually computing the value if we notice we did not parse the whole range. Both of these APIs support a very lenient format to be usable in as many places as possible. Also it does not check for "named" values like "nan", "inf", "NAN" etc. Because this can be different for every usage. For integers and small values this new method is not faster and often even a tiny bit slower than the current strtod implementation. However the strtod implementation is wrong for a lot of values and has a much less predictable running time. For correctness this method was tested against known string -> double datasets from https://github.com/nigeltao/parse-number-fxx-test-data This method gives 100% accuracy. The old strtod gave an incorrect value in over 50% of the numbers tested.
2022-10-13 00:18:56 +00:00
auto b = 2 * z + 128;
u4096 base2 { 1u };
for (auto i = 0u; i < b; ++i) {
base2 *= 2u;
}
base2 /= base;
base2 += 1u;
while (base2 >= bit129)
base2 >>= 1u;
return u128 { base2 };
}
static constexpr i64 lowest_exponent = -342;
static constexpr i64 highest_exponent = 308;
constexpr auto pre_compute_table()
{
// Computing this entire table at compile time is slow and hits constexpr
// limits, so we just compute a (the simplest) value to make sure the
// function is used. This table can thus be generated with the function
// `u128 compute_power_of_five(i64 exponent)` above.
AK::Array<u128, highest_exponent - lowest_exponent + 1> values = {
u128 { 0x113faa2906a13b3fULL, 0xeef453d6923bd65aULL },
u128 { 0x4ac7ca59a424c507ULL, 0x9558b4661b6565f8ULL },
u128 { 0x5d79bcf00d2df649ULL, 0xbaaee17fa23ebf76ULL },
u128 { 0xf4d82c2c107973dcULL, 0xe95a99df8ace6f53ULL },
u128 { 0x79071b9b8a4be869ULL, 0x91d8a02bb6c10594ULL },
u128 { 0x9748e2826cdee284ULL, 0xb64ec836a47146f9ULL },
u128 { 0xfd1b1b2308169b25ULL, 0xe3e27a444d8d98b7ULL },
u128 { 0xfe30f0f5e50e20f7ULL, 0x8e6d8c6ab0787f72ULL },
u128 { 0xbdbd2d335e51a935ULL, 0xb208ef855c969f4fULL },
u128 { 0xad2c788035e61382ULL, 0xde8b2b66b3bc4723ULL },
u128 { 0x4c3bcb5021afcc31ULL, 0x8b16fb203055ac76ULL },
u128 { 0xdf4abe242a1bbf3dULL, 0xaddcb9e83c6b1793ULL },
u128 { 0xd71d6dad34a2af0dULL, 0xd953e8624b85dd78ULL },
u128 { 0x8672648c40e5ad68ULL, 0x87d4713d6f33aa6bULL },
u128 { 0x680efdaf511f18c2ULL, 0xa9c98d8ccb009506ULL },
u128 { 0x212bd1b2566def2ULL, 0xd43bf0effdc0ba48ULL },
u128 { 0x14bb630f7604b57ULL, 0x84a57695fe98746dULL },
u128 { 0x419ea3bd35385e2dULL, 0xa5ced43b7e3e9188ULL },
u128 { 0x52064cac828675b9ULL, 0xcf42894a5dce35eaULL },
u128 { 0x7343efebd1940993ULL, 0x818995ce7aa0e1b2ULL },
u128 { 0x1014ebe6c5f90bf8ULL, 0xa1ebfb4219491a1fULL },
u128 { 0xd41a26e077774ef6ULL, 0xca66fa129f9b60a6ULL },
u128 { 0x8920b098955522b4ULL, 0xfd00b897478238d0ULL },
u128 { 0x55b46e5f5d5535b0ULL, 0x9e20735e8cb16382ULL },
u128 { 0xeb2189f734aa831dULL, 0xc5a890362fddbc62ULL },
u128 { 0xa5e9ec7501d523e4ULL, 0xf712b443bbd52b7bULL },
u128 { 0x47b233c92125366eULL, 0x9a6bb0aa55653b2dULL },
u128 { 0x999ec0bb696e840aULL, 0xc1069cd4eabe89f8ULL },
u128 { 0xc00670ea43ca250dULL, 0xf148440a256e2c76ULL },
u128 { 0x380406926a5e5728ULL, 0x96cd2a865764dbcaULL },
u128 { 0xc605083704f5ecf2ULL, 0xbc807527ed3e12bcULL },
u128 { 0xf7864a44c633682eULL, 0xeba09271e88d976bULL },
u128 { 0x7ab3ee6afbe0211dULL, 0x93445b8731587ea3ULL },
u128 { 0x5960ea05bad82964ULL, 0xb8157268fdae9e4cULL },
u128 { 0x6fb92487298e33bdULL, 0xe61acf033d1a45dfULL },
u128 { 0xa5d3b6d479f8e056ULL, 0x8fd0c16206306babULL },
u128 { 0x8f48a4899877186cULL, 0xb3c4f1ba87bc8696ULL },
u128 { 0x331acdabfe94de87ULL, 0xe0b62e2929aba83cULL },
u128 { 0x9ff0c08b7f1d0b14ULL, 0x8c71dcd9ba0b4925ULL },
u128 { 0x7ecf0ae5ee44dd9ULL, 0xaf8e5410288e1b6fULL },
u128 { 0xc9e82cd9f69d6150ULL, 0xdb71e91432b1a24aULL },
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u128 { 0xaab8f01e6e10b4a6ULL, 0xe9d71b689dde71afULL },
u128 { 0xcab3961304ca70e8ULL, 0x9226712162ab070dULL },
u128 { 0x3d607b97c5fd0d22ULL, 0xb6b00d69bb55c8d1ULL },
u128 { 0x8cb89a7db77c506aULL, 0xe45c10c42a2b3b05ULL },
u128 { 0x77f3608e92adb242ULL, 0x8eb98a7a9a5b04e3ULL },
u128 { 0x55f038b237591ed3ULL, 0xb267ed1940f1c61cULL },
u128 { 0x6b6c46dec52f6688ULL, 0xdf01e85f912e37a3ULL },
u128 { 0x2323ac4b3b3da015ULL, 0x8b61313bbabce2c6ULL },
u128 { 0xabec975e0a0d081aULL, 0xae397d8aa96c1b77ULL },
u128 { 0x96e7bd358c904a21ULL, 0xd9c7dced53c72255ULL },
u128 { 0x7e50d64177da2e54ULL, 0x881cea14545c7575ULL },
u128 { 0xdde50bd1d5d0b9e9ULL, 0xaa242499697392d2ULL },
u128 { 0x955e4ec64b44e864ULL, 0xd4ad2dbfc3d07787ULL },
u128 { 0xbd5af13bef0b113eULL, 0x84ec3c97da624ab4ULL },
u128 { 0xecb1ad8aeacdd58eULL, 0xa6274bbdd0fadd61ULL },
u128 { 0x67de18eda5814af2ULL, 0xcfb11ead453994baULL },
u128 { 0x80eacf948770ced7ULL, 0x81ceb32c4b43fcf4ULL },
u128 { 0xa1258379a94d028dULL, 0xa2425ff75e14fc31ULL },
u128 { 0x96ee45813a04330ULL, 0xcad2f7f5359a3b3eULL },
u128 { 0x8bca9d6e188853fcULL, 0xfd87b5f28300ca0dULL },
u128 { 0x775ea264cf55347eULL, 0x9e74d1b791e07e48ULL },
u128 { 0x95364afe032a819eULL, 0xc612062576589ddaULL },
u128 { 0x3a83ddbd83f52205ULL, 0xf79687aed3eec551ULL },
u128 { 0xc4926a9672793543ULL, 0x9abe14cd44753b52ULL },
u128 { 0x75b7053c0f178294ULL, 0xc16d9a0095928a27ULL },
u128 { 0x5324c68b12dd6339ULL, 0xf1c90080baf72cb1ULL },
u128 { 0xd3f6fc16ebca5e04ULL, 0x971da05074da7beeULL },
u128 { 0x88f4bb1ca6bcf585ULL, 0xbce5086492111aeaULL },
u128 { 0x2b31e9e3d06c32e6ULL, 0xec1e4a7db69561a5ULL },
u128 { 0x3aff322e62439fd0ULL, 0x9392ee8e921d5d07ULL },
u128 { 0x9befeb9fad487c3ULL, 0xb877aa3236a4b449ULL },
u128 { 0x4c2ebe687989a9b4ULL, 0xe69594bec44de15bULL },
u128 { 0xf9d37014bf60a11ULL, 0x901d7cf73ab0acd9ULL },
u128 { 0x538484c19ef38c95ULL, 0xb424dc35095cd80fULL },
u128 { 0x2865a5f206b06fbaULL, 0xe12e13424bb40e13ULL },
u128 { 0xf93f87b7442e45d4ULL, 0x8cbccc096f5088cbULL },
u128 { 0xf78f69a51539d749ULL, 0xafebff0bcb24aafeULL },
u128 { 0xb573440e5a884d1cULL, 0xdbe6fecebdedd5beULL },
u128 { 0x31680a88f8953031ULL, 0x89705f4136b4a597ULL },
u128 { 0xfdc20d2b36ba7c3eULL, 0xabcc77118461cefcULL },
u128 { 0x3d32907604691b4dULL, 0xd6bf94d5e57a42bcULL },
u128 { 0xa63f9a49c2c1b110ULL, 0x8637bd05af6c69b5ULL },
u128 { 0xfcf80dc33721d54ULL, 0xa7c5ac471b478423ULL },
u128 { 0xd3c36113404ea4a9ULL, 0xd1b71758e219652bULL },
u128 { 0x645a1cac083126eaULL, 0x83126e978d4fdf3bULL },
u128 { 0x3d70a3d70a3d70a4ULL, 0xa3d70a3d70a3d70aULL },
u128 { 0xcccccccccccccccdULL, 0xccccccccccccccccULL },
compute_power_of_five(0),
u128 { 0x0ULL, 0xa000000000000000ULL },
u128 { 0x0ULL, 0xc800000000000000ULL },
u128 { 0x0ULL, 0xfa00000000000000ULL },
u128 { 0x0ULL, 0x9c40000000000000ULL },
u128 { 0x0ULL, 0xc350000000000000ULL },
u128 { 0x0ULL, 0xf424000000000000ULL },
u128 { 0x0ULL, 0x9896800000000000ULL },
u128 { 0x0ULL, 0xbebc200000000000ULL },
u128 { 0x0ULL, 0xee6b280000000000ULL },
u128 { 0x0ULL, 0x9502f90000000000ULL },
u128 { 0x0ULL, 0xba43b74000000000ULL },
u128 { 0x0ULL, 0xe8d4a51000000000ULL },
u128 { 0x0ULL, 0x9184e72a00000000ULL },
u128 { 0x0ULL, 0xb5e620f480000000ULL },
u128 { 0x0ULL, 0xe35fa931a0000000ULL },
u128 { 0x0ULL, 0x8e1bc9bf04000000ULL },
u128 { 0x0ULL, 0xb1a2bc2ec5000000ULL },
u128 { 0x0ULL, 0xde0b6b3a76400000ULL },
u128 { 0x0ULL, 0x8ac7230489e80000ULL },
u128 { 0x0ULL, 0xad78ebc5ac620000ULL },
u128 { 0x0ULL, 0xd8d726b7177a8000ULL },
u128 { 0x0ULL, 0x878678326eac9000ULL },
u128 { 0x0ULL, 0xa968163f0a57b400ULL },
u128 { 0x0ULL, 0xd3c21bcecceda100ULL },
u128 { 0x0ULL, 0x84595161401484a0ULL },
u128 { 0x0ULL, 0xa56fa5b99019a5c8ULL },
u128 { 0x0ULL, 0xcecb8f27f4200f3aULL },
u128 { 0x4000000000000000ULL, 0x813f3978f8940984ULL },
u128 { 0x5000000000000000ULL, 0xa18f07d736b90be5ULL },
u128 { 0xa400000000000000ULL, 0xc9f2c9cd04674edeULL },
u128 { 0x4d00000000000000ULL, 0xfc6f7c4045812296ULL },
u128 { 0xf020000000000000ULL, 0x9dc5ada82b70b59dULL },
u128 { 0x6c28000000000000ULL, 0xc5371912364ce305ULL },
u128 { 0xc732000000000000ULL, 0xf684df56c3e01bc6ULL },
u128 { 0x3c7f400000000000ULL, 0x9a130b963a6c115cULL },
u128 { 0x4b9f100000000000ULL, 0xc097ce7bc90715b3ULL },
u128 { 0x1e86d40000000000ULL, 0xf0bdc21abb48db20ULL },
u128 { 0x1314448000000000ULL, 0x96769950b50d88f4ULL },
u128 { 0x17d955a000000000ULL, 0xbc143fa4e250eb31ULL },
u128 { 0x5dcfab0800000000ULL, 0xeb194f8e1ae525fdULL },
u128 { 0x5aa1cae500000000ULL, 0x92efd1b8d0cf37beULL },
u128 { 0xf14a3d9e40000000ULL, 0xb7abc627050305adULL },
u128 { 0x6d9ccd05d0000000ULL, 0xe596b7b0c643c719ULL },
u128 { 0xe4820023a2000000ULL, 0x8f7e32ce7bea5c6fULL },
u128 { 0xdda2802c8a800000ULL, 0xb35dbf821ae4f38bULL },
u128 { 0xd50b2037ad200000ULL, 0xe0352f62a19e306eULL },
u128 { 0x4526f422cc340000ULL, 0x8c213d9da502de45ULL },
u128 { 0x9670b12b7f410000ULL, 0xaf298d050e4395d6ULL },
u128 { 0x3c0cdd765f114000ULL, 0xdaf3f04651d47b4cULL },
u128 { 0xa5880a69fb6ac800ULL, 0x88d8762bf324cd0fULL },
u128 { 0x8eea0d047a457a00ULL, 0xab0e93b6efee0053ULL },
u128 { 0x72a4904598d6d880ULL, 0xd5d238a4abe98068ULL },
u128 { 0x47a6da2b7f864750ULL, 0x85a36366eb71f041ULL },
u128 { 0x999090b65f67d924ULL, 0xa70c3c40a64e6c51ULL },
u128 { 0xfff4b4e3f741cf6dULL, 0xd0cf4b50cfe20765ULL },
u128 { 0xbff8f10e7a8921a4ULL, 0x82818f1281ed449fULL },
u128 { 0xaff72d52192b6a0dULL, 0xa321f2d7226895c7ULL },
u128 { 0x9bf4f8a69f764490ULL, 0xcbea6f8ceb02bb39ULL },
u128 { 0x2f236d04753d5b4ULL, 0xfee50b7025c36a08ULL },
u128 { 0x1d762422c946590ULL, 0x9f4f2726179a2245ULL },
u128 { 0x424d3ad2b7b97ef5ULL, 0xc722f0ef9d80aad6ULL },
u128 { 0xd2e0898765a7deb2ULL, 0xf8ebad2b84e0d58bULL },
u128 { 0x63cc55f49f88eb2fULL, 0x9b934c3b330c8577ULL },
u128 { 0x3cbf6b71c76b25fbULL, 0xc2781f49ffcfa6d5ULL },
u128 { 0x8bef464e3945ef7aULL, 0xf316271c7fc3908aULL },
u128 { 0x97758bf0e3cbb5acULL, 0x97edd871cfda3a56ULL },
u128 { 0x3d52eeed1cbea317ULL, 0xbde94e8e43d0c8ecULL },
u128 { 0x4ca7aaa863ee4bddULL, 0xed63a231d4c4fb27ULL },
u128 { 0x8fe8caa93e74ef6aULL, 0x945e455f24fb1cf8ULL },
u128 { 0xb3e2fd538e122b44ULL, 0xb975d6b6ee39e436ULL },
u128 { 0x60dbbca87196b616ULL, 0xe7d34c64a9c85d44ULL },
u128 { 0xbc8955e946fe31cdULL, 0x90e40fbeea1d3a4aULL },
u128 { 0x6babab6398bdbe41ULL, 0xb51d13aea4a488ddULL },
u128 { 0xc696963c7eed2dd1ULL, 0xe264589a4dcdab14ULL },
u128 { 0xfc1e1de5cf543ca2ULL, 0x8d7eb76070a08aecULL },
u128 { 0x3b25a55f43294bcbULL, 0xb0de65388cc8ada8ULL },
u128 { 0x49ef0eb713f39ebeULL, 0xdd15fe86affad912ULL },
u128 { 0x6e3569326c784337ULL, 0x8a2dbf142dfcc7abULL },
u128 { 0x49c2c37f07965404ULL, 0xacb92ed9397bf996ULL },
u128 { 0xdc33745ec97be906ULL, 0xd7e77a8f87daf7fbULL },
u128 { 0x69a028bb3ded71a3ULL, 0x86f0ac99b4e8dafdULL },
u128 { 0xc40832ea0d68ce0cULL, 0xa8acd7c0222311bcULL },
u128 { 0xf50a3fa490c30190ULL, 0xd2d80db02aabd62bULL },
u128 { 0x792667c6da79e0faULL, 0x83c7088e1aab65dbULL },
u128 { 0x577001b891185938ULL, 0xa4b8cab1a1563f52ULL },
u128 { 0xed4c0226b55e6f86ULL, 0xcde6fd5e09abcf26ULL },
u128 { 0x544f8158315b05b4ULL, 0x80b05e5ac60b6178ULL },
u128 { 0x696361ae3db1c721ULL, 0xa0dc75f1778e39d6ULL },
u128 { 0x3bc3a19cd1e38e9ULL, 0xc913936dd571c84cULL },
u128 { 0x4ab48a04065c723ULL, 0xfb5878494ace3a5fULL },
u128 { 0x62eb0d64283f9c76ULL, 0x9d174b2dcec0e47bULL },
u128 { 0x3ba5d0bd324f8394ULL, 0xc45d1df942711d9aULL },
u128 { 0xca8f44ec7ee36479ULL, 0xf5746577930d6500ULL },
u128 { 0x7e998b13cf4e1ecbULL, 0x9968bf6abbe85f20ULL },
u128 { 0x9e3fedd8c321a67eULL, 0xbfc2ef456ae276e8ULL },
u128 { 0xc5cfe94ef3ea101eULL, 0xefb3ab16c59b14a2ULL },
u128 { 0xbba1f1d158724a12ULL, 0x95d04aee3b80ece5ULL },
u128 { 0x2a8a6e45ae8edc97ULL, 0xbb445da9ca61281fULL },
u128 { 0xf52d09d71a3293bdULL, 0xea1575143cf97226ULL },
u128 { 0x593c2626705f9c56ULL, 0x924d692ca61be758ULL },
u128 { 0x6f8b2fb00c77836cULL, 0xb6e0c377cfa2e12eULL },
u128 { 0xb6dfb9c0f956447ULL, 0xe498f455c38b997aULL },
u128 { 0x4724bd4189bd5eacULL, 0x8edf98b59a373fecULL },
u128 { 0x58edec91ec2cb657ULL, 0xb2977ee300c50fe7ULL },
u128 { 0x2f2967b66737e3edULL, 0xdf3d5e9bc0f653e1ULL },
u128 { 0xbd79e0d20082ee74ULL, 0x8b865b215899f46cULL },
u128 { 0xecd8590680a3aa11ULL, 0xae67f1e9aec07187ULL },
u128 { 0xe80e6f4820cc9495ULL, 0xda01ee641a708de9ULL },
u128 { 0x3109058d147fdcddULL, 0x884134fe908658b2ULL },
u128 { 0xbd4b46f0599fd415ULL, 0xaa51823e34a7eedeULL },
u128 { 0x6c9e18ac7007c91aULL, 0xd4e5e2cdc1d1ea96ULL },
u128 { 0x3e2cf6bc604ddb0ULL, 0x850fadc09923329eULL },
u128 { 0x84db8346b786151cULL, 0xa6539930bf6bff45ULL },
u128 { 0xe612641865679a63ULL, 0xcfe87f7cef46ff16ULL },
u128 { 0x4fcb7e8f3f60c07eULL, 0x81f14fae158c5f6eULL },
u128 { 0xe3be5e330f38f09dULL, 0xa26da3999aef7749ULL },
u128 { 0x5cadf5bfd3072cc5ULL, 0xcb090c8001ab551cULL },
u128 { 0x73d9732fc7c8f7f6ULL, 0xfdcb4fa002162a63ULL },
u128 { 0x2867e7fddcdd9afaULL, 0x9e9f11c4014dda7eULL },
u128 { 0xb281e1fd541501b8ULL, 0xc646d63501a1511dULL },
u128 { 0x1f225a7ca91a4226ULL, 0xf7d88bc24209a565ULL },
u128 { 0x3375788de9b06958ULL, 0x9ae757596946075fULL },
u128 { 0x52d6b1641c83aeULL, 0xc1a12d2fc3978937ULL },
u128 { 0xc0678c5dbd23a49aULL, 0xf209787bb47d6b84ULL },
u128 { 0xf840b7ba963646e0ULL, 0x9745eb4d50ce6332ULL },
u128 { 0xb650e5a93bc3d898ULL, 0xbd176620a501fbffULL },
u128 { 0xa3e51f138ab4cebeULL, 0xec5d3fa8ce427affULL },
u128 { 0xc66f336c36b10137ULL, 0x93ba47c980e98cdfULL },
u128 { 0xb80b0047445d4184ULL, 0xb8a8d9bbe123f017ULL },
u128 { 0xa60dc059157491e5ULL, 0xe6d3102ad96cec1dULL },
u128 { 0x87c89837ad68db2fULL, 0x9043ea1ac7e41392ULL },
u128 { 0x29babe4598c311fbULL, 0xb454e4a179dd1877ULL },
u128 { 0xf4296dd6fef3d67aULL, 0xe16a1dc9d8545e94ULL },
u128 { 0x1899e4a65f58660cULL, 0x8ce2529e2734bb1dULL },
u128 { 0x5ec05dcff72e7f8fULL, 0xb01ae745b101e9e4ULL },
u128 { 0x76707543f4fa1f73ULL, 0xdc21a1171d42645dULL },
u128 { 0x6a06494a791c53a8ULL, 0x899504ae72497ebaULL },
u128 { 0x487db9d17636892ULL, 0xabfa45da0edbde69ULL },
u128 { 0x45a9d2845d3c42b6ULL, 0xd6f8d7509292d603ULL },
u128 { 0xb8a2392ba45a9b2ULL, 0x865b86925b9bc5c2ULL },
u128 { 0x8e6cac7768d7141eULL, 0xa7f26836f282b732ULL },
u128 { 0x3207d795430cd926ULL, 0xd1ef0244af2364ffULL },
u128 { 0x7f44e6bd49e807b8ULL, 0x8335616aed761f1fULL },
u128 { 0x5f16206c9c6209a6ULL, 0xa402b9c5a8d3a6e7ULL },
u128 { 0x36dba887c37a8c0fULL, 0xcd036837130890a1ULL },
u128 { 0xc2494954da2c9789ULL, 0x802221226be55a64ULL },
u128 { 0xf2db9baa10b7bd6cULL, 0xa02aa96b06deb0fdULL },
u128 { 0x6f92829494e5acc7ULL, 0xc83553c5c8965d3dULL },
u128 { 0xcb772339ba1f17f9ULL, 0xfa42a8b73abbf48cULL },
u128 { 0xff2a760414536efbULL, 0x9c69a97284b578d7ULL },
u128 { 0xfef5138519684abaULL, 0xc38413cf25e2d70dULL },
u128 { 0x7eb258665fc25d69ULL, 0xf46518c2ef5b8cd1ULL },
u128 { 0xef2f773ffbd97a61ULL, 0x98bf2f79d5993802ULL },
u128 { 0xaafb550ffacfd8faULL, 0xbeeefb584aff8603ULL },
u128 { 0x95ba2a53f983cf38ULL, 0xeeaaba2e5dbf6784ULL },
u128 { 0xdd945a747bf26183ULL, 0x952ab45cfa97a0b2ULL },
u128 { 0x94f971119aeef9e4ULL, 0xba756174393d88dfULL },
u128 { 0x7a37cd5601aab85dULL, 0xe912b9d1478ceb17ULL },
u128 { 0xac62e055c10ab33aULL, 0x91abb422ccb812eeULL },
u128 { 0x577b986b314d6009ULL, 0xb616a12b7fe617aaULL },
u128 { 0xed5a7e85fda0b80bULL, 0xe39c49765fdf9d94ULL },
u128 { 0x14588f13be847307ULL, 0x8e41ade9fbebc27dULL },
u128 { 0x596eb2d8ae258fc8ULL, 0xb1d219647ae6b31cULL },
u128 { 0x6fca5f8ed9aef3bbULL, 0xde469fbd99a05fe3ULL },
u128 { 0x25de7bb9480d5854ULL, 0x8aec23d680043beeULL },
u128 { 0xaf561aa79a10ae6aULL, 0xada72ccc20054ae9ULL },
u128 { 0x1b2ba1518094da04ULL, 0xd910f7ff28069da4ULL },
u128 { 0x90fb44d2f05d0842ULL, 0x87aa9aff79042286ULL },
u128 { 0x353a1607ac744a53ULL, 0xa99541bf57452b28ULL },
u128 { 0x42889b8997915ce8ULL, 0xd3fa922f2d1675f2ULL },
u128 { 0x69956135febada11ULL, 0x847c9b5d7c2e09b7ULL },
u128 { 0x43fab9837e699095ULL, 0xa59bc234db398c25ULL },
u128 { 0x94f967e45e03f4bbULL, 0xcf02b2c21207ef2eULL },
u128 { 0x1d1be0eebac278f5ULL, 0x8161afb94b44f57dULL },
u128 { 0x6462d92a69731732ULL, 0xa1ba1ba79e1632dcULL },
u128 { 0x7d7b8f7503cfdcfeULL, 0xca28a291859bbf93ULL },
u128 { 0x5cda735244c3d43eULL, 0xfcb2cb35e702af78ULL },
u128 { 0x3a0888136afa64a7ULL, 0x9defbf01b061adabULL },
u128 { 0x88aaa1845b8fdd0ULL, 0xc56baec21c7a1916ULL },
u128 { 0x8aad549e57273d45ULL, 0xf6c69a72a3989f5bULL },
u128 { 0x36ac54e2f678864bULL, 0x9a3c2087a63f6399ULL },
u128 { 0x84576a1bb416a7ddULL, 0xc0cb28a98fcf3c7fULL },
u128 { 0x656d44a2a11c51d5ULL, 0xf0fdf2d3f3c30b9fULL },
u128 { 0x9f644ae5a4b1b325ULL, 0x969eb7c47859e743ULL },
u128 { 0x873d5d9f0dde1feeULL, 0xbc4665b596706114ULL },
u128 { 0xa90cb506d155a7eaULL, 0xeb57ff22fc0c7959ULL },
u128 { 0x9a7f12442d588f2ULL, 0x9316ff75dd87cbd8ULL },
u128 { 0xc11ed6d538aeb2fULL, 0xb7dcbf5354e9beceULL },
u128 { 0x8f1668c8a86da5faULL, 0xe5d3ef282a242e81ULL },
u128 { 0xf96e017d694487bcULL, 0x8fa475791a569d10ULL },
u128 { 0x37c981dcc395a9acULL, 0xb38d92d760ec4455ULL },
u128 { 0x85bbe253f47b1417ULL, 0xe070f78d3927556aULL },
u128 { 0x93956d7478ccec8eULL, 0x8c469ab843b89562ULL },
u128 { 0x387ac8d1970027b2ULL, 0xaf58416654a6babbULL },
u128 { 0x6997b05fcc0319eULL, 0xdb2e51bfe9d0696aULL },
u128 { 0x441fece3bdf81f03ULL, 0x88fcf317f22241e2ULL },
u128 { 0xd527e81cad7626c3ULL, 0xab3c2fddeeaad25aULL },
u128 { 0x8a71e223d8d3b074ULL, 0xd60b3bd56a5586f1ULL },
u128 { 0xf6872d5667844e49ULL, 0x85c7056562757456ULL },
u128 { 0xb428f8ac016561dbULL, 0xa738c6bebb12d16cULL },
u128 { 0xe13336d701beba52ULL, 0xd106f86e69d785c7ULL },
u128 { 0xecc0024661173473ULL, 0x82a45b450226b39cULL },
u128 { 0x27f002d7f95d0190ULL, 0xa34d721642b06084ULL },
u128 { 0x31ec038df7b441f4ULL, 0xcc20ce9bd35c78a5ULL },
u128 { 0x7e67047175a15271ULL, 0xff290242c83396ceULL },
u128 { 0xf0062c6e984d386ULL, 0x9f79a169bd203e41ULL },
u128 { 0x52c07b78a3e60868ULL, 0xc75809c42c684dd1ULL },
u128 { 0xa7709a56ccdf8a82ULL, 0xf92e0c3537826145ULL },
u128 { 0x88a66076400bb691ULL, 0x9bbcc7a142b17ccbULL },
u128 { 0x6acff893d00ea435ULL, 0xc2abf989935ddbfeULL },
u128 { 0x583f6b8c4124d43ULL, 0xf356f7ebf83552feULL },
u128 { 0xc3727a337a8b704aULL, 0x98165af37b2153deULL },
u128 { 0x744f18c0592e4c5cULL, 0xbe1bf1b059e9a8d6ULL },
u128 { 0x1162def06f79df73ULL, 0xeda2ee1c7064130cULL },
u128 { 0x8addcb5645ac2ba8ULL, 0x9485d4d1c63e8be7ULL },
u128 { 0x6d953e2bd7173692ULL, 0xb9a74a0637ce2ee1ULL },
u128 { 0xc8fa8db6ccdd0437ULL, 0xe8111c87c5c1ba99ULL },
u128 { 0x1d9c9892400a22a2ULL, 0x910ab1d4db9914a0ULL },
u128 { 0x2503beb6d00cab4bULL, 0xb54d5e4a127f59c8ULL },
u128 { 0x2e44ae64840fd61dULL, 0xe2a0b5dc971f303aULL },
u128 { 0x5ceaecfed289e5d2ULL, 0x8da471a9de737e24ULL },
u128 { 0x7425a83e872c5f47ULL, 0xb10d8e1456105dadULL },
u128 { 0xd12f124e28f77719ULL, 0xdd50f1996b947518ULL },
u128 { 0x82bd6b70d99aaa6fULL, 0x8a5296ffe33cc92fULL },
u128 { 0x636cc64d1001550bULL, 0xace73cbfdc0bfb7bULL },
u128 { 0x3c47f7e05401aa4eULL, 0xd8210befd30efa5aULL },
u128 { 0x65acfaec34810a71ULL, 0x8714a775e3e95c78ULL },
u128 { 0x7f1839a741a14d0dULL, 0xa8d9d1535ce3b396ULL },
u128 { 0x1ede48111209a050ULL, 0xd31045a8341ca07cULL },
u128 { 0x934aed0aab460432ULL, 0x83ea2b892091e44dULL },
u128 { 0xf81da84d5617853fULL, 0xa4e4b66b68b65d60ULL },
u128 { 0x36251260ab9d668eULL, 0xce1de40642e3f4b9ULL },
u128 { 0xc1d72b7c6b426019ULL, 0x80d2ae83e9ce78f3ULL },
u128 { 0xb24cf65b8612f81fULL, 0xa1075a24e4421730ULL },
u128 { 0xdee033f26797b627ULL, 0xc94930ae1d529cfcULL },
u128 { 0x169840ef017da3b1ULL, 0xfb9b7cd9a4a7443cULL },
u128 { 0x8e1f289560ee864eULL, 0x9d412e0806e88aa5ULL },
u128 { 0xf1a6f2bab92a27e2ULL, 0xc491798a08a2ad4eULL },
u128 { 0xae10af696774b1dbULL, 0xf5b5d7ec8acb58a2ULL },
u128 { 0xacca6da1e0a8ef29ULL, 0x9991a6f3d6bf1765ULL },
u128 { 0x17fd090a58d32af3ULL, 0xbff610b0cc6edd3fULL },
u128 { 0xddfc4b4cef07f5b0ULL, 0xeff394dcff8a948eULL },
u128 { 0x4abdaf101564f98eULL, 0x95f83d0a1fb69cd9ULL },
u128 { 0x9d6d1ad41abe37f1ULL, 0xbb764c4ca7a4440fULL },
u128 { 0x84c86189216dc5edULL, 0xea53df5fd18d5513ULL },
u128 { 0x32fd3cf5b4e49bb4ULL, 0x92746b9be2f8552cULL },
u128 { 0x3fbc8c33221dc2a1ULL, 0xb7118682dbb66a77ULL },
u128 { 0xfabaf3feaa5334aULL, 0xe4d5e82392a40515ULL },
u128 { 0x29cb4d87f2a7400eULL, 0x8f05b1163ba6832dULL },
u128 { 0x743e20e9ef511012ULL, 0xb2c71d5bca9023f8ULL },
u128 { 0x914da9246b255416ULL, 0xdf78e4b2bd342cf6ULL },
u128 { 0x1ad089b6c2f7548eULL, 0x8bab8eefb6409c1aULL },
u128 { 0xa184ac2473b529b1ULL, 0xae9672aba3d0c320ULL },
u128 { 0xc9e5d72d90a2741eULL, 0xda3c0f568cc4f3e8ULL },
u128 { 0x7e2fa67c7a658892ULL, 0x8865899617fb1871ULL },
u128 { 0xddbb901b98feeab7ULL, 0xaa7eebfb9df9de8dULL },
u128 { 0x552a74227f3ea565ULL, 0xd51ea6fa85785631ULL },
u128 { 0xd53a88958f87275fULL, 0x8533285c936b35deULL },
u128 { 0x8a892abaf368f137ULL, 0xa67ff273b8460356ULL },
u128 { 0x2d2b7569b0432d85ULL, 0xd01fef10a657842cULL },
u128 { 0x9c3b29620e29fc73ULL, 0x8213f56a67f6b29bULL },
u128 { 0x8349f3ba91b47b8fULL, 0xa298f2c501f45f42ULL },
u128 { 0x241c70a936219a73ULL, 0xcb3f2f7642717713ULL },
u128 { 0xed238cd383aa0110ULL, 0xfe0efb53d30dd4d7ULL },
u128 { 0xf4363804324a40aaULL, 0x9ec95d1463e8a506ULL },
u128 { 0xb143c6053edcd0d5ULL, 0xc67bb4597ce2ce48ULL },
u128 { 0xdd94b7868e94050aULL, 0xf81aa16fdc1b81daULL },
u128 { 0xca7cf2b4191c8326ULL, 0x9b10a4e5e9913128ULL },
u128 { 0xfd1c2f611f63a3f0ULL, 0xc1d4ce1f63f57d72ULL },
u128 { 0xbc633b39673c8cecULL, 0xf24a01a73cf2dccfULL },
u128 { 0xd5be0503e085d813ULL, 0x976e41088617ca01ULL },
u128 { 0x4b2d8644d8a74e18ULL, 0xbd49d14aa79dbc82ULL },
u128 { 0xddf8e7d60ed1219eULL, 0xec9c459d51852ba2ULL },
u128 { 0xcabb90e5c942b503ULL, 0x93e1ab8252f33b45ULL },
u128 { 0x3d6a751f3b936243ULL, 0xb8da1662e7b00a17ULL },
u128 { 0xcc512670a783ad4ULL, 0xe7109bfba19c0c9dULL },
u128 { 0x27fb2b80668b24c5ULL, 0x906a617d450187e2ULL },
u128 { 0xb1f9f660802dedf6ULL, 0xb484f9dc9641e9daULL },
u128 { 0x5e7873f8a0396973ULL, 0xe1a63853bbd26451ULL },
u128 { 0xdb0b487b6423e1e8ULL, 0x8d07e33455637eb2ULL },
u128 { 0x91ce1a9a3d2cda62ULL, 0xb049dc016abc5e5fULL },
u128 { 0x7641a140cc7810fbULL, 0xdc5c5301c56b75f7ULL },
u128 { 0xa9e904c87fcb0a9dULL, 0x89b9b3e11b6329baULL },
u128 { 0x546345fa9fbdcd44ULL, 0xac2820d9623bf429ULL },
u128 { 0xa97c177947ad4095ULL, 0xd732290fbacaf133ULL },
u128 { 0x49ed8eabcccc485dULL, 0x867f59a9d4bed6c0ULL },
u128 { 0x5c68f256bfff5a74ULL, 0xa81f301449ee8c70ULL },
u128 { 0x73832eec6fff3111ULL, 0xd226fc195c6a2f8cULL },
u128 { 0xc831fd53c5ff7eabULL, 0x83585d8fd9c25db7ULL },
u128 { 0xba3e7ca8b77f5e55ULL, 0xa42e74f3d032f525ULL },
u128 { 0x28ce1bd2e55f35ebULL, 0xcd3a1230c43fb26fULL },
u128 { 0x7980d163cf5b81b3ULL, 0x80444b5e7aa7cf85ULL },
u128 { 0xd7e105bcc332621fULL, 0xa0555e361951c366ULL },
u128 { 0x8dd9472bf3fefaa7ULL, 0xc86ab5c39fa63440ULL },
u128 { 0xb14f98f6f0feb951ULL, 0xfa856334878fc150ULL },
u128 { 0x6ed1bf9a569f33d3ULL, 0x9c935e00d4b9d8d2ULL },
u128 { 0xa862f80ec4700c8ULL, 0xc3b8358109e84f07ULL },
u128 { 0xcd27bb612758c0faULL, 0xf4a642e14c6262c8ULL },
u128 { 0x8038d51cb897789cULL, 0x98e7e9cccfbd7dbdULL },
u128 { 0xe0470a63e6bd56c3ULL, 0xbf21e44003acdd2cULL },
u128 { 0x1858ccfce06cac74ULL, 0xeeea5d5004981478ULL },
u128 { 0xf37801e0c43ebc8ULL, 0x95527a5202df0ccbULL },
u128 { 0xd30560258f54e6baULL, 0xbaa718e68396cffdULL },
u128 { 0x47c6b82ef32a2069ULL, 0xe950df20247c83fdULL },
u128 { 0x4cdc331d57fa5441ULL, 0x91d28b7416cdd27eULL },
u128 { 0xe0133fe4adf8e952ULL, 0xb6472e511c81471dULL },
u128 { 0x58180fddd97723a6ULL, 0xe3d8f9e563a198e5ULL },
u128 { 0x570f09eaa7ea7648ULL, 0x8e679c2f5e44ff8fULL },
};
return values;
}
static constexpr auto pre_computed_powers_of_five = pre_compute_table();
static constexpr u128 power_of_five(i64 exponent)
{
return pre_computed_powers_of_five[exponent - lowest_exponent];
}
struct FloatingPointBuilder {
u64 mantissa = 0;
// This exponent is power of 2 and with the bias already added.
i32 exponent = 0;
static constexpr i32 invalid_exponent_offset = 32768;
static FloatingPointBuilder zero()
{
return { 0, 0 };
}
template<typename T>
static FloatingPointBuilder infinity()
{
return { 0, FloatingPointInfo<T>::infinity_exponent() };
}
template<typename T>
static FloatingPointBuilder nan()
{
return { 1ull << (FloatingPointInfo<T>::mantissa_bits() - 1), FloatingPointInfo<T>::infinity_exponent() };
}
template<typename T>
static FloatingPointBuilder from_value(T value)
{
using BitDetails = FloatingPointInfo<T>;
auto bits = bit_cast<typename BitDetails::SameSizeUnsigned>(value);
// we ignore negative
FloatingPointBuilder result;
i32 bias = BitDetails::mantissa_bits() + BitDetails::exponent_bias();
if ((bits & BitDetails::exponent_mask()) == 0) {
// 0 exponent -> denormal (or zero)
result.exponent = 1 - bias;
// Denormal so _DON'T_ add the implicit 1
result.mantissa = bits & BitDetails::mantissa_mask();
} else {
result.exponent = (bits & BitDetails::exponent_mask()) >> BitDetails::mantissa_bits();
result.exponent -= bias;
result.mantissa = (bits & BitDetails::mantissa_mask()) | (1ull << BitDetails::mantissa_bits());
}
return result;
}
template<typename T>
T to_value(bool is_negative) const
{
if constexpr (IsSame<double, T>) {
VERIFY((mantissa & 0xffe0'0000'0000'0000) == 0);
VERIFY((mantissa & 0xfff0'0000'0000'0000) == 0 || exponent == 1);
VERIFY((exponent & ~(0x7ff)) == 0);
} else {
static_assert(IsSame<float, T>);
VERIFY((mantissa & 0xff00'0000) == 0);
VERIFY((mantissa & 0xff80'0000) == 0 || exponent == 1);
VERIFY((exponent & ~(0xff)) == 0);
}
using BitSizedUnsigened = BitSizedUnsignedForFloatingPoint<T>;
BitSizedUnsigened raw_bits = mantissa;
raw_bits |= BitSizedUnsigened(exponent) << FloatingPointInfo<T>::mantissa_bits();
raw_bits |= BitSizedUnsigened(is_negative) << FloatingPointInfo<T>::sign_bit_index();
return bit_cast<T>(raw_bits);
}
};
template<typename T>
static FloatingPointBuilder parse_arbitrarily_long_floating_point(BasicParseResult& result, FloatingPointBuilder initial);
static i32 decimal_exponent_to_binary_exponent(i32 exponent)
{
return ((((152170 + 65536) * exponent) >> 16) + 63);
}
static u128 multiply(u64 a, u64 b)
{
AK+LibCrypto: Delete 64x64 wide multiplication workarounds Now UFixedBigInt exposes API to do wide multiplications of this kind efficiently.
2023-02-04 16:03:52 +00:00
return UFixedBigInt<64>(a).wide_multiply(b);
AK: Add an exact and fast floating point parsing algorithm This is based on the paper by Daniel Lemire called "Number parsing at a Gigabyte per second", currently available at https://arxiv.org/abs/2101.11408 An implementation can be found at https://github.com/fastfloat/fast_float To support both strtod like methods and String::to_double we have two different APIs. The parse_first_floating_point gives back both the result, next character to read and the error/out of range status. Out of range here means we rounded to infinity 0. The other API, parse_floating_point_completely, will return a floating point only if the given character range contains just the floating point and nothing else. This can be much faster as we can skip actually computing the value if we notice we did not parse the whole range. Both of these APIs support a very lenient format to be usable in as many places as possible. Also it does not check for "named" values like "nan", "inf", "NAN" etc. Because this can be different for every usage. For integers and small values this new method is not faster and often even a tiny bit slower than the current strtod implementation. However the strtod implementation is wrong for a lot of values and has a much less predictable running time. For correctness this method was tested against known string -> double datasets from https://github.com/nigeltao/parse-number-fxx-test-data This method gives 100% accuracy. The old strtod gave an incorrect value in over 50% of the numbers tested.
2022-10-13 00:18:56 +00:00
}
template<unsigned Precision>
u128 multiplication_approximation(u64 value, i32 exponent)
{
auto z = power_of_five(exponent);
static_assert(Precision < 64);
constexpr u64 mask = NumericLimits<u64>::max() >> Precision;
auto lower_result = multiply(z.high(), value);
if ((lower_result.high() & mask) == mask) {
auto upper_result = multiply(z.low(), value);
AK+LibCrypto: Delete 64x64 wide multiplication workarounds Now UFixedBigInt exposes API to do wide multiplications of this kind efficiently.
2023-02-04 16:03:52 +00:00
lower_result += upper_result.high();
AK: Add an exact and fast floating point parsing algorithm This is based on the paper by Daniel Lemire called "Number parsing at a Gigabyte per second", currently available at https://arxiv.org/abs/2101.11408 An implementation can be found at https://github.com/fastfloat/fast_float To support both strtod like methods and String::to_double we have two different APIs. The parse_first_floating_point gives back both the result, next character to read and the error/out of range status. Out of range here means we rounded to infinity 0. The other API, parse_floating_point_completely, will return a floating point only if the given character range contains just the floating point and nothing else. This can be much faster as we can skip actually computing the value if we notice we did not parse the whole range. Both of these APIs support a very lenient format to be usable in as many places as possible. Also it does not check for "named" values like "nan", "inf", "NAN" etc. Because this can be different for every usage. For integers and small values this new method is not faster and often even a tiny bit slower than the current strtod implementation. However the strtod implementation is wrong for a lot of values and has a much less predictable running time. For correctness this method was tested against known string -> double datasets from https://github.com/nigeltao/parse-number-fxx-test-data This method gives 100% accuracy. The old strtod gave an incorrect value in over 50% of the numbers tested.
2022-10-13 00:18:56 +00:00
}
return lower_result;
}
template<typename T>
static FloatingPointBuilder not_enough_precision_binary_to_decimal(i64 exponent, u64 mantissa, int leading_zeros)
{
using FloatingPointRepr = FloatingPointInfo<T>;
i32 did_not_have_upper_bit = static_cast<i32>(mantissa >> 63) ^ 1;
FloatingPointBuilder answer;
answer.mantissa = mantissa << did_not_have_upper_bit;
i32 bias = FloatingPointRepr::mantissa_bits() + FloatingPointRepr::exponent_bias();
answer.exponent = decimal_exponent_to_binary_exponent(static_cast<i32>(exponent)) - leading_zeros - did_not_have_upper_bit - 62 + bias;
// Make it negative to show we need more precision.
answer.exponent -= FloatingPointBuilder::invalid_exponent_offset;
VERIFY(answer.exponent < 0);
return answer;
}
template<typename T>
static FloatingPointBuilder fallback_binary_to_decimal(u64 mantissa, i64 exponent)
{
// We should have caught huge exponents already
VERIFY(exponent >= -400 && exponent <= 400);
// Perform the initial steps of binary_to_decimal.
auto w = mantissa;
auto leading_zeros = count_leading_zeroes(mantissa);
w <<= leading_zeros;
auto product = multiplication_approximation<FloatingPointInfo<T>::mantissa_bits() + 3>(w, exponent);
return not_enough_precision_binary_to_decimal<T>(exponent, product.high(), leading_zeros);
}
template<typename T>
static FloatingPointBuilder binary_to_decimal(u64 mantissa, i64 exponent)
{
using FloatingPointRepr = FloatingPointInfo<T>;
if (mantissa == 0 || exponent < FloatingPointRepr::min_power_of_10())
return FloatingPointBuilder::zero();
// Max double value which isn't negative is xe308
if (exponent > FloatingPointRepr::max_power_of_10())
return FloatingPointBuilder::infinity<T>();
auto w = mantissa;
// Normalize the decimal significand w by shifting it so that w ∈ [2^63, 2^64)
auto leading_zeros = count_leading_zeroes(mantissa);
w <<= leading_zeros;
// We need at least mantissa bits + 1 for the implicit bit + 1 for the implicit 0 top bit and one extra for rounding
u128 approximation_of_product_with_power_of_five = multiplication_approximation<FloatingPointRepr::mantissa_bits() + 3>(w, exponent);
// The paper (and code of fastfloat/fast_float as of writing) mention that the low part
// of approximation_of_product_with_power_of_five can be 2^64 - 1 here in which case we need more
// precision if the exponent lies outside of [-27, 55]. However the authors of the paper have
// shown that this case cannot actually occur. See https://github.com/fastfloat/fast_float/issues/146#issuecomment-1262527329
u8 upperbit = approximation_of_product_with_power_of_five.high() >> 63;
auto real_mantissa = approximation_of_product_with_power_of_five.high() >> (upperbit + 64 - FloatingPointRepr::mantissa_bits() - 3);
// We immediately normalize the exponent to 0 - max else we have to add the bias in most following calculations
i32 power_of_two_with_bias = decimal_exponent_to_binary_exponent(exponent) - leading_zeros + upperbit + FloatingPointRepr::exponent_bias();
if (power_of_two_with_bias <= 0) {
// If the exponent is less than the bias we might have a denormal on our hands
// A denormal is a float with exponent zero, which means it doesn't have the implicit
// 1 at the top of the mantissa.
// If the top bit would be below the bottom of the mantissa we have to round to zero
if (power_of_two_with_bias <= -63)
return FloatingPointBuilder::zero();
// Otherwise, we have to shift the mantissa to be a denormal
auto s = -power_of_two_with_bias + 1;
real_mantissa = real_mantissa >> s;
// And then round ties to even
real_mantissa += real_mantissa & 1;
real_mantissa >>= 1;
// Check for subnormal by checking if the 53th bit of the mantissa it set in which case exponent is 1 not 0
// It is only a real subnormal if the top bit isn't set
power_of_two_with_bias = real_mantissa < (1ull << FloatingPointRepr::mantissa_bits()) ? 0 : 1;
return { real_mantissa, power_of_two_with_bias };
}
if (approximation_of_product_with_power_of_five.low() <= 1 && (real_mantissa & 0b11) == 0b01
&& exponent >= FloatingPointRepr::min_exponent_round_to_even()
&& exponent <= FloatingPointRepr::max_exponent_round_to_even()) {
// If the lowest bit is set but the one above it isn't this is a value exactly halfway
// between two floating points
// if (z ÷ 264 )/m is a power of two then m ← m − 1
// effectively all discard bits from z.high are 0
if (approximation_of_product_with_power_of_five.high() == (real_mantissa << (upperbit + 64 - FloatingPointRepr::mantissa_bits() - 3))) {
real_mantissa &= ~u64(1);
}
}
real_mantissa += real_mantissa & 1;
real_mantissa >>= 1;
// If we overflowed the mantissa round up the exponent
if (real_mantissa >= (2ull << FloatingPointRepr::mantissa_bits())) {
real_mantissa = 1ull << FloatingPointRepr::mantissa_bits();
++power_of_two_with_bias;
}
real_mantissa &= ~(1ull << FloatingPointRepr::mantissa_bits());
// We might have rounded exponent up to infinity
if (power_of_two_with_bias >= FloatingPointRepr::infinity_exponent())
return FloatingPointBuilder::infinity<T>();
return {
real_mantissa, power_of_two_with_bias
};
}
static constexpr u64 multiply_with_carry(u64 x, u64 y, u64& carry)
{
u128 result = (u128 { x } * y) + carry;
carry = result.high();
return result.low();
}
static constexpr u64 add_with_overflow(u64 x, u64 y, bool& did_overflow)
{
u64 value;
did_overflow = __builtin_add_overflow(x, y, &value);
return value;
}
class MinimalBigInt {
public:
MinimalBigInt() = default;
MinimalBigInt(u64 value)
{
append(value);
}
static MinimalBigInt from_decimal_floating_point(BasicParseResult const& parse_result, size_t& digits_parsed, size_t max_total_digits)
{
size_t current_word_counter = 0;
// 10**19 is the biggest power of ten which fits in 64 bit
constexpr size_t max_word_counter = max_representable_power_of_ten_in_u64;
u64 current_word = 0;
enum AddDigitResult {
DidNotHitMaxDigits,
HitMaxDigits,
};
auto does_truncate_non_zero = [](char const* parse_head, char const* parse_end) {
while (parse_end - parse_head >= 8) {
static_assert('0' == 0x30);
if (read_eight_digits(parse_head) != 0x3030303030303030)
return true;
parse_head += 8;
}
while (parse_head != parse_end) {
if (*parse_head != '0')
return true;
++parse_head;
}
return false;
};
MinimalBigInt value;
auto add_digits = [&](StringView digits, bool check_fraction_for_truncation = false) {
char const* parse_head = digits.characters_without_null_termination();
char const* parse_end = digits.characters_without_null_termination() + digits.length();
if (digits_parsed == 0) {
// Skip all leading zeros as long as we haven't hit a non zero
while (parse_head != parse_end && *parse_head == '0')
++parse_head;
}
while (parse_head != parse_end) {
while (max_word_counter - current_word_counter >= 8
&& parse_end - parse_head >= 8
&& max_total_digits - digits_parsed >= 8) {
current_word = current_word * 100'000'000 + eight_digits_to_value(read_eight_digits(parse_head));
digits_parsed += 8;
current_word_counter += 8;
parse_head += 8;
}
while (current_word_counter < max_word_counter
&& parse_head != parse_end
&& digits_parsed < max_total_digits) {
current_word = current_word * 10 + (*parse_head - '0');
++digits_parsed;
++current_word_counter;
++parse_head;
}
if (digits_parsed == max_total_digits) {
// Check if we are leaving behind any non zero
bool truncated = does_truncate_non_zero(parse_head, parse_end);
if (auto fraction = parse_result.fractional_part; check_fraction_for_truncation && !fraction.is_empty())
truncated = truncated || does_truncate_non_zero(fraction.characters_without_null_termination(), fraction.characters_without_null_termination() + fraction.length());
// If we truncated we just pretend there is another 1 after the already parsed digits
if (truncated && current_word_counter != max_word_counter) {
// If it still fits in the current add it there, this saves a wide multiply
current_word = current_word * 10 + 1;
++current_word_counter;
truncated = false;
}
value.add_digits(current_word, current_word_counter);
// If it didn't fit just do * 10 + 1
if (truncated)
value.add_digits(1, 1);
return HitMaxDigits;
} else {
value.add_digits(current_word, current_word_counter);
current_word = 0;
current_word_counter = 0;
}
}
return DidNotHitMaxDigits;
};
if (add_digits(parse_result.whole_part, true) == HitMaxDigits)
return value;
add_digits(parse_result.fractional_part);
return value;
}
u64 top_64_bits(bool& has_truncated_bits) const
{
if (m_used_length == 0)
return 0;
// Top word should be non-zero
VERIFY(m_words[m_used_length - 1] != 0);
auto leading_zeros = count_leading_zeroes(m_words[m_used_length - 1]);
if (m_used_length == 1)
return m_words[0] << leading_zeros;
for (size_t i = 0; i < m_used_length - 2; i++) {
if (m_words[i] != 0) {
has_truncated_bits = true;
break;
}
}
if (leading_zeros == 0) {
// Shift of 64+ is undefined so this has to be a separate case
has_truncated_bits |= m_words[m_used_length - 2] != 0;
return m_words[m_used_length - 1] << leading_zeros;
}
auto bits_from_second = 64 - leading_zeros;
has_truncated_bits |= (m_words[m_used_length - 2] << leading_zeros) != 0;
return (m_words[m_used_length - 1] << leading_zeros) | (m_words[m_used_length - 2] >> bits_from_second);
}
i32 size_in_bits() const
{
if (m_used_length == 0)
return 0;
// This is guaranteed to be at most max_size_in_words * 64 so not above i32 max
return static_cast<i32>(64 * (m_used_length)-count_leading_zeroes(m_words[m_used_length - 1]));
}
void multiply_with_power_of_10(u32 exponent)
{
multiply_with_power_of_5(exponent);
multiply_with_power_of_2(exponent);
}
void multiply_with_power_of_5(u32 exponent)
{
// FIXME: We might be able to store a bigger power of 5 but this would
// require a wide multiply, so perhaps using u4096 would be
// better to get wide multiply and not duplicate logic.
static constexpr Array<u64, 28> power_of_5 = {
1ul,
5ul,
25ul,
125ul,
625ul,
3125ul,
15625ul,
78125ul,
390625ul,
1953125ul,
9765625ul,
48828125ul,
244140625ul,
1220703125ul,
6103515625ul,
30517578125ul,
152587890625ul,
762939453125ul,
3814697265625ul,
19073486328125ul,
95367431640625ul,
476837158203125ul,
2384185791015625ul,
11920928955078125ul,
59604644775390625ul,
298023223876953125ul,
1490116119384765625ul,
7450580596923828125ul,
};
static constexpr u32 max_step = power_of_5.size() - 1;
static constexpr u64 max_power = power_of_5[max_step];
while (exponent >= max_step) {
multiply_with_small(max_power);
exponent -= max_step;
}
if (exponent > 0)
multiply_with_small(power_of_5[exponent]);
}
void multiply_with_power_of_2(u32 exponent)
{
// It's cheaper to shift bits first since that creates at most 1 new word
shift_bits(exponent % 64);
shift_words(exponent / 64);
}
enum class CompareResult {
Equal,
GreaterThan,
LessThan
};
CompareResult compare_to(MinimalBigInt const& other) const
{
if (m_used_length > other.m_used_length)
return CompareResult::GreaterThan;
if (m_used_length < other.m_used_length)
return CompareResult::LessThan;
// Now we know it's the same size
for (size_t i = m_used_length; i > 0; --i) {
auto our_word = m_words[i - 1];
auto their_word = other.m_words[i - 1];
if (our_word > their_word)
return CompareResult::GreaterThan;
if (their_word > our_word)
return CompareResult::LessThan;
}
return CompareResult::Equal;
}
private:
void shift_words(u32 amount)
{
if (amount == 0)
return;
VERIFY(amount + m_used_length <= max_words_needed);
for (size_t i = m_used_length + amount - 1; i > amount - 1; --i)
m_words[i] = m_words[i - amount];
for (size_t i = 0; i < amount; ++i)
m_words[i] = 0;
m_used_length += amount;
}
void shift_bits(u32 amount)
{
if (amount == 0)
return;
VERIFY(amount < 64);
u32 inverse = 64 - amount;
u64 last_word = 0;
for (size_t i = 0; i < m_used_length; ++i) {
u64 word = m_words[i];
m_words[i] = (word << amount) | (last_word >> inverse);
last_word = word;
}
u64 carry = last_word >> inverse;
if (carry != 0)
append(carry);
}
static constexpr Array<u64, 20> powers_of_ten_uint64 = {
1UL, 10UL, 100UL, 1000UL, 10000UL, 100000UL, 1000000UL, 10000000UL, 100000000UL,
1000000000UL, 10000000000UL, 100000000000UL, 1000000000000UL, 10000000000000UL,
100000000000000UL, 1000000000000000UL, 10000000000000000UL, 100000000000000000UL,
1000000000000000000UL, 10000000000000000000UL
};
void multiply_with_small(u64 value)
{
u64 carry = 0;
for (size_t i = 0; i < m_used_length; ++i)
m_words[i] = multiply_with_carry(m_words[i], value, carry);
if (carry != 0)
append(carry);
}
void add_small(u64 value)
{
bool overflow;
size_t index = 0;
while (value != 0 && index < m_used_length) {
m_words[index] = add_with_overflow(m_words[index], value, overflow);
value = overflow ? 1 : 0;
++index;
}
if (value != 0)
append(value);
}
void add_digits(u64 value, size_t digits_for_value)
{
VERIFY(digits_for_value < powers_of_ten_uint64.size());
multiply_with_small(powers_of_ten_uint64[digits_for_value]);
add_small(value);
}
void append(u64 word)
{
VERIFY(m_used_length <= max_words_needed);
m_words[m_used_length] = word;
++m_used_length;
}
// The max valid words we might need are log2(10^(769 + 342)), max digits + max exponent
static constexpr size_t max_words_needed = 58;
size_t m_used_length = 0;
// FIXME: This is an array just to avoid allocations, but the max size is only needed for
// massive amount of digits, so a smaller vector would work for most cases.
Array<u64, max_words_needed> m_words {};
};
static bool round_nearest_tie_even(FloatingPointBuilder& value, bool did_truncate_bits, i32 shift)
{
VERIFY(shift == 11 || shift == 40);
u64 mask = (1ull << shift) - 1;
u64 halfway = 1ull << (shift - 1);
u64 truncated_bits = value.mantissa & mask;
bool is_halfway = truncated_bits == halfway;
bool is_above = truncated_bits > halfway;
value.mantissa >>= shift;
value.exponent += shift;
bool is_odd = (value.mantissa & 1) == 1;
return is_above || (is_halfway && did_truncate_bits) || (is_halfway && is_odd);
}
template<typename T, typename Callback>
static void round(FloatingPointBuilder& value, Callback&& should_round_up)
{
using FloatingRepr = FloatingPointInfo<T>;
i32 mantissa_shift = 64 - FloatingRepr::mantissa_bits() - 1;
if (-value.exponent >= mantissa_shift) {
// This is a denormal so we have to shift????
mantissa_shift = min(-value.exponent + 1, 64);
if (should_round_up(value, mantissa_shift))
++value.mantissa;
value.exponent = (value.mantissa < (1ull << FloatingRepr::mantissa_bits())) ? 0 : 1;
return;
}
if (should_round_up(value, mantissa_shift))
++value.mantissa;
// Mantissa might have been rounded so if it overflowed increase the exponent
if (value.mantissa >= (2ull << FloatingRepr::mantissa_bits())) {
value.mantissa = 0;
++value.exponent;
} else {
// Clear the implicit top bit
value.mantissa &= ~(1ull << FloatingRepr::mantissa_bits());
}
// If we also overflowed the exponent make it infinity!
if (value.exponent >= FloatingRepr::infinity_exponent()) {
value.exponent = FloatingRepr::infinity_exponent();
value.mantissa = 0;
}
}
template<typename T>
static FloatingPointBuilder build_positive_double(MinimalBigInt& mantissa, i32 exponent)
{
mantissa.multiply_with_power_of_10(exponent);
FloatingPointBuilder result {};
bool should_round_up_ties = false;
// First we get the 64 most significant bits WARNING not masked to real mantissa yet
result.mantissa = mantissa.top_64_bits(should_round_up_ties);
i32 bias = FloatingPointInfo<T>::mantissa_bits() + FloatingPointInfo<T>::exponent_bias();
result.exponent = mantissa.size_in_bits() - 64 + bias;
round<T>(result, [should_round_up_ties](FloatingPointBuilder& value, i32 shift) {
return round_nearest_tie_even(value, should_round_up_ties, shift);
});
return result;
}
template<ParseableFloatingPoint T>
static FloatingPointBuilder build_negative_exponent_double(MinimalBigInt& mantissa, i32 exponent, FloatingPointBuilder initial)
{
VERIFY(exponent < 0);
// Building a fraction from a big integer is harder to understand
// But fundamentely we have mantissa * 10^-e and so divide by 5^f
auto parts_copy = initial;
round<T>(parts_copy, [](FloatingPointBuilder& value, i32 shift) {
if (shift == 64)
value.mantissa = 0;
else
value.mantissa >>= shift;
value.exponent += shift;
return false;
});
T rounded_down_double_value = parts_copy.template to_value<T>(false);
auto exact_halfway_builder = FloatingPointBuilder::from_value(rounded_down_double_value);
// halfway is exactly just the next bit 1 (rest implicit zeros)
exact_halfway_builder.mantissa <<= 1;
exact_halfway_builder.mantissa += 1;
--exact_halfway_builder.exponent;
MinimalBigInt rounded_down_full_mantissa { exact_halfway_builder.mantissa };
// Scale halfway up with 5**(-e)
if (u32 power_of_5 = -exponent; power_of_5 != 0)
rounded_down_full_mantissa.multiply_with_power_of_5(power_of_5);
i32 power_of_2 = exact_halfway_builder.exponent - exponent;
if (power_of_2 > 0) {
// halfway has lower exponent scale up to real exponent
rounded_down_full_mantissa.multiply_with_power_of_2(power_of_2);
} else if (power_of_2 < 0) {
// halfway has higher exponent scale original mantissa up to real halfway
mantissa.multiply_with_power_of_2(-power_of_2);
}
auto compared_to_halfway = mantissa.compare_to(rounded_down_full_mantissa);
round<T>(initial, [compared_to_halfway](FloatingPointBuilder& value, i32 shift) {
if (shift == 64) {
value.mantissa = 0;
} else {
value.mantissa >>= shift;
}
value.exponent += shift;
if (compared_to_halfway == MinimalBigInt::CompareResult::GreaterThan)
return true;
if (compared_to_halfway == MinimalBigInt::CompareResult::LessThan)
return false;
return (value.mantissa & 1) == 1;
});
return initial;
}
template<typename T>
static FloatingPointBuilder parse_arbitrarily_long_floating_point(BasicParseResult& result, FloatingPointBuilder initial)
{
VERIFY(initial.exponent < 0);
initial.exponent += FloatingPointBuilder::invalid_exponent_offset;
VERIFY(result.exponent >= NumericLimits<i32>::min() && result.exponent <= NumericLimits<i32>::max());
i32 exponent = static_cast<i32>(result.exponent);
{
u64 mantissa_copy = result.mantissa;
while (mantissa_copy >= 10000) {
mantissa_copy /= 10000;
exponent += 4;
}
while (mantissa_copy >= 10) {
mantissa_copy /= 10;
++exponent;
}
}
size_t digits = 0;
constexpr auto max_digits_to_parse = FloatingPointInfo<T>::max_possible_digits_needed_for_parsing();
// Reparse mantissa into big int
auto mantissa = MinimalBigInt::from_decimal_floating_point(result, digits, max_digits_to_parse);
VERIFY(digits <= 1024);
exponent += 1 - static_cast<i32>(digits);
if (exponent >= 0)
return build_positive_double<T>(mantissa, exponent);
return build_negative_exponent_double<T>(mantissa, exponent, initial);
}
template<FloatingPoint T>
T parse_result_to_value(BasicParseResult& parse_result)
{
using FloatingPointRepr = FloatingPointInfo<T>;
if (parse_result.mantissa <= u64(2) << FloatingPointRepr::mantissa_bits()
&& parse_result.exponent >= -FloatingPointRepr::max_exact_power_of_10() && parse_result.exponent <= FloatingPointRepr::max_exact_power_of_10()
&& !parse_result.more_than_19_digits_with_overflow) {
T value = parse_result.mantissa;
VERIFY(u64(value) == parse_result.mantissa);
if (parse_result.exponent < 0)
value = value / FloatingPointRepr::power_of_ten(-parse_result.exponent);
else
value = value * FloatingPointRepr::power_of_ten(parse_result.exponent);
if (parse_result.negative)
value = -value;
return value;
}
auto floating_point_parts = binary_to_decimal<T>(parse_result.mantissa, parse_result.exponent);
if (parse_result.more_than_19_digits_with_overflow && floating_point_parts.exponent >= 0) {
auto rounded_up_double_build = binary_to_decimal<T>(parse_result.mantissa + 1, parse_result.exponent);
if (floating_point_parts.mantissa != rounded_up_double_build.mantissa || floating_point_parts.exponent != rounded_up_double_build.exponent) {
floating_point_parts = fallback_binary_to_decimal<T>(parse_result.mantissa, parse_result.exponent);
VERIFY(floating_point_parts.exponent < 0);
}
}
if (floating_point_parts.exponent < 0) {
// Invalid have to parse perfectly
floating_point_parts = parse_arbitrarily_long_floating_point<T>(parse_result, floating_point_parts);
}
return floating_point_parts.template to_value<T>(parse_result.negative);
}
template<FloatingPoint T>
constexpr FloatingPointParseResults<T> parse_result_to_full_result(BasicParseResult parse_result)
{
if (!parse_result.valid)
return { nullptr, FloatingPointError::NoOrInvalidInput, __builtin_nan("") };
FloatingPointParseResults<T> full_result {};
full_result.end_ptr = parse_result.last_parsed;
// We special case this to be able to differentiate between 0 and values rounded down to 0
if (parse_result.mantissa == 0) {
full_result.value = parse_result.negative ? -0. : 0.;
return full_result;
}
full_result.value = parse_result_to_value<T>(parse_result);
// The only way we can get infinity is from rounding up/down to it.
if (__builtin_isinf(full_result.value))
full_result.error = FloatingPointError::OutOfRange;
else if (full_result.value == T(0.))
full_result.error = FloatingPointError::RoundedDownToZero;
return full_result;
}
template<FloatingPoint T>
FloatingPointParseResults<T> parse_first_floating_point(char const* start, char const* end)
{
auto parse_result = parse_numbers(
start,
[end](char const* head) { return head == end; },
[end](char const* head) { return head - end >= 8; });
return parse_result_to_full_result<T>(parse_result);
}
template FloatingPointParseResults<double> parse_first_floating_point(char const* start, char const* end);
template FloatingPointParseResults<float> parse_first_floating_point(char const* start, char const* end);
template<FloatingPoint T>
FloatingPointParseResults<T> parse_first_floating_point_until_zero_character(char const* start)
{
auto parse_result = parse_numbers(
start,
[](char const* head) { return *head == '\0'; },
[](char const*) { return false; });
return parse_result_to_full_result<T>(parse_result);
}
template FloatingPointParseResults<double> parse_first_floating_point_until_zero_character(char const* start);
template FloatingPointParseResults<float> parse_first_floating_point_until_zero_character(char const* start);
template<FloatingPoint T>
Optional<T> parse_floating_point_completely(char const* start, char const* end)
{
auto parse_result = parse_numbers(
start,
[end](char const* head) { return head == end; },
[end](char const* head) { return head - end >= 8; });
if (!parse_result.valid || parse_result.last_parsed != end)
return {};
return parse_result_to_value<T>(parse_result);
}
template Optional<double> parse_floating_point_completely(char const* start, char const* end);
template Optional<float> parse_floating_point_completely(char const* start, char const* end);
AK: Add an exact and fast hex float parsing algorithm Similar to decimal floating point parsing the current strtod hex float parsing gives a lot of incorrect results. We can use a similar technique as with decimal parsing however hex floats are much simpler as we don't need to scale with a power of 5. For hex floats we just provide the parse_first_hexfloat API as there is currently no need for a parse_hexfloat_completely API. Again the accepted input for parse_first_hexfloat is very lenient and any validation should be done before calling this method.
2022-10-13 00:23:07 +00:00
struct HexFloatParseResult {
bool is_negative = false;
bool valid = false;
char const* last_parsed = nullptr;
u64 mantissa = 0;
i64 exponent = 0;
};
static HexFloatParseResult parse_hexfloat(char const* start)
{
HexFloatParseResult result {};
if (start == nullptr || *start == '\0')
return result;
char const* parse_head = start;
bool any_digits = false;
bool truncated_non_zero = false;
if (*parse_head == '-') {
result.is_negative = true;
++parse_head;
if (*parse_head == '\0' || (!is_ascii_hex_digit(*parse_head) && *parse_head != floating_point_decimal_separator))
return result;
} else if (*parse_head == '+') {
++parse_head;
if (*parse_head == '\0' || (!is_ascii_hex_digit(*parse_head) && *parse_head != floating_point_decimal_separator))
return result;
}
if (*parse_head == '0' && (*(parse_head + 1) != '\0') && (*(parse_head + 1) == 'x' || *(parse_head + 1) == 'X')) {
// Skip potential 0[xX], we have to do this here since the sign comes at the front
parse_head += 2;
}
auto add_mantissa_digit = [&] {
any_digits = true;
// We assume you already checked this is actually a digit
auto digit = parse_ascii_hex_digit(*parse_head);
// Because the power of sixteen is just scaling of power of two we don't
// need to keep all the remaining digits beyond the first 52 bits, just because
// it's easy we store the first 16 digits. However for rounding we do need to parse
// all the digits and keep track if we see any non zero one.
if (result.mantissa < (1ull << 60)) {
result.mantissa = (result.mantissa * 16) + digit;
return true;
}
if (digit != 0)
truncated_non_zero = true;
return false;
};
while (*parse_head != '\0' && is_ascii_hex_digit(*parse_head)) {
add_mantissa_digit();
++parse_head;
}
if (*parse_head != '\0' && *parse_head == floating_point_decimal_separator) {
++parse_head;
i64 digits_after_separator = 0;
while (*parse_head != '\0' && is_ascii_hex_digit(*parse_head)) {
// Track how many characters we actually read into the mantissa
digits_after_separator += add_mantissa_digit() ? 1 : 0;
++parse_head;
}
// We parsed x digits after the dot so need to multiply with 2^(-x * 4)
// Since every digit is 4 bits
result.exponent = -digits_after_separator * 4;
}
if (!any_digits)
return result;
if (*parse_head != '\0' && (*parse_head == 'p' || *parse_head == 'P')) {
[&] {
auto const* head_before_p = parse_head;
ArmedScopeGuard reset_ptr { [&] { parse_head = head_before_p; } };
++parse_head;
if (*parse_head == '\0')
return;
bool exponent_is_negative = false;
i64 explicit_exponent = 0;
if (*parse_head == '-' || *parse_head == '+') {
exponent_is_negative = *parse_head == '-';
++parse_head;
if (*parse_head == '\0')
return;
}
if (!is_ascii_digit(*parse_head))
return;
// We have at least one digit (with optional preceding sign) so we will not reset
reset_ptr.disarm();
while (*parse_head != '\0' && is_ascii_digit(*parse_head)) {
// If we hit exponent overflow the number is so huge we are in trouble anyway, see
// a comment in parse_numbers.
if (explicit_exponent < 0x10000000)
explicit_exponent = 10 * explicit_exponent + (*parse_head - '0');
++parse_head;
}
if (exponent_is_negative)
explicit_exponent = -explicit_exponent;
result.exponent += explicit_exponent;
}();
}
result.valid = true;
// Round up exactly halfway with truncated non zeros, but don't if it would cascade up
if (truncated_non_zero && (result.mantissa & 0xF) != 0xF) {
VERIFY(result.mantissa >= 0x1000'0000'0000'0000);
result.mantissa |= 1;
}
result.last_parsed = parse_head;
return result;
}
template<FloatingPoint T>
static FloatingPointBuilder build_hex_float(HexFloatParseResult& parse_result)
{
using FloatingPointRepr = FloatingPointInfo<T>;
VERIFY(parse_result.mantissa != 0);
if (parse_result.exponent >= FloatingPointRepr::infinity_exponent())
return FloatingPointBuilder::infinity<T>();
auto leading_zeros = count_leading_zeroes(parse_result.mantissa);
u64 normalized_mantissa = parse_result.mantissa << leading_zeros;
// No need to multiply with some power of 5 here the exponent is already a power of 2.
u8 upperbit = normalized_mantissa >> 63;
FloatingPointBuilder parts;
parts.mantissa = normalized_mantissa >> (upperbit + 64 - FloatingPointRepr::mantissa_bits() - 3);
parts.exponent = parse_result.exponent + upperbit - leading_zeros + FloatingPointRepr::exponent_bias() + 62;
if (parts.exponent <= 0) {
// subnormal
if (-parts.exponent + 1 >= 64) {
parts.mantissa = 0;
parts.exponent = 0;
return parts;
}
parts.mantissa >>= -parts.exponent + 1;
parts.mantissa += parts.mantissa & 1;
parts.mantissa >>= 1;
if (parts.mantissa < (1ull << FloatingPointRepr::mantissa_bits())) {
parts.exponent = 0;
} else {
parts.exponent = 1;
}
return parts;
}
// Here we don't have to only do this halfway check for some exponents
if ((parts.mantissa & 0b11) == 0b01) {
// effectively all discard bits from z.high are 0
if (normalized_mantissa == (parts.mantissa << (upperbit + 64 - FloatingPointRepr::mantissa_bits() - 3)))
parts.mantissa &= ~u64(1);
}
parts.mantissa += parts.mantissa & 1;
parts.mantissa >>= 1;
if (parts.mantissa >= (2ull << FloatingPointRepr::mantissa_bits())) {
parts.mantissa = 1ull << FloatingPointRepr::mantissa_bits();
++parts.exponent;
}
parts.mantissa &= ~(1ull << FloatingPointRepr::mantissa_bits());
if (parts.exponent >= FloatingPointRepr::infinity_exponent()) {
parts.mantissa = 0;
parts.exponent = FloatingPointRepr::infinity_exponent();
}
return parts;
}
template<FloatingPoint T>
FloatingPointParseResults<T> parse_first_hexfloat_until_zero_character(char const* start)
{
using FloatingPointRepr = FloatingPointInfo<T>;
auto parse_result = parse_hexfloat(start);
if (!parse_result.valid)
return { nullptr, FloatingPointError::NoOrInvalidInput, __builtin_nan("") };
FloatingPointParseResults<T> full_result {};
full_result.end_ptr = parse_result.last_parsed;
// We special case this to be able to differentiate between 0 and values rounded down to 0
if (parse_result.mantissa == 0) {
full_result.value = 0.;
return full_result;
}
auto result = build_hex_float<T>(parse_result);
full_result.value = result.template to_value<T>(parse_result.is_negative);
if (result.exponent == FloatingPointRepr::infinity_exponent()) {
VERIFY(result.mantissa == 0);
full_result.error = FloatingPointError::OutOfRange;
} else if (result.mantissa == 0 && result.exponent == 0) {
full_result.error = FloatingPointError::RoundedDownToZero;
}
return full_result;
}
template FloatingPointParseResults<double> parse_first_hexfloat_until_zero_character(char const* start);
template FloatingPointParseResults<float> parse_first_hexfloat_until_zero_character(char const* start);
AK: Add an exact and fast floating point parsing algorithm This is based on the paper by Daniel Lemire called "Number parsing at a Gigabyte per second", currently available at https://arxiv.org/abs/2101.11408 An implementation can be found at https://github.com/fastfloat/fast_float To support both strtod like methods and String::to_double we have two different APIs. The parse_first_floating_point gives back both the result, next character to read and the error/out of range status. Out of range here means we rounded to infinity 0. The other API, parse_floating_point_completely, will return a floating point only if the given character range contains just the floating point and nothing else. This can be much faster as we can skip actually computing the value if we notice we did not parse the whole range. Both of these APIs support a very lenient format to be usable in as many places as possible. Also it does not check for "named" values like "nan", "inf", "NAN" etc. Because this can be different for every usage. For integers and small values this new method is not faster and often even a tiny bit slower than the current strtod implementation. However the strtod implementation is wrong for a lot of values and has a much less predictable running time. For correctness this method was tested against known string -> double datasets from https://github.com/nigeltao/parse-number-fxx-test-data This method gives 100% accuracy. The old strtod gave an incorrect value in over 50% of the numbers tested.
2022-10-13 00:18:56 +00:00
}
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