ladybird/AK/FloatingPointStringConversions.cpp
Ali Mohammad Pur 94f5389934 AK: Add a 'HostIsLittleEndian' constant and use it instead of BYTE_ORDER
Previously we were using the preprocessor everywhere we needed this
constant, so let's move away from that and use a constexpr constant.
2023-08-21 13:39:32 +03:30

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/*
* Copyright (c) 2022, David Tuin <davidot@serenityos.org>
*
* SPDX-License-Identifier: BSD-2-Clause
*/
#include <AK/BigIntBase.h>
#include <AK/CharacterTypes.h>
#include <AK/FloatingPointStringConversions.h>
#include <AK/Format.h>
#include <AK/ScopeGuard.h>
#include <AK/StringView.h>
#include <AK/UFixedBigInt.h>
#include <AK/UFixedBigIntDivision.h>
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>
concept ParseableFloatingPoint = IsFloatingPoint<T> && (sizeof(T) == sizeof(u32) || sizeof(T) == sizeof(u64));
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()));
static_assert(HostIsLittleEndian, "Float parsing currently assumes little endian, this fact is only used in fast parsing of 8 digits at a time"
"\nyou _should_ only need to change read eight_digits to make this big endian compatible.");
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;
}
auto z = 4096 - base.clz();
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;
}
auto z = 4096 - base.clz();
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 },
u128 { 0xbe311c083a225cd2ULL, 0x892731ac9faf056eULL },
u128 { 0x6dbd630a48aaf406ULL, 0xab70fe17c79ac6caULL },
u128 { 0x92cbbccdad5b108ULL, 0xd64d3d9db981787dULL },
u128 { 0x25bbf56008c58ea5ULL, 0x85f0468293f0eb4eULL },
u128 { 0xaf2af2b80af6f24eULL, 0xa76c582338ed2621ULL },
u128 { 0x1af5af660db4aee1ULL, 0xd1476e2c07286faaULL },
u128 { 0x50d98d9fc890ed4dULL, 0x82cca4db847945caULL },
u128 { 0xe50ff107bab528a0ULL, 0xa37fce126597973cULL },
u128 { 0x1e53ed49a96272c8ULL, 0xcc5fc196fefd7d0cULL },
u128 { 0x25e8e89c13bb0f7aULL, 0xff77b1fcbebcdc4fULL },
u128 { 0x77b191618c54e9acULL, 0x9faacf3df73609b1ULL },
u128 { 0xd59df5b9ef6a2417ULL, 0xc795830d75038c1dULL },
u128 { 0x4b0573286b44ad1dULL, 0xf97ae3d0d2446f25ULL },
u128 { 0x4ee367f9430aec32ULL, 0x9becce62836ac577ULL },
u128 { 0x229c41f793cda73fULL, 0xc2e801fb244576d5ULL },
u128 { 0x6b43527578c1110fULL, 0xf3a20279ed56d48aULL },
u128 { 0x830a13896b78aaa9ULL, 0x9845418c345644d6ULL },
u128 { 0x23cc986bc656d553ULL, 0xbe5691ef416bd60cULL },
u128 { 0x2cbfbe86b7ec8aa8ULL, 0xedec366b11c6cb8fULL },
u128 { 0x7bf7d71432f3d6a9ULL, 0x94b3a202eb1c3f39ULL },
u128 { 0xdaf5ccd93fb0cc53ULL, 0xb9e08a83a5e34f07ULL },
u128 { 0xd1b3400f8f9cff68ULL, 0xe858ad248f5c22c9ULL },
u128 { 0x23100809b9c21fa1ULL, 0x91376c36d99995beULL },
u128 { 0xabd40a0c2832a78aULL, 0xb58547448ffffb2dULL },
u128 { 0x16c90c8f323f516cULL, 0xe2e69915b3fff9f9ULL },
u128 { 0xae3da7d97f6792e3ULL, 0x8dd01fad907ffc3bULL },
u128 { 0x99cd11cfdf41779cULL, 0xb1442798f49ffb4aULL },
u128 { 0x40405643d711d583ULL, 0xdd95317f31c7fa1dULL },
u128 { 0x482835ea666b2572ULL, 0x8a7d3eef7f1cfc52ULL },
u128 { 0xda3243650005eecfULL, 0xad1c8eab5ee43b66ULL },
u128 { 0x90bed43e40076a82ULL, 0xd863b256369d4a40ULL },
u128 { 0x5a7744a6e804a291ULL, 0x873e4f75e2224e68ULL },
u128 { 0x711515d0a205cb36ULL, 0xa90de3535aaae202ULL },
u128 { 0xd5a5b44ca873e03ULL, 0xd3515c2831559a83ULL },
u128 { 0xe858790afe9486c2ULL, 0x8412d9991ed58091ULL },
u128 { 0x626e974dbe39a872ULL, 0xa5178fff668ae0b6ULL },
u128 { 0xfb0a3d212dc8128fULL, 0xce5d73ff402d98e3ULL },
u128 { 0x7ce66634bc9d0b99ULL, 0x80fa687f881c7f8eULL },
u128 { 0x1c1fffc1ebc44e80ULL, 0xa139029f6a239f72ULL },
u128 { 0xa327ffb266b56220ULL, 0xc987434744ac874eULL },
u128 { 0x4bf1ff9f0062baa8ULL, 0xfbe9141915d7a922ULL },
u128 { 0x6f773fc3603db4a9ULL, 0x9d71ac8fada6c9b5ULL },
u128 { 0xcb550fb4384d21d3ULL, 0xc4ce17b399107c22ULL },
u128 { 0x7e2a53a146606a48ULL, 0xf6019da07f549b2bULL },
u128 { 0x2eda7444cbfc426dULL, 0x99c102844f94e0fbULL },
u128 { 0xfa911155fefb5308ULL, 0xc0314325637a1939ULL },
u128 { 0x793555ab7eba27caULL, 0xf03d93eebc589f88ULL },
u128 { 0x4bc1558b2f3458deULL, 0x96267c7535b763b5ULL },
u128 { 0x9eb1aaedfb016f16ULL, 0xbbb01b9283253ca2ULL },
u128 { 0x465e15a979c1cadcULL, 0xea9c227723ee8bcbULL },
u128 { 0xbfacd89ec191ec9ULL, 0x92a1958a7675175fULL },
u128 { 0xcef980ec671f667bULL, 0xb749faed14125d36ULL },
u128 { 0x82b7e12780e7401aULL, 0xe51c79a85916f484ULL },
u128 { 0xd1b2ecb8b0908810ULL, 0x8f31cc0937ae58d2ULL },
u128 { 0x861fa7e6dcb4aa15ULL, 0xb2fe3f0b8599ef07ULL },
u128 { 0x67a791e093e1d49aULL, 0xdfbdcece67006ac9ULL },
u128 { 0xe0c8bb2c5c6d24e0ULL, 0x8bd6a141006042bdULL },
u128 { 0x58fae9f773886e18ULL, 0xaecc49914078536dULL },
u128 { 0xaf39a475506a899eULL, 0xda7f5bf590966848ULL },
u128 { 0x6d8406c952429603ULL, 0x888f99797a5e012dULL },
u128 { 0xc8e5087ba6d33b83ULL, 0xaab37fd7d8f58178ULL },
u128 { 0xfb1e4a9a90880a64ULL, 0xd5605fcdcf32e1d6ULL },
u128 { 0x5cf2eea09a55067fULL, 0x855c3be0a17fcd26ULL },
u128 { 0xf42faa48c0ea481eULL, 0xa6b34ad8c9dfc06fULL },
u128 { 0xf13b94daf124da26ULL, 0xd0601d8efc57b08bULL },
u128 { 0x76c53d08d6b70858ULL, 0x823c12795db6ce57ULL },
u128 { 0x54768c4b0c64ca6eULL, 0xa2cb1717b52481edULL },
u128 { 0xa9942f5dcf7dfd09ULL, 0xcb7ddcdda26da268ULL },
u128 { 0xd3f93b35435d7c4cULL, 0xfe5d54150b090b02ULL },
u128 { 0xc47bc5014a1a6dafULL, 0x9efa548d26e5a6e1ULL },
u128 { 0x359ab6419ca1091bULL, 0xc6b8e9b0709f109aULL },
u128 { 0xc30163d203c94b62ULL, 0xf867241c8cc6d4c0ULL },
u128 { 0x79e0de63425dcf1dULL, 0x9b407691d7fc44f8ULL },
u128 { 0x985915fc12f542e4ULL, 0xc21094364dfb5636ULL },
u128 { 0x3e6f5b7b17b2939dULL, 0xf294b943e17a2bc4ULL },
u128 { 0xa705992ceecf9c42ULL, 0x979cf3ca6cec5b5aULL },
u128 { 0x50c6ff782a838353ULL, 0xbd8430bd08277231ULL },
u128 { 0xa4f8bf5635246428ULL, 0xece53cec4a314ebdULL },
u128 { 0x871b7795e136be99ULL, 0x940f4613ae5ed136ULL },
u128 { 0x28e2557b59846e3fULL, 0xb913179899f68584ULL },
u128 { 0x331aeada2fe589cfULL, 0xe757dd7ec07426e5ULL },
u128 { 0x3ff0d2c85def7621ULL, 0x9096ea6f3848984fULL },
u128 { 0xfed077a756b53a9ULL, 0xb4bca50b065abe63ULL },
u128 { 0xd3e8495912c62894ULL, 0xe1ebce4dc7f16dfbULL },
u128 { 0x64712dd7abbbd95cULL, 0x8d3360f09cf6e4bdULL },
u128 { 0xbd8d794d96aacfb3ULL, 0xb080392cc4349decULL },
u128 { 0xecf0d7a0fc5583a0ULL, 0xdca04777f541c567ULL },
u128 { 0xf41686c49db57244ULL, 0x89e42caaf9491b60ULL },
u128 { 0x311c2875c522ced5ULL, 0xac5d37d5b79b6239ULL },
u128 { 0x7d633293366b828bULL, 0xd77485cb25823ac7ULL },
u128 { 0xae5dff9c02033197ULL, 0x86a8d39ef77164bcULL },
u128 { 0xd9f57f830283fdfcULL, 0xa8530886b54dbdebULL },
u128 { 0xd072df63c324fd7bULL, 0xd267caa862a12d66ULL },
u128 { 0x4247cb9e59f71e6dULL, 0x8380dea93da4bc60ULL },
u128 { 0x52d9be85f074e608ULL, 0xa46116538d0deb78ULL },
u128 { 0x67902e276c921f8bULL, 0xcd795be870516656ULL },
u128 { 0xba1cd8a3db53b6ULL, 0x806bd9714632dff6ULL },
u128 { 0x80e8a40eccd228a4ULL, 0xa086cfcd97bf97f3ULL },
u128 { 0x6122cd128006b2cdULL, 0xc8a883c0fdaf7df0ULL },
u128 { 0x796b805720085f81ULL, 0xfad2a4b13d1b5d6cULL },
u128 { 0xcbe3303674053bb0ULL, 0x9cc3a6eec6311a63ULL },
u128 { 0xbedbfc4411068a9cULL, 0xc3f490aa77bd60fcULL },
u128 { 0xee92fb5515482d44ULL, 0xf4f1b4d515acb93bULL },
u128 { 0x751bdd152d4d1c4aULL, 0x991711052d8bf3c5ULL },
u128 { 0xd262d45a78a0635dULL, 0xbf5cd54678eef0b6ULL },
u128 { 0x86fb897116c87c34ULL, 0xef340a98172aace4ULL },
u128 { 0xd45d35e6ae3d4da0ULL, 0x9580869f0e7aac0eULL },
u128 { 0x8974836059cca109ULL, 0xbae0a846d2195712ULL },
u128 { 0x2bd1a438703fc94bULL, 0xe998d258869facd7ULL },
u128 { 0x7b6306a34627ddcfULL, 0x91ff83775423cc06ULL },
u128 { 0x1a3bc84c17b1d542ULL, 0xb67f6455292cbf08ULL },
u128 { 0x20caba5f1d9e4a93ULL, 0xe41f3d6a7377eecaULL },
u128 { 0x547eb47b7282ee9cULL, 0x8e938662882af53eULL },
u128 { 0xe99e619a4f23aa43ULL, 0xb23867fb2a35b28dULL },
u128 { 0x6405fa00e2ec94d4ULL, 0xdec681f9f4c31f31ULL },
u128 { 0xde83bc408dd3dd04ULL, 0x8b3c113c38f9f37eULL },
u128 { 0x9624ab50b148d445ULL, 0xae0b158b4738705eULL },
u128 { 0x3badd624dd9b0957ULL, 0xd98ddaee19068c76ULL },
u128 { 0xe54ca5d70a80e5d6ULL, 0x87f8a8d4cfa417c9ULL },
u128 { 0x5e9fcf4ccd211f4cULL, 0xa9f6d30a038d1dbcULL },
u128 { 0x7647c3200069671fULL, 0xd47487cc8470652bULL },
u128 { 0x29ecd9f40041e073ULL, 0x84c8d4dfd2c63f3bULL },
u128 { 0xf468107100525890ULL, 0xa5fb0a17c777cf09ULL },
u128 { 0x7182148d4066eeb4ULL, 0xcf79cc9db955c2ccULL },
u128 { 0xc6f14cd848405530ULL, 0x81ac1fe293d599bfULL },
u128 { 0xb8ada00e5a506a7cULL, 0xa21727db38cb002fULL },
u128 { 0xa6d90811f0e4851cULL, 0xca9cf1d206fdc03bULL },
u128 { 0x908f4a166d1da663ULL, 0xfd442e4688bd304aULL },
u128 { 0x9a598e4e043287feULL, 0x9e4a9cec15763e2eULL },
u128 { 0x40eff1e1853f29fdULL, 0xc5dd44271ad3cdbaULL },
u128 { 0xd12bee59e68ef47cULL, 0xf7549530e188c128ULL },
u128 { 0x82bb74f8301958ceULL, 0x9a94dd3e8cf578b9ULL },
u128 { 0xe36a52363c1faf01ULL, 0xc13a148e3032d6e7ULL },
u128 { 0xdc44e6c3cb279ac1ULL, 0xf18899b1bc3f8ca1ULL },
u128 { 0x29ab103a5ef8c0b9ULL, 0x96f5600f15a7b7e5ULL },
u128 { 0x7415d448f6b6f0e7ULL, 0xbcb2b812db11a5deULL },
u128 { 0x111b495b3464ad21ULL, 0xebdf661791d60f56ULL },
u128 { 0xcab10dd900beec34ULL, 0x936b9fcebb25c995ULL },
u128 { 0x3d5d514f40eea742ULL, 0xb84687c269ef3bfbULL },
u128 { 0xcb4a5a3112a5112ULL, 0xe65829b3046b0afaULL },
u128 { 0x47f0e785eaba72abULL, 0x8ff71a0fe2c2e6dcULL },
u128 { 0x59ed216765690f56ULL, 0xb3f4e093db73a093ULL },
u128 { 0x306869c13ec3532cULL, 0xe0f218b8d25088b8ULL },
u128 { 0x1e414218c73a13fbULL, 0x8c974f7383725573ULL },
u128 { 0xe5d1929ef90898faULL, 0xafbd2350644eeacfULL },
u128 { 0xdf45f746b74abf39ULL, 0xdbac6c247d62a583ULL },
u128 { 0x6b8bba8c328eb783ULL, 0x894bc396ce5da772ULL },
u128 { 0x66ea92f3f326564ULL, 0xab9eb47c81f5114fULL },
u128 { 0xc80a537b0efefebdULL, 0xd686619ba27255a2ULL },
u128 { 0xbd06742ce95f5f36ULL, 0x8613fd0145877585ULL },
u128 { 0x2c48113823b73704ULL, 0xa798fc4196e952e7ULL },
u128 { 0xf75a15862ca504c5ULL, 0xd17f3b51fca3a7a0ULL },
u128 { 0x9a984d73dbe722fbULL, 0x82ef85133de648c4ULL },
u128 { 0xc13e60d0d2e0ebbaULL, 0xa3ab66580d5fdaf5ULL },
u128 { 0x318df905079926a8ULL, 0xcc963fee10b7d1b3ULL },
u128 { 0xfdf17746497f7052ULL, 0xffbbcfe994e5c61fULL },
u128 { 0xfeb6ea8bedefa633ULL, 0x9fd561f1fd0f9bd3ULL },
u128 { 0xfe64a52ee96b8fc0ULL, 0xc7caba6e7c5382c8ULL },
u128 { 0x3dfdce7aa3c673b0ULL, 0xf9bd690a1b68637bULL },
u128 { 0x6bea10ca65c084eULL, 0x9c1661a651213e2dULL },
u128 { 0x486e494fcff30a62ULL, 0xc31bfa0fe5698db8ULL },
u128 { 0x5a89dba3c3efccfaULL, 0xf3e2f893dec3f126ULL },
u128 { 0xf89629465a75e01cULL, 0x986ddb5c6b3a76b7ULL },
u128 { 0xf6bbb397f1135823ULL, 0xbe89523386091465ULL },
u128 { 0x746aa07ded582e2cULL, 0xee2ba6c0678b597fULL },
u128 { 0xa8c2a44eb4571cdcULL, 0x94db483840b717efULL },
u128 { 0x92f34d62616ce413ULL, 0xba121a4650e4ddebULL },
u128 { 0x77b020baf9c81d17ULL, 0xe896a0d7e51e1566ULL },
u128 { 0xace1474dc1d122eULL, 0x915e2486ef32cd60ULL },
u128 { 0xd819992132456baULL, 0xb5b5ada8aaff80b8ULL },
u128 { 0x10e1fff697ed6c69ULL, 0xe3231912d5bf60e6ULL },
u128 { 0xca8d3ffa1ef463c1ULL, 0x8df5efabc5979c8fULL },
u128 { 0xbd308ff8a6b17cb2ULL, 0xb1736b96b6fd83b3ULL },
u128 { 0xac7cb3f6d05ddbdeULL, 0xddd0467c64bce4a0ULL },
u128 { 0x6bcdf07a423aa96bULL, 0x8aa22c0dbef60ee4ULL },
u128 { 0x86c16c98d2c953c6ULL, 0xad4ab7112eb3929dULL },
u128 { 0xe871c7bf077ba8b7ULL, 0xd89d64d57a607744ULL },
u128 { 0x11471cd764ad4972ULL, 0x87625f056c7c4a8bULL },
u128 { 0xd598e40d3dd89bcfULL, 0xa93af6c6c79b5d2dULL },
u128 { 0x4aff1d108d4ec2c3ULL, 0xd389b47879823479ULL },
u128 { 0xcedf722a585139baULL, 0x843610cb4bf160cbULL },
u128 { 0xc2974eb4ee658828ULL, 0xa54394fe1eedb8feULL },
u128 { 0x733d226229feea32ULL, 0xce947a3da6a9273eULL },
u128 { 0x806357d5a3f525fULL, 0x811ccc668829b887ULL },
u128 { 0xca07c2dcb0cf26f7ULL, 0xa163ff802a3426a8ULL },
u128 { 0xfc89b393dd02f0b5ULL, 0xc9bcff6034c13052ULL },
u128 { 0xbbac2078d443ace2ULL, 0xfc2c3f3841f17c67ULL },
u128 { 0xd54b944b84aa4c0dULL, 0x9d9ba7832936edc0ULL },
u128 { 0xa9e795e65d4df11ULL, 0xc5029163f384a931ULL },
u128 { 0x4d4617b5ff4a16d5ULL, 0xf64335bcf065d37dULL },
u128 { 0x504bced1bf8e4e45ULL, 0x99ea0196163fa42eULL },
u128 { 0xe45ec2862f71e1d6ULL, 0xc06481fb9bcf8d39ULL },
u128 { 0x5d767327bb4e5a4cULL, 0xf07da27a82c37088ULL },
u128 { 0x3a6a07f8d510f86fULL, 0x964e858c91ba2655ULL },
u128 { 0x890489f70a55368bULL, 0xbbe226efb628afeaULL },
u128 { 0x2b45ac74ccea842eULL, 0xeadab0aba3b2dbe5ULL },
u128 { 0x3b0b8bc90012929dULL, 0x92c8ae6b464fc96fULL },
u128 { 0x9ce6ebb40173744ULL, 0xb77ada0617e3bbcbULL },
u128 { 0xcc420a6a101d0515ULL, 0xe55990879ddcaabdULL },
u128 { 0x9fa946824a12232dULL, 0x8f57fa54c2a9eab6ULL },
u128 { 0x47939822dc96abf9ULL, 0xb32df8e9f3546564ULL },
u128 { 0x59787e2b93bc56f7ULL, 0xdff9772470297ebdULL },
u128 { 0x57eb4edb3c55b65aULL, 0x8bfbea76c619ef36ULL },
u128 { 0xede622920b6b23f1ULL, 0xaefae51477a06b03ULL },
u128 { 0xe95fab368e45ecedULL, 0xdab99e59958885c4ULL },
u128 { 0x11dbcb0218ebb414ULL, 0x88b402f7fd75539bULL },
u128 { 0xd652bdc29f26a119ULL, 0xaae103b5fcd2a881ULL },
u128 { 0x4be76d3346f0495fULL, 0xd59944a37c0752a2ULL },
u128 { 0x6f70a4400c562ddbULL, 0x857fcae62d8493a5ULL },
u128 { 0xcb4ccd500f6bb952ULL, 0xa6dfbd9fb8e5b88eULL },
u128 { 0x7e2000a41346a7a7ULL, 0xd097ad07a71f26b2ULL },
u128 { 0x8ed400668c0c28c8ULL, 0x825ecc24c873782fULL },
u128 { 0x728900802f0f32faULL, 0xa2f67f2dfa90563bULL },
u128 { 0x4f2b40a03ad2ffb9ULL, 0xcbb41ef979346bcaULL },
u128 { 0xe2f610c84987bfa8ULL, 0xfea126b7d78186bcULL },
u128 { 0xdd9ca7d2df4d7c9ULL, 0x9f24b832e6b0f436ULL },
u128 { 0x91503d1c79720dbbULL, 0xc6ede63fa05d3143ULL },
u128 { 0x75a44c6397ce912aULL, 0xf8a95fcf88747d94ULL },
u128 { 0xc986afbe3ee11abaULL, 0x9b69dbe1b548ce7cULL },
u128 { 0xfbe85badce996168ULL, 0xc24452da229b021bULL },
u128 { 0xfae27299423fb9c3ULL, 0xf2d56790ab41c2a2ULL },
u128 { 0xdccd879fc967d41aULL, 0x97c560ba6b0919a5ULL },
u128 { 0x5400e987bbc1c920ULL, 0xbdb6b8e905cb600fULL },
u128 { 0x290123e9aab23b68ULL, 0xed246723473e3813ULL },
u128 { 0xf9a0b6720aaf6521ULL, 0x9436c0760c86e30bULL },
u128 { 0xf808e40e8d5b3e69ULL, 0xb94470938fa89bceULL },
u128 { 0xb60b1d1230b20e04ULL, 0xe7958cb87392c2c2ULL },
u128 { 0xb1c6f22b5e6f48c2ULL, 0x90bd77f3483bb9b9ULL },
u128 { 0x1e38aeb6360b1af3ULL, 0xb4ecd5f01a4aa828ULL },
u128 { 0x25c6da63c38de1b0ULL, 0xe2280b6c20dd5232ULL },
u128 { 0x579c487e5a38ad0eULL, 0x8d590723948a535fULL },
u128 { 0x2d835a9df0c6d851ULL, 0xb0af48ec79ace837ULL },
u128 { 0xf8e431456cf88e65ULL, 0xdcdb1b2798182244ULL },
u128 { 0x1b8e9ecb641b58ffULL, 0x8a08f0f8bf0f156bULL },
u128 { 0xe272467e3d222f3fULL, 0xac8b2d36eed2dac5ULL },
u128 { 0x5b0ed81dcc6abb0fULL, 0xd7adf884aa879177ULL },
u128 { 0x98e947129fc2b4e9ULL, 0x86ccbb52ea94baeaULL },
u128 { 0x3f2398d747b36224ULL, 0xa87fea27a539e9a5ULL },
u128 { 0x8eec7f0d19a03aadULL, 0xd29fe4b18e88640eULL },
u128 { 0x1953cf68300424acULL, 0x83a3eeeef9153e89ULL },
u128 { 0x5fa8c3423c052dd7ULL, 0xa48ceaaab75a8e2bULL },
u128 { 0x3792f412cb06794dULL, 0xcdb02555653131b6ULL },
u128 { 0xe2bbd88bbee40bd0ULL, 0x808e17555f3ebf11ULL },
u128 { 0x5b6aceaeae9d0ec4ULL, 0xa0b19d2ab70e6ed6ULL },
u128 { 0xf245825a5a445275ULL, 0xc8de047564d20a8bULL },
u128 { 0xeed6e2f0f0d56712ULL, 0xfb158592be068d2eULL },
u128 { 0x55464dd69685606bULL, 0x9ced737bb6c4183dULL },
u128 { 0xaa97e14c3c26b886ULL, 0xc428d05aa4751e4cULL },
u128 { 0xd53dd99f4b3066a8ULL, 0xf53304714d9265dfULL },
u128 { 0xe546a8038efe4029ULL, 0x993fe2c6d07b7fabULL },
u128 { 0xde98520472bdd033ULL, 0xbf8fdb78849a5f96ULL },
u128 { 0x963e66858f6d4440ULL, 0xef73d256a5c0f77cULL },
u128 { 0xdde7001379a44aa8ULL, 0x95a8637627989aadULL },
u128 { 0x5560c018580d5d52ULL, 0xbb127c53b17ec159ULL },
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 },
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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)
{
return UFixedBigInt<64>(a).wide_multiply(b);
}
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);
lower_result += upper_result.high();
}
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
};
}
class MinimalBigInt {
public:
MinimalBigInt() = default;
MinimalBigInt(u64 value)
{
StorageOperations::copy(Detail::get_storage_of(value), get_storage(words_in_u64));
}
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 top_u64_start = static_cast<size_t>(size_in_bits()) - 64;
u64 top_u64 = 0;
for (size_t i = 0; i < m_used_length; ++i) {
size_t word_start = i * word_size;
size_t word_end = word_start + word_size;
if (top_u64_start < word_end) {
if (top_u64_start >= word_start) {
auto shift = top_u64_start - word_start;
top_u64 = m_words[i] >> shift;
has_truncated_bits |= m_words[i] ^ (top_u64 << shift);
} else {
top_u64 |= static_cast<u64>(m_words[i]) << (word_start - top_u64_start);
}
} else {
has_truncated_bits |= m_words[i];
}
}
return top_u64;
}
i32 size_in_bits() const
{
if (m_used_length == 0)
return 0;
// This is guaranteed to be at most max_words_needed * word_size so not above i32 max
return static_cast<i32>(word_size * 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: Using UFixedBigInt here does not feel right. We need to nicely (without
// reinterpret_cast and ifs) convert u64 to NativeWord (if NativeWord == u32)
// and one of UFixedBigInt constructors happens to do this exact job.
//
// To calculate this lookup table, we compute 5 ** (2 ** i) for all i \in [0; 10], split
// numbers into 64-bit words and concatenate the results, writing corresponding sizes to
// `sizes` array in `power_of_5` lambda.
static constexpr UFixedBigInt<82 * 64> power_of_5_coefficients = { {
5ull,
25ull,
625ull,
390625ull,
152587890625ull,
3273344365508751233ull,
1262ull,
7942358959831785217ull,
16807427164405733357ull,
1593091ull,
279109966635548161ull,
2554917779393558781ull,
14124656261812188652ull,
11976055582626787546ull,
2537941837315ull,
13750482914757213185ull,
1302999927698857842ull,
14936872543252795590ull,
2788415840139466767ull,
2095640732773017264ull,
7205570348933370714ull,
7348167152523113408ull,
9285516396840364274ull,
6907659600622710236ull,
349175ull,
8643096425819600897ull,
6743743997439985372ull,
14059704609098336919ull,
10729359125898331411ull,
4933048501514368705ull,
12258131603170554683ull,
2172371001088594721ull,
13569903330219142946ull,
13809142207969578845ull,
16716360519037769646ull,
9631256923806107285ull,
12866941232305103710ull,
1397931361048440292ull,
7619627737732970332ull,
12725409486282665900ull,
11703051443360963910ull,
9947078370803086083ull,
13966287901448440471ull,
121923442132ull,
17679772531488845825ull,
2216509366347768155ull,
1568689219195129479ull,
5511594616325588277ull,
1067709417009240089ull,
9070650952098657518ull,
11515285870634858015ull,
2539561553659505564ull,
17604889300961091799ull,
14511540856854204724ull,
12099083339557485471ull,
7115240299237943815ull,
313979240050606788ull,
10004784664717172195ull,
15570268847930131473ull,
10359715202835930803ull,
17685054012115162812ull,
13183273382855797757ull,
7743260039872919062ull,
9284593436392572926ull,
11105921222066415013ull,
18198799323400703846ull,
16314988383739458320ull,
4387527177871570570ull,
8476708682254672590ull,
4925096874831034057ull,
14075687868072027455ull,
112866656203221926ull,
9852830467773230418ull,
25755239915196746ull,
2201493076310172510ull,
8342165458688466438ull,
13954006576066379050ull,
15193819059903295636ull,
12565616718911389531ull,
3815854855847885129ull,
15696762163583540628ull,
805ull,
} };
// power_of_5[i] = 5 ** (2 ** i)
static constexpr auto power_of_5 = [&] {
constexpr size_t powers_count = 11;
Array<UnsignedStorageReadonlySpan, powers_count> result;
Array<size_t, powers_count> sizes = { 1, 1, 1, 1, 1, 2, 3, 5, 10, 19, 38 };
size_t offset = 0;
for (size_t i = 0; i < powers_count; ++i) {
if constexpr (SameAs<NativeWord, u32>)
sizes[i] *= 2;
result[i] = UnsignedStorageReadonlySpan(power_of_5_coefficients.span().slice(offset, sizes[i]));
offset += sizes[i];
}
return result;
}();
VERIFY(exponent < (1 << power_of_5.size()));
// Binary exponentiation
for (size_t i = 0; i < power_of_5.size(); ++i) {
if (exponent >> i & 1) {
if (power_of_5[i].size() == 1) {
multiply_with_small(power_of_5[i][0]);
} else {
auto copy = *this;
StorageOperations::baseline_mul(copy.get_storage(), power_of_5[i],
get_storage(m_used_length + power_of_5[i].size()), g_null_allocator);
trim_last_word_if_zero();
}
}
}
}
void multiply_with_power_of_2(u32 exponent)
{
if (exponent) {
size_t max_new_length = m_used_length + (exponent + word_size - 1) / word_size;
if (m_used_length != max_words_needed)
m_words[m_used_length] = 0;
auto storage = get_storage(max_new_length);
StorageOperations::shift_left(storage, exponent, storage);
trim_last_word_if_zero();
}
}
enum class CompareResult {
Equal = 0,
GreaterThan = 1,
LessThan = -1
};
CompareResult compare_to(MinimalBigInt const& other) const
{
return static_cast<CompareResult>(StorageOperations::compare(get_storage(), other.get_storage(), false));
}
private:
UnsignedStorageSpan get_storage(size_t new_length = 0)
{
if (new_length > m_used_length)
m_used_length = min(max_words_needed, new_length);
return { m_words.data(), m_used_length };
}
UnsignedStorageReadonlySpan get_storage() const
{
return { m_words.data(), m_used_length };
}
void trim_last_word_if_zero()
{
if (m_used_length > 0 && !m_words[m_used_length - 1])
--m_used_length;
}
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)
{
if (value <= max_word) {
auto native_value = static_cast<NativeWord>(value);
NativeWord carry = 0;
for (size_t i = 0; i < m_used_length; ++i) {
auto result = UFixedBigInt<word_size>(m_words[i]).wide_multiply(native_value) + carry;
carry = result.high();
m_words[i] = result.low();
}
if (carry != 0)
m_words[m_used_length++] = carry;
} else {
// word_size == 32 && value > NumericLimits<u32>::max()
auto copy = *this;
StorageOperations::baseline_mul(copy.get_storage(), Detail::get_storage_of(value), get_storage(m_used_length + 2), g_null_allocator);
trim_last_word_if_zero();
}
}
void add_small(u64 value)
{
if (m_used_length == 0 && value <= max_word) {
m_words[m_used_length++] = static_cast<NativeWord>(value);
return;
}
auto initial_storage = get_storage();
auto expanded_storage = get_storage(max(m_used_length, words_in_u64));
if (StorageOperations::add<false>(initial_storage, Detail::get_storage_of(value), expanded_storage))
m_words[m_used_length++] = 1;
}
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);
}
// The max valid words we might need are log2(10^(769 + 342)), max digits + max exponent
static constexpr size_t words_in_u64 = word_size == 64 ? 1 : 2;
static constexpr size_t max_words_needed = 58 * words_in_u64;
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<NativeWord, 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);
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);
}