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ladybird/AK/StringFloatingPointConversions.cpp
Nico Weber 6911c5545c Everywhere: Fix a few comment typos
2022-11-09 16:00:32 +00:00

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/*
* Copyright (c) 2022, Dan Klishch <danilklishch@gmail.com>
*
* SPDX-License-Identifier: BSD-2-Clause
*/
#include <AK/Array.h>
#include <AK/BuiltinWrappers.h>
#include <AK/FloatingPoint.h>
#include <AK/StringFloatingPointConversions.h>
#include <AK/UFixedBigInt.h>
namespace AK {
// This entire algorithm is an implementation of the paper: Ryu: Fast Float-to-String Conversion
// by Ulf Adams, available at https://dl.acm.org/doi/pdf/10.1145/3192366.3192369 and an implementation
// at https://github.com/ulfjack/ryu . A lot of possible mistakes from the article were corrected, see
// discussion at https://github.com/SerenityOS/serenity/pull/15796 .
//
// Not implemented for float80, as it will require an insane lookup table size (193Kb).
//
// Run stress tests from https://github.com/DanShaders/serenity-arithmetic-benchmark after non-trivial
// modifications.
// These approximations should match the ones used in the Python script.
static constexpr i64 log10_5_num = 10043;
static constexpr i64 log10_5_denum = 14369;
static constexpr i64 log10_2_num = 1406;
static constexpr i64 log10_2_denum = 4671;
static constexpr i64 log2_5_num = 8245;
static constexpr i64 log2_5_denum = 3551;
template<typename Number, size_t Size1, size_t Size2>
struct LookupInformation {
i32 b0, b1; // B0 and B1 from the paper (accidentally swapped)
Number lt[Size1];
Number ge[Size2];
};
template<FloatingPoint>
int lookup_table;
template<typename FloatingPoint, typename MultiplyAndShiftFunction>
FloatingPointExponentialForm inner_convert_floating_point_to_decimal_exponential_form(FloatingPoint value, MultiplyAndShiftFunction const& multiply_and_shift)
{
using Extractor = FloatExtractor<FloatingPoint>;
Extractor bit_representation { .d = value };
bool sign = bit_representation.sign;
i32 exponent = bit_representation.exponent;
u64 mantissa = bit_representation.mantissa;
// For +0, it is {.sign = 0, fraction = 0, exponent = 0},
// for -0, is {.sign = 1, fraction = 0, exponent = 0},
if (exponent == 0 && mantissa == 0)
return { sign, 0, 0 };
// for +inf, -inf, and NaN is undefined.
VERIFY(exponent != Extractor::exponent_max);
// Step 1. Decode the floating point number, and unify normalized and subnormal cases.
u64 real_mantissa = (exponent == 0 ? 0 : (1ull << Extractor::mantissa_bits)) + mantissa;
i32 real_exponent = (exponent == 0 ? 1 : exponent) - Extractor::exponent_bias - Extractor::mantissa_bits;
// abs(value) = real_mantissa * 2 ^ real_exponent
// Step 2. Determine the interval of information-preserving outputs.
// u, v, w are, respectively, lower bound for answer, exact value and upper bound for answer.
i32 synthetic_exponent = real_exponent - 2;
u64 u = 4 * real_mantissa - (mantissa == 0 && exponent > 1 ? 1 : 2);
u64 v = 4 * real_mantissa;
u64 w = 4 * real_mantissa + 2;
// u * 2 ^ synthetic_exponent < abs(answer) < w * 2 ^ synthetic_exponent (1)
// abs(value) = v * 2 ^ synthetic_exponent (yet another representation)
// Step 3'. Convert to a decimal power base and simultaneously remove most digits.
// We want to skip `skipped_iters' iterations of the main conversion loop and find out if
// last `skipped_iters' digits of u, v and w would have been zeroes.
i32 skipped_iters;
bool all_u_zero, all_v_zero, all_w_zero;
if (synthetic_exponent < 0) {
skipped_iters = max(0, -synthetic_exponent * log10_5_num / log10_5_denum - 1);
all_u_zero = count_trailing_zeroes(u) >= skipped_iters;
all_v_zero = count_trailing_zeroes(v) >= skipped_iters;
all_w_zero = count_trailing_zeroes(w) >= skipped_iters;
auto multiplier = lookup_table<FloatingPoint>.lt[-synthetic_exponent - skipped_iters];
i32 k_numerator = (log2_5_num + 1) * (-synthetic_exponent - skipped_iters);
i32 k = max(0, (k_numerator + log2_5_denum - 1) / log2_5_denum + lookup_table<FloatingPoint>.b0);
u = multiply_and_shift(u, multiplier, skipped_iters - k);
v = multiply_and_shift(v, multiplier, skipped_iters - k);
w = multiply_and_shift(w, multiplier, skipped_iters - k);
} else {
skipped_iters = max(0, synthetic_exponent * log10_2_num / log10_2_denum - 1);
// Checks if value is divisible by 5 ^ power.
auto is_divisible_by_pow_5 = [](u64 value, i32 power) {
constexpr Array<u64, 5> powers_of_five = { { 5, 25, 625, 390625, 152587890625 } };
if (power <= 0 || value == 0)
return true;
if (power >= 28) // 2 ^ 64 - 1 < 5 ^ 28
return false;
i32 result = 0;
for (i32 i = 5; i--;) {
if (value % powers_of_five[i] == 0) {
value /= powers_of_five[i];
result += 1 << i;
}
}
return result >= power;
};
all_u_zero = is_divisible_by_pow_5(u, skipped_iters);
all_v_zero = is_divisible_by_pow_5(v, skipped_iters);
all_w_zero = is_divisible_by_pow_5(w, skipped_iters);
auto multiplier = lookup_table<FloatingPoint>.ge[skipped_iters];
i32 k = log2_5_num * skipped_iters / log2_5_denum + lookup_table<FloatingPoint>.b1;
u = multiply_and_shift(u, multiplier, skipped_iters + k - synthetic_exponent);
v = multiply_and_shift(v, multiplier, skipped_iters + k - synthetic_exponent);
w = multiply_and_shift(w, multiplier, skipped_iters + k - synthetic_exponent);
}
// Step 4'. Find the shortest, correctly-rounded decimal representation in the interval.
bool is_even = ~mantissa & 1;
bool accept_smaller = is_even && all_u_zero;
bool accept_larger = is_even || !all_w_zero;
if (!accept_larger)
--w;
bool all_a_zero = accept_smaller;
bool all_b_zero = all_v_zero;
int last_digit = 0;
int exponent10 = skipped_iters - max(-synthetic_exponent, 0);
while (u / 10 < w / 10) {
all_a_zero &= u % 10 == 0;
all_b_zero &= last_digit == 0;
last_digit = v % 10;
u /= 10;
v /= 10;
w /= 10;
++exponent10;
}
if (all_a_zero) {
while (u % 10 == 0) {
all_b_zero &= last_digit == 0;
last_digit = v % 10;
u /= 10;
v /= 10;
w /= 10;
++exponent10;
}
}
bool is_tie = all_b_zero && last_digit == 5;
bool want_round_down = last_digit < 5 || (is_tie && v % 2 == 0);
bool round_down = (want_round_down && (u != v || all_a_zero)) || (v + 1 > w);
return { sign, round_down ? v : v + 1, exponent10 };
}
static u128 multiply(u64 a, u64 b)
{
#ifdef __SIZEOF_INT128__
unsigned __int128 result = (unsigned __int128)a * b;
u64 low = result;
u64 high = result >> 64;
return u128 { low, high };
#else
return u128 { a }.wide_multiply(u128 { b }).low;
#endif
}
template<>
FloatingPointExponentialForm convert_floating_point_to_decimal_exponential_form<float>(float value)
{
auto multiply_and_shift = [](u64 operand, u64 multiplier, i32 shift) {
#ifdef __SIZEOF_INT128__
auto result = (unsigned __int128)operand * multiplier;
#else
auto result = multiply(operand, multiplier);
#endif
if (shift < 0)
return static_cast<u64>(result << static_cast<u32>(-shift));
else
return static_cast<u64>(result >> static_cast<u32>(shift));
};
return inner_convert_floating_point_to_decimal_exponential_form(value, multiply_and_shift);
}
template<>
FloatingPointExponentialForm convert_floating_point_to_decimal_exponential_form<double>(double value)
{
auto multiply_and_shift = [](u64 operand, u64 const multiplier[2], i32 shift) {
u128 a = multiply(operand, multiplier[0]);
u128 b = multiply(operand, multiplier[1]) + a.high();
u64 c = a.low();
if (0 <= shift && shift < 64) {
return (c >> shift) | (b << static_cast<u32>(64 - shift)).low();
} else if (shift < 0) {
return c << static_cast<u32>(-shift);
} else {
VERIFY(64 <= shift && shift <= 128);
return (b >> static_cast<u32>(shift - 64)).low();
}
};
return inner_convert_floating_point_to_decimal_exponential_form(value, multiply_and_shift);
}
// Step 0. Precompute lookup tables for the given floating point type.
// Lookup tables was generated using the following Python script.
/*
from math import *
from more_itertools import chunked
def ifloor(x, y):
assert y > 0
if x < 0:
return (x - y + 1) // y
else:
return x // y
def iceil(x, y):
assert y > 0
if x < 0:
return x // y
else:
return (x + y - 1) // y
# Finds X = min(a * x % b) and Y = max(a * x % b) where 1 <= x <= N and returns (X, Y)
# Algorithm is from https://github.com/jk-jeon/Grisu-Exact/blob/master/other_files/Grisu-Exact.pdf , p. 22
def minmax_euclid(a, b, N):
a_i, b_i = a, b
s_i, u_i = 1, 0
while True:
q_i = iceil(b_i, a_i) - 1
b_i1 = b_i - q_i * a_i
u_i1 = u_i + q_i * s_i
if N < u_i1:
k = ifloor(N - u_i, s_i)
return (a_i, b - b_i + k * a_i)
p_i = iceil(a_i, b_i1) - 1
a_i1 = a_i - p_i * b_i1
s_i1 = s_i + p_i * u_i1
if N < s_i1:
k = ifloor(N - s_i, u_i1)
return (a_i - k * b_i1, b - b_i1)
if b_i1 == b_i and a_i1 == a_i:
if N < s_i1 + u_i1:
return (a_i1, b - b_i1)
else:
return (0, b - b_i1)
b_i, u_i, a_i, s_i = b_i1, u_i1, a_i1, s_i1
assert minmax_euclid(3, 8, 5) == (1, 7)
def calculate_lookup_tables(mantissa_bits, exponent_bits, nibbles_per_wide_digit, wide_digits_count, digit_suffix):
def split_by_wide_digits_and_print(value):
length = wide_digits_count * nibbles_per_wide_digit
number = reversed(list(chunked(f"{value:0{length}x}", nibbles_per_wide_digit)))
number = ", ".join(map(lambda x: "0x" + "".join(x) + digit_suffix, number))
print(f"{{ {number} }},")
mantissa_bias = 1 << mantissa_bits
mantissa_max = (1 << mantissa_bits) - 1
exponent_bias = (1 << (exponent_bits - 1)) - 1
exponent_max = (1 << exponent_bits) - 1
real_exponent_min = 1 - exponent_bias - mantissa_bits
real_exponent_max = exponent_max - exponent_bias - mantissa_bits
# real_exponent_min <= ef < real_exponent_max
synthetic_exponent_min = real_exponent_min - 2
synthetic_exponent_max = real_exponent_max - 2
# synthetic_exponent_min <= e2 < synthetic_exponent_max
max_synthetic_mantissa = 4 * (mantissa_bias + mantissa_max) + 2
# The following are some random approximations. Absolutely nothing special with these exact numbers.
LOG10_5_NUM = 10043
LOG10_5_DENUM = 14369
assert LOG10_5_NUM / LOG10_5_DENUM < log(5, 10)
LOG10_2_NUM = 1406
LOG10_2_DENUM = 4671
assert LOG10_2_NUM / LOG10_2_DENUM < log(2, 10)
LOG2_5_NUM = 8245
LOG2_5_DENUM = 3551
assert LOG2_5_NUM / LOG2_5_DENUM < log(5, 2)
assert (LOG2_5_NUM + 1) / LOG2_5_DENUM > log(5, 2)
# We want to find maximal b0, such that ceil(log(5, 2) * (-e2 - q)) + b0 <= k. One might plot (-e2 - q, k) from the
# iterations of the following loop and k = (-e2 - q) * log(5, 2) to understand the motivation behind this.
b0 = 0
q0max = 0
for e2 in range(synthetic_exponent_min, 0):
# q = max(0, floor(-e2 * log(5, 10)) - 1)
q = max(0, ifloor(-e2 * LOG10_5_NUM, LOG10_5_DENUM) - 1)
q0max = max(q0max, -e2 - q)
a = 5 ** (-e2 - q)
b = 2 ** q
[min_modular_product, _] = minmax_euclid(a, b, max_synthetic_mantissa)
# Directly via lemma 3.4 we obtain
# k = floor(log2(min_modular_product / max_synthetic_mantissa))
# But computing this directly might result in OverflowError, so we approximate the value
k = (min_modular_product.bit_length() - 1) - max_synthetic_mantissa.bit_length()
# "It is never wrong just to use 0"
# -- Some Guy
k = max(k, 0)
# coefficient = 5 ** (-e2 - q) // 2 ** k
# ceil(log(5, 2) * (-e2 - q)) + b0 <= k
# b0 <= k - ceil(log(5, 2) * (-e2 - q))
b0 = min(b0, k - iceil((-e2 - q) * (LOG2_5_NUM + 1), LOG2_5_DENUM))
print('b0 =', b0)
print('q0max =', q0max)
for q in range(0, q0max + 1):
k = max(0, iceil((LOG2_5_NUM + 1) * q, LOG2_5_DENUM) + b0)
coefficient = 5 ** q // 2 ** k
split_by_wide_digits_and_print(coefficient)
# Finding minimal b1, such that floor(log(5, 2) * q) + b1 >= k.
b1 = 0
q1max = 0
for e2 in range(0, synthetic_exponent_max):
# q = max(0, floor(e2 * log(2, 10)) - 1)
q = max(0, ifloor(e2 * LOG10_2_NUM, LOG10_2_DENUM) - 1)
q1max = max(q1max, q)
a = 2 ** (e2 - q)
b = 5 ** q
[_, max_modular_product] = minmax_euclid(a, b, max_synthetic_mantissa)
# Via lemma 3.3:
# k = ceil(log2(max_synthetic_mantissa * a * b / (b - max_modular_product)))
numerator = max_synthetic_mantissa * a * b
denumerator = b - max_modular_product
k = numerator.bit_length() - denumerator.bit_length() + 1
# coefficient = 2 ** k // 5 ** q + 1
# b1 = max(b1, k - floor(log(5, 2) * q))
b1 = max(b1, k - ifloor(q * LOG2_5_NUM, LOG2_5_DENUM))
print('b1 =', b1)
print('q1max =', q1max)
for q in range(0, q1max + 1):
k = ifloor(LOG2_5_NUM * q, LOG2_5_DENUM) + b1
coefficient = 2 ** k // 5 ** q + 1
split_by_wide_digits_and_print(coefficient)
# float:
print("float:")
calculate_lookup_tables(
23, 8,
16, 1, "ULL"
)
# double:
print("double:")
calculate_lookup_tables(
52, 11,
16, 2, "ULL"
)
# long double:
# print("long double:")
# calculate_lookup_tables(
# 64, 15,
# 8, 5, "U"
# )
*/
template<>
constexpr LookupInformation<u64, 48, 30> lookup_table<float> {
.b0 = -64,
.b1 = 62,
.lt = {
0x0000000000000001ULL,
0x0000000000000005ULL,
0x0000000000000019ULL,
0x000000000000007dULL,
0x0000000000000271ULL,
0x0000000000000c35ULL,
0x0000000000003d09ULL,
0x000000000001312dULL,
0x000000000005f5e1ULL,
0x00000000001dcd65ULL,
0x00000000009502f9ULL,
0x0000000002e90eddULL,
0x000000000e8d4a51ULL,
0x0000000048c27395ULL,
0x000000016bcc41e9ULL,
0x000000071afd498dULL,
0x0000002386f26fc1ULL,
0x000000b1a2bc2ec5ULL,
0x000003782dace9d9ULL,
0x00001158e460913dULL,
0x000056bc75e2d631ULL,
0x0001b1ae4d6e2ef5ULL,
0x000878678326eac9ULL,
0x002a5a058fc295edULL,
0x00d3c21bcecceda1ULL,
0x0422ca8b0a00a425ULL,
0x14adf4b7320334b9ULL,
0x6765c793fa10079dULL,
0x813f3978f8940984ULL,
0xa18f07d736b90be5ULL,
0xc9f2c9cd04674edeULL,
0xfc6f7c4045812296ULL,
0x9dc5ada82b70b59dULL,
0xc5371912364ce305ULL,
0xf684df56c3e01bc6ULL,
0x9a130b963a6c115cULL,
0xc097ce7bc90715b3ULL,
0xf0bdc21abb48db20ULL,
0x96769950b50d88f4ULL,
0xbc143fa4e250eb31ULL,
0xeb194f8e1ae525fdULL,
0x92efd1b8d0cf37beULL,
0xb7abc627050305adULL,
0xe596b7b0c643c719ULL,
0x8f7e32ce7bea5c6fULL,
0xb35dbf821ae4f38bULL,
0xe0352f62a19e306eULL,
0x8c213d9da502de45ULL,
},
.ge = {
0x4000000000000001ULL,
0x3333333333333334ULL,
0x28f5c28f5c28f5c3ULL,
0x20c49ba5e353f7cfULL,
0x346dc5d63886594bULL,
0x29f16b11c6d1e109ULL,
0x218def416bdb1a6eULL,
0x35afe535795e90b0ULL,
0x2af31dc4611873c0ULL,
0x225c17d04dad2966ULL,
0x36f9bfb3af7b7570ULL,
0x2bfaffc2f2c92ac0ULL,
0x232f33025bd42233ULL,
0x384b84d092ed0385ULL,
0x2d09370d42573604ULL,
0x24075f3dceac2b37ULL,
0x39a5652fb1137857ULL,
0x2e1dea8c8da92d13ULL,
0x24e4bba3a4875742ULL,
0x3b07929f6da5586aULL,
0x2f394219248446bbULL,
0x25c768141d369efcULL,
0x3c7240202ebdcb2dULL,
0x305b66802564a28aULL,
0x26af8533511d4ed5ULL,
0x3de5a1ebb4fbb155ULL,
0x318481895d962777ULL,
0x279d346de4781f93ULL,
0x3f61ed7ca0c03284ULL,
0x32b4bdfd4d668ed0ULL,
},
};
template<>
constexpr LookupInformation<u64[2], 326, 291> lookup_table<double> {
.b0 = -125,
.b1 = 125,
.lt = {
{ 0x0000000000000001ULL, 0x0000000000000000ULL },
{ 0x0000000000000005ULL, 0x0000000000000000ULL },
{ 0x0000000000000019ULL, 0x0000000000000000ULL },
{ 0x000000000000007dULL, 0x0000000000000000ULL },
{ 0x0000000000000271ULL, 0x0000000000000000ULL },
{ 0x0000000000000c35ULL, 0x0000000000000000ULL },
{ 0x0000000000003d09ULL, 0x0000000000000000ULL },
{ 0x000000000001312dULL, 0x0000000000000000ULL },
{ 0x000000000005f5e1ULL, 0x0000000000000000ULL },
{ 0x00000000001dcd65ULL, 0x0000000000000000ULL },
{ 0x00000000009502f9ULL, 0x0000000000000000ULL },
{ 0x0000000002e90eddULL, 0x0000000000000000ULL },
{ 0x000000000e8d4a51ULL, 0x0000000000000000ULL },
{ 0x0000000048c27395ULL, 0x0000000000000000ULL },
{ 0x000000016bcc41e9ULL, 0x0000000000000000ULL },
{ 0x000000071afd498dULL, 0x0000000000000000ULL },
{ 0x0000002386f26fc1ULL, 0x0000000000000000ULL },
{ 0x000000b1a2bc2ec5ULL, 0x0000000000000000ULL },
{ 0x000003782dace9d9ULL, 0x0000000000000000ULL },
{ 0x00001158e460913dULL, 0x0000000000000000ULL },
{ 0x000056bc75e2d631ULL, 0x0000000000000000ULL },
{ 0x0001b1ae4d6e2ef5ULL, 0x0000000000000000ULL },
{ 0x000878678326eac9ULL, 0x0000000000000000ULL },
{ 0x002a5a058fc295edULL, 0x0000000000000000ULL },
{ 0x00d3c21bcecceda1ULL, 0x0000000000000000ULL },
{ 0x0422ca8b0a00a425ULL, 0x0000000000000000ULL },
{ 0x14adf4b7320334b9ULL, 0x0000000000000000ULL },
{ 0x6765c793fa10079dULL, 0x0000000000000000ULL },
{ 0x04fce5e3e2502611ULL, 0x0000000000000002ULL },
{ 0x18f07d736b90be55ULL, 0x000000000000000aULL },
{ 0x7cb2734119d3b7a9ULL, 0x0000000000000032ULL },
{ 0x6f7c40458122964dULL, 0x00000000000000fcULL },
{ 0x2d6d415b85acef81ULL, 0x00000000000004eeULL },
{ 0xe32246c99c60ad85ULL, 0x00000000000018a6ULL },
{ 0x6fab61f00de36399ULL, 0x0000000000007b42ULL },
{ 0x2e58e9b04570f1fdULL, 0x000000000002684cULL },
{ 0xe7bc90715b34b9f1ULL, 0x00000000000c097cULL },
{ 0x86aed236c807a1b5ULL, 0x00000000003c2f70ULL },
{ 0xa16a1b11e8262889ULL, 0x00000000012ced32ULL },
{ 0x2712875988becaadULL, 0x0000000005e0a1fdULL },
{ 0xc35ca4bfabb9f561ULL, 0x000000001d6329f1ULL },
{ 0xd0cf37be5aa1cae5ULL, 0x0000000092efd1b8ULL },
{ 0x140c16b7c528f679ULL, 0x00000002deaf189cULL },
{ 0x643c7196d9ccd05dULL, 0x0000000e596b7b0cULL },
{ 0xf52e37f2410011d1ULL, 0x00000047bf19673dULL },
{ 0xc9e717bb45005915ULL, 0x00000166bb7f0435ULL },
{ 0xf18376a85901bd69ULL, 0x00000701a97b150cULL },
{ 0xb7915149bd08b30dULL, 0x000023084f676940ULL },
{ 0x95d69670b12b7f41ULL, 0x0000af298d050e43ULL },
{ 0xed30f03375d97c45ULL, 0x00036bcfc1194751ULL },
{ 0xa1f4b1014d3f6d59ULL, 0x00111b0ec57e6499ULL },
{ 0x29c77506823d22bdULL, 0x00558749db77f700ULL },
{ 0xd0e549208b31adb1ULL, 0x01aba4714957d300ULL },
{ 0x147a6da2b7f86475ULL, 0x085a36366eb71f04ULL },
{ 0x33321216cbecfb24ULL, 0x14e1878814c9cd8aULL },
{ 0xbffe969c7ee839edULL, 0x1a19e96a19fc40ecULL },
{ 0xf7ff1e21cf512434ULL, 0x105031e2503da893ULL },
{ 0xf5fee5aa43256d41ULL, 0x14643e5ae44d12b8ULL },
{ 0x337e9f14d3eec892ULL, 0x197d4df19d605767ULL },
{ 0x802f236d04753d5bULL, 0x0fee50b7025c36a0ULL },
{ 0xa03aec4845928cb2ULL, 0x13e9e4e4c2f34448ULL },
{ 0xc849a75a56f72fdeULL, 0x18e45e1df3b0155aULL },
{ 0x7a5c1130ecb4fbd6ULL, 0x1f1d75a5709c1ab1ULL },
{ 0xec798abe93f11d65ULL, 0x13726987666190aeULL },
{ 0xa797ed6e38ed64bfULL, 0x184f03e93ff9f4daULL },
{ 0x517de8c9c728bdefULL, 0x1e62c4e38ff87211ULL },
{ 0xd2eeb17e1c7976b5ULL, 0x12fdbb0e39fb474aULL },
{ 0x87aa5ddda397d462ULL, 0x17bd29d1c87a191dULL },
{ 0xe994f5550c7dc97bULL, 0x1dac74463a989f64ULL },
{ 0x11fd195527ce9dedULL, 0x128bc8abe49f639fULL },
{ 0xd67c5faa71c24568ULL, 0x172ebad6ddc73c86ULL },
{ 0x8c1b77950e32d6c2ULL, 0x1cfa698c95390ba8ULL },
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{ 0xd06142712d6f1556ULL, 0x1308a831868ac89aULL },
{ 0x4d686a4eaf182222ULL, 0x1e74404f3daada91ULL },
{ 0xa453883ef279b4e8ULL, 0x185d003f6488aedaULL },
{ 0xe9dc6cff28615d87ULL, 0x137d99cc506d58aeULL },
{ 0xa960ae650d6895a4ULL, 0x1f2f5c7a1a488de4ULL },
{ 0xbab3beb73ded4483ULL, 0x18f2b061aea07183ULL },
},
};
}
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