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// Copyright 2020-2023 Daniel Lemire
// Copyright 2023 Matt Borland
// Distributed under the Boost Software License, Version 1.0.
// https://www.boost.org/LICENSE_1_0.txt
#ifndef BOOST_CHARCONV_DETAIL_COMPUTE_FLOAT64_HPP
#define BOOST_CHARCONV_DETAIL_COMPUTE_FLOAT64_HPP
#include <boost/charconv/detail/config.hpp>
#include <boost/charconv/detail/significand_tables.hpp>
#include <boost/charconv/detail/emulated128.hpp>
#include <boost/core/bit.hpp>
#include <cstdint>
#include <cfloat>
#include <cstring>
#include <cmath>
namespace boost { namespace charconv { namespace detail {
static constexpr double powers_of_ten[] = {
1e0, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9, 1e10, 1e11,
1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19, 1e20, 1e21, 1e22
};
// Attempts to compute i * 10^(power) exactly; and if "negative" is true, negate the result.
//
// This function will only work in some cases, when it does not work, success is
// set to false. This should work *most of the time* (like 99% of the time).
// We assume that power is in the [-325, 308] interval.
inline double compute_float64(std::int64_t power, std::uint64_t i, bool negative, bool& success) noexcept
{
static constexpr auto smallest_power = -325;
static constexpr auto largest_power = 308;
// We start with a fast path
// It was described in Clinger WD.
// How to read floating point numbers accurately.
// ACM SIGPLAN Notices. 1990
#if (FLT_EVAL_METHOD != 1) && (FLT_EVAL_METHOD != 0)
if (0 <= power && power <= 22 && i <= UINT64_C(9007199254740991))
#else
if (-22 <= power && power <= 22 && i <= UINT64_C(9007199254740991))
#endif
{
// The general idea is as follows.
// If 0 <= s < 2^53 and if 10^0 <= p <= 10^22 then
// 1) Both s and p can be represented exactly as 64-bit floating-point
// values
// (binary64).
// 2) Because s and p can be represented exactly as floating-point values,
// then s * p
// and s / p will produce correctly rounded values.
auto d = static_cast<double>(i);
if (power < 0)
{
d = d / powers_of_ten[-power];
}
else
{
d = d * powers_of_ten[power];
}
if (negative)
{
d = -d;
}
success = true;
return d;
}
// When 22 < power && power < 22 + 16, we could
// hope for another, secondary fast path. It was
// described by David M. Gay in "Correctly rounded
// binary-decimal and decimal-binary conversions." (1990)
// If you need to compute i * 10^(22 + x) for x < 16,
// first compute i * 10^x, if you know that result is exact
// (e.g., when i * 10^x < 2^53),
// then you can still proceed and do (i * 10^x) * 10^22.
// Is this worth your time?
// You need 22 < power *and* power < 22 + 16 *and* (i * 10^(x-22) < 2^53)
// for this second fast path to work.
// If you have 22 < power *and* power < 22 + 16, and then you
// optimistically compute "i * 10^(x-22)", there is still a chance that you
// have wasted your time if i * 10^(x-22) >= 2^53. It makes the use cases of
// this optimization maybe less common than we would like. Source:
// http://www.exploringbinary.com/fast-path-decimal-to-floating-point-conversion/
// also used in RapidJSON: https://rapidjson.org/strtod_8h_source.html
if (i == 0 || power < smallest_power)
{
return negative ? -0.0 : 0.0;
}
else if (power > largest_power)
{
return negative ? -HUGE_VAL : HUGE_VAL;
}
const std::uint64_t factor_significand = significands_table::significand_64[power - smallest_power];
const std::int64_t exponent = (((152170 + 65536) * power) >> 16) + 1024 + 63;
int leading_zeros = boost::core::countl_zero(i);
i <<= static_cast<std::uint64_t>(leading_zeros);
uint128 product = umul128(i, factor_significand);
std::uint64_t low = product.low;
std::uint64_t high = product.high;
// We know that upper has at most one leading zero because
// both i and factor_mantissa have a leading one. This means
// that the result is at least as large as ((1<<63)*(1<<63))/(1<<64).
//
// As long as the first 9 bits of "upper" are not "1", then we
// know that we have an exact computed value for the leading
// 55 bits because any imprecision would play out as a +1, in the worst case.
// Having 55 bits is necessary because we need 53 bits for the mantissa,
// but we have to have one rounding bit and, we can waste a bit if the most
// significant bit of the product is zero.
//
// We expect this next branch to be rarely taken (say 1% of the time).
// When (upper & 0x1FF) == 0x1FF, it can be common for
// lower + i < lower to be true (proba. much higher than 1%).
if (BOOST_UNLIKELY((high & 0x1FF) == 0x1FF) && (low + i < low))
{
const std::uint64_t factor_significand_low = significands_table::significand_128[power - smallest_power];
product = umul128(i, factor_significand_low);
//const std::uint64_t product_low = product.low;
const std::uint64_t product_middle2 = product.high;
const std::uint64_t product_middle1 = low;
std::uint64_t product_high = high;
const std::uint64_t product_middle = product_middle1 + product_middle2;
if (product_middle < product_middle1)
{
product_high++;
}
// Commented out because possibly unneeded
// See: https://arxiv.org/pdf/2212.06644.pdf
/*
// we want to check whether mantissa *i + i would affect our result
// This does happen, e.g. with 7.3177701707893310e+15
if (((product_middle + 1 == 0) && ((product_high & 0x1FF) == 0x1FF) && (product_low + i < product_low)))
{
success = false;
return 0;
}
*/
low = product_middle;
high = product_high;
}
// The final significand should be 53 bits with a leading 1
// We shift it so that it occupies 54 bits with a leading 1
const std::uint64_t upper_bit = high >> 63;
std::uint64_t significand = high >> (upper_bit + 9);
leading_zeros += static_cast<int>(1 ^ upper_bit);
// If we have lots of trailing zeros we may fall between two values
if (BOOST_UNLIKELY((low == 0) && ((high & 0x1FF) == 0) && ((significand & 3) == 1)))
{
// if significand & 1 == 1 we might need to round up
success = false;
return 0;
}
significand += significand & 1;
significand >>= 1;
// Here the significand < (1<<53), unless there is an overflow
if (significand >= (UINT64_C(1) << 53))
{
significand = (UINT64_C(1) << 52);
leading_zeros--;
}
significand &= ~(UINT64_C(1) << 52);
const auto real_exponent = static_cast<std::uint64_t>(exponent - leading_zeros);
// We have to check that real_exponent is in range, otherwise fail
if (BOOST_UNLIKELY((real_exponent < 1) || (real_exponent > 2046)))
{
success = false;
return 0;
}
significand |= real_exponent << 52;
significand |= ((static_cast<std::uint64_t>(negative) << 63));
double d;
std::memcpy(&d, &significand, sizeof(d));
success = true;
return d;
}
}}} // Namespaces
#endif // BOOST_CHARCONV_DETAIL_COMPUTE_FLOAT64_HPP
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