1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
|
//===-- Common header for FMA implementations -------------------*- C++ -*-===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//
#ifndef LLVM_LIBC_UTILS_FPUTIL_GENERIC_FMA_H
#define LLVM_LIBC_UTILS_FPUTIL_GENERIC_FMA_H
#include "utils/CPP/TypeTraits.h"
namespace __llvm_libc {
namespace fputil {
namespace generic {
template <typename T>
static inline cpp::EnableIfType<cpp::IsSame<T, float>::Value, T> fma(T x, T y,
T z) {
// Product is exact.
double prod = static_cast<double>(x) * static_cast<double>(y);
double z_d = static_cast<double>(z);
double sum = prod + z_d;
fputil::FPBits<double> bit_prod(prod), bitz(z_d), bit_sum(sum);
if (!(bit_sum.isInfOrNaN() || bit_sum.isZero())) {
// Since the sum is computed in double precision, rounding might happen
// (for instance, when bitz.exponent > bit_prod.exponent + 5, or
// bit_prod.exponent > bitz.exponent + 40). In that case, when we round
// the sum back to float, double rounding error might occur.
// A concrete example of this phenomenon is as follows:
// x = y = 1 + 2^(-12), z = 2^(-53)
// The exact value of x*y + z is 1 + 2^(-11) + 2^(-24) + 2^(-53)
// So when rounding to float, fmaf(x, y, z) = 1 + 2^(-11) + 2^(-23)
// On the other hand, with the default rounding mode,
// double(x*y + z) = 1 + 2^(-11) + 2^(-24)
// and casting again to float gives us:
// float(double(x*y + z)) = 1 + 2^(-11).
//
// In order to correct this possible double rounding error, first we use
// Dekker's 2Sum algorithm to find t such that sum - t = prod + z exactly,
// assuming the (default) rounding mode is round-to-the-nearest,
// tie-to-even. Moreover, t satisfies the condition that t < eps(sum),
// i.e., t.exponent < sum.exponent - 52. So if t is not 0, meaning rounding
// occurs when computing the sum, we just need to use t to adjust (any) last
// bit of sum, so that the sticky bits used when rounding sum to float are
// correct (when it matters).
fputil::FPBits<double> t(
(bit_prod.getUnbiasedExponent() >= bitz.getUnbiasedExponent())
? ((double(bit_sum) - double(bit_prod)) - double(bitz))
: ((double(bit_sum) - double(bitz)) - double(bit_prod)));
// Update sticky bits if t != 0.0 and the least (52 - 23 - 1 = 28) bits are
// zero.
if (!t.isZero() && ((bit_sum.getMantissa() & 0xfff'ffffULL) == 0)) {
if (bit_sum.getSign() != t.getSign()) {
bit_sum.setMantissa(bit_sum.getMantissa() + 1);
} else if (bit_sum.getMantissa()) {
bit_sum.setMantissa(bit_sum.getMantissa() - 1);
}
}
}
return static_cast<float>(static_cast<double>(bit_sum));
}
} // namespace generic
} // namespace fputil
} // namespace __llvm_libc
#endif // LLVM_LIBC_UTILS_FPUTIL_GENERIC_FMA_H
|
