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| 1 | //===-- Implementation of hypotf function ---------------------------------===// |
|---|---|
| 2 | // |
| 3 | // Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions. |
| 4 | // See https://llvm.org/LICENSE.txt for license information. |
| 5 | // SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception |
| 6 | // |
| 7 | //===----------------------------------------------------------------------===// |
| 8 | |
| 9 | #ifndef LLVM_LIBC_SRC___SUPPORT_FPUTIL_HYPOT_H |
| 10 | #define LLVM_LIBC_SRC___SUPPORT_FPUTIL_HYPOT_H |
| 11 | |
| 12 | #include "BasicOperations.h" |
| 13 | #include "FEnvImpl.h" |
| 14 | #include "FPBits.h" |
| 15 | #include "rounding_mode.h" |
| 16 | #include "src/__support/CPP/bit.h" |
| 17 | #include "src/__support/CPP/type_traits.h" |
| 18 | #include "src/__support/common.h" |
| 19 | #include "src/__support/macros/config.h" |
| 20 | #include "src/__support/uint128.h" |
| 21 | |
| 22 | namespace LIBC_NAMESPACE_DECL { |
| 23 | namespace fputil { |
| 24 | |
| 25 | namespace internal { |
| 26 | |
| 27 | template <typename T> |
| 28 | LIBC_INLINE T find_leading_one(T mant, int &shift_length) { |
| 29 | shift_length = 0; |
| 30 | if (mant > 0) { |
| 31 | shift_length = (sizeof(mant) * 8) - 1 - cpp::countl_zero(mant); |
| 32 | } |
| 33 | return static_cast<T>((T(1) << shift_length)); |
| 34 | } |
| 35 | |
| 36 | } // namespace internal |
| 37 | |
| 38 | template <typename T> struct DoubleLength; |
| 39 | |
| 40 | template <> struct DoubleLength<uint16_t> { |
| 41 | using Type = uint32_t; |
| 42 | }; |
| 43 | |
| 44 | template <> struct DoubleLength<uint32_t> { |
| 45 | using Type = uint64_t; |
| 46 | }; |
| 47 | |
| 48 | template <> struct DoubleLength<uint64_t> { |
| 49 | using Type = UInt128; |
| 50 | }; |
| 51 | |
| 52 | // Correctly rounded IEEE 754 HYPOT(x, y) with round to nearest, ties to even. |
| 53 | // |
| 54 | // Algorithm: |
| 55 | // - Let a = max(|x|, |y|), b = min(|x|, |y|), then we have that: |
| 56 | // a <= sqrt(a^2 + b^2) <= min(a + b, a*sqrt(2)) |
| 57 | // 1. So if b < eps(a)/2, then HYPOT(x, y) = a. |
| 58 | // |
| 59 | // - Moreover, the exponent part of HYPOT(x, y) is either the same or 1 more |
| 60 | // than the exponent part of a. |
| 61 | // |
| 62 | // 2. For the remaining cases, we will use the digit-by-digit (shift-and-add) |
| 63 | // algorithm to compute SQRT(Z): |
| 64 | // |
| 65 | // - For Y = y0.y1...yn... = SQRT(Z), |
| 66 | // let Y(n) = y0.y1...yn be the first n fractional digits of Y. |
| 67 | // |
| 68 | // - The nth scaled residual R(n) is defined to be: |
| 69 | // R(n) = 2^n * (Z - Y(n)^2) |
| 70 | // |
| 71 | // - Since Y(n) = Y(n - 1) + yn * 2^(-n), the scaled residual |
| 72 | // satisfies the following recurrence formula: |
| 73 | // R(n) = 2*R(n - 1) - yn*(2*Y(n - 1) + 2^(-n)), |
| 74 | // with the initial conditions: |
| 75 | // Y(0) = y0, and R(0) = Z - y0. |
| 76 | // |
| 77 | // - So the nth fractional digit of Y = SQRT(Z) can be decided by: |
| 78 | // yn = 1 if 2*R(n - 1) >= 2*Y(n - 1) + 2^(-n), |
| 79 | // 0 otherwise. |
| 80 | // |
| 81 | // 3. Precision analysis: |
| 82 | // |
| 83 | // - Notice that in the decision function: |
| 84 | // 2*R(n - 1) >= 2*Y(n - 1) + 2^(-n), |
| 85 | // the right hand side only uses up to the 2^(-n)-bit, and both sides are |
| 86 | // non-negative, so R(n - 1) can be truncated at the 2^(-(n + 1))-bit, so |
| 87 | // that 2*R(n - 1) is corrected up to the 2^(-n)-bit. |
| 88 | // |
| 89 | // - Thus, in order to round SQRT(a^2 + b^2) correctly up to n-fractional |
| 90 | // bits, we need to perform the summation (a^2 + b^2) correctly up to (2n + |
| 91 | // 2)-fractional bits, and the remaining bits are sticky bits (i.e. we only |
| 92 | // care if they are 0 or > 0), and the comparisons, additions/subtractions |
| 93 | // can be done in n-fractional bits precision. |
| 94 | // |
| 95 | // - For single precision (float), we can use uint64_t to store the sum a^2 + |
| 96 | // b^2 exact up to (2n + 2)-fractional bits. |
| 97 | // |
| 98 | // - Then we can feed this sum into the digit-by-digit algorithm for SQRT(Z) |
| 99 | // described above. |
| 100 | // |
| 101 | // |
| 102 | // Special cases: |
| 103 | // - HYPOT(x, y) is +Inf if x or y is +Inf or -Inf; else |
| 104 | // - HYPOT(x, y) is NaN if x or y is NaN. |
| 105 | // |
| 106 | template <typename T, cpp::enable_if_t<cpp::is_floating_point_v<T>, int> = 0> |
| 107 | LIBC_INLINE T hypot(T x, T y) { |
| 108 | using FPBits_t = FPBits<T>; |
| 109 | using StorageType = typename FPBits<T>::StorageType; |
| 110 | using DStorageType = typename DoubleLength<StorageType>::Type; |
| 111 | |
| 112 | FPBits_t x_abs = FPBits_t(x).abs(); |
| 113 | FPBits_t y_abs = FPBits_t(y).abs(); |
| 114 | |
| 115 | bool x_abs_larger = x_abs.uintval() >= y_abs.uintval(); |
| 116 | |
| 117 | FPBits_t a_bits = x_abs_larger ? x_abs : y_abs; |
| 118 | FPBits_t b_bits = x_abs_larger ? y_abs : x_abs; |
| 119 | |
| 120 | if (LIBC_UNLIKELY(a_bits.is_inf_or_nan())) { |
| 121 | if (x_abs.is_signaling_nan() || y_abs.is_signaling_nan()) { |
| 122 | fputil::raise_except_if_required(FE_INVALID); |
| 123 | return FPBits_t::quiet_nan().get_val(); |
| 124 | } |
| 125 | if (x_abs.is_inf() || y_abs.is_inf()) |
| 126 | return FPBits_t::inf().get_val(); |
| 127 | if (x_abs.is_nan()) |
| 128 | return x; |
| 129 | // y is nan |
| 130 | return y; |
| 131 | } |
| 132 | |
| 133 | uint16_t a_exp = a_bits.get_biased_exponent(); |
| 134 | uint16_t b_exp = b_bits.get_biased_exponent(); |
| 135 | |
| 136 | if ((a_exp - b_exp >= FPBits_t::FRACTION_LEN + 2) || (x == 0) || (y == 0)) |
| 137 | return x_abs.get_val() + y_abs.get_val(); |
| 138 | |
| 139 | uint64_t out_exp = a_exp; |
| 140 | StorageType a_mant = a_bits.get_mantissa(); |
| 141 | StorageType b_mant = b_bits.get_mantissa(); |
| 142 | DStorageType a_mant_sq, b_mant_sq; |
| 143 | bool sticky_bits; |
| 144 | |
| 145 | // Add an extra bit to simplify the final rounding bit computation. |
| 146 | constexpr StorageType ONE = StorageType(1) << (FPBits_t::FRACTION_LEN + 1); |
| 147 | |
| 148 | a_mant <<= 1; |
| 149 | b_mant <<= 1; |
| 150 | |
| 151 | StorageType leading_one; |
| 152 | int y_mant_width; |
| 153 | if (a_exp != 0) { |
| 154 | leading_one = ONE; |
| 155 | a_mant |= ONE; |
| 156 | y_mant_width = FPBits_t::FRACTION_LEN + 1; |
| 157 | } else { |
| 158 | leading_one = internal::find_leading_one(a_mant, y_mant_width); |
| 159 | a_exp = 1; |
| 160 | } |
| 161 | |
| 162 | if (b_exp != 0) |
| 163 | b_mant |= ONE; |
| 164 | else |
| 165 | b_exp = 1; |
| 166 | |
| 167 | a_mant_sq = static_cast<DStorageType>(a_mant) * a_mant; |
| 168 | b_mant_sq = static_cast<DStorageType>(b_mant) * b_mant; |
| 169 | |
| 170 | // At this point, a_exp >= b_exp > a_exp - 25, so in order to line up aSqMant |
| 171 | // and bSqMant, we need to shift bSqMant to the right by (a_exp - b_exp) bits. |
| 172 | // But before that, remember to store the losing bits to sticky. |
| 173 | // The shift length is for a^2 and b^2, so it's double of the exponent |
| 174 | // difference between a and b. |
| 175 | uint16_t shift_length = static_cast<uint16_t>(2 * (a_exp - b_exp)); |
| 176 | sticky_bits = |
| 177 | ((b_mant_sq & ((DStorageType(1) << shift_length) - DStorageType(1))) != |
| 178 | DStorageType(0)); |
| 179 | b_mant_sq >>= shift_length; |
| 180 | |
| 181 | DStorageType sum = a_mant_sq + b_mant_sq; |
| 182 | if (sum >= (DStorageType(1) << (2 * y_mant_width + 2))) { |
| 183 | // a^2 + b^2 >= 4* leading_one^2, so we will need an extra bit to the left. |
| 184 | if (leading_one == ONE) { |
| 185 | // For normal result, we discard the last 2 bits of the sum and increase |
| 186 | // the exponent. |
| 187 | sticky_bits = sticky_bits || ((sum & 0x3U) != 0); |
| 188 | sum >>= 2; |
| 189 | ++out_exp; |
| 190 | if (out_exp >= FPBits_t::MAX_BIASED_EXPONENT) { |
| 191 | if (int round_mode = quick_get_round(); |
| 192 | round_mode == FE_TONEAREST || round_mode == FE_UPWARD) |
| 193 | return FPBits_t::inf().get_val(); |
| 194 | return FPBits_t::max_normal().get_val(); |
| 195 | } |
| 196 | } else { |
| 197 | // For denormal result, we simply move the leading bit of the result to |
| 198 | // the left by 1. |
| 199 | leading_one <<= 1; |
| 200 | ++y_mant_width; |
| 201 | } |
| 202 | } |
| 203 | |
| 204 | StorageType y_new = leading_one; |
| 205 | StorageType r = static_cast<StorageType>(sum >> y_mant_width) - leading_one; |
| 206 | StorageType tail_bits = static_cast<StorageType>(sum) & (leading_one - 1); |
| 207 | |
| 208 | for (StorageType current_bit = leading_one >> 1; current_bit; |
| 209 | current_bit >>= 1) { |
| 210 | r = static_cast<StorageType>((r << 1)) + |
| 211 | ((tail_bits & current_bit) ? 1 : 0); |
| 212 | StorageType tmp = static_cast<StorageType>((y_new << 1)) + |
| 213 | current_bit; // 2*y_new(n - 1) + 2^(-n) |
| 214 | if (r >= tmp) { |
| 215 | r -= tmp; |
| 216 | y_new += current_bit; |
| 217 | } |
| 218 | } |
| 219 | |
| 220 | bool round_bit = y_new & StorageType(1); |
| 221 | bool lsb = y_new & StorageType(2); |
| 222 | |
| 223 | if (y_new >= ONE) { |
| 224 | y_new -= ONE; |
| 225 | |
| 226 | if (out_exp == 0) { |
| 227 | out_exp = 1; |
| 228 | } |
| 229 | } |
| 230 | |
| 231 | y_new >>= 1; |
| 232 | |
| 233 | // Round to the nearest, tie to even. |
| 234 | int round_mode = quick_get_round(); |
| 235 | switch (round_mode) { |
| 236 | case FE_TONEAREST: |
| 237 | // Round to nearest, ties to even |
| 238 | if (round_bit && (lsb || sticky_bits || (r != 0))) |
| 239 | ++y_new; |
| 240 | break; |
| 241 | case FE_UPWARD: |
| 242 | if (round_bit || sticky_bits || (r != 0)) |
| 243 | ++y_new; |
| 244 | break; |
| 245 | } |
| 246 | |
| 247 | if (y_new >= (ONE >> 1)) { |
| 248 | y_new -= ONE >> 1; |
| 249 | ++out_exp; |
| 250 | if (out_exp >= FPBits_t::MAX_BIASED_EXPONENT) { |
| 251 | if (round_mode == FE_TONEAREST || round_mode == FE_UPWARD) |
| 252 | return FPBits_t::inf().get_val(); |
| 253 | return FPBits_t::max_normal().get_val(); |
| 254 | } |
| 255 | } |
| 256 | |
| 257 | y_new |= static_cast<StorageType>(out_exp) << FPBits_t::FRACTION_LEN; |
| 258 | |
| 259 | if (!(round_bit || sticky_bits || (r != 0))) |
| 260 | fputil::clear_except_if_required(FE_INEXACT); |
| 261 | |
| 262 | return cpp::bit_cast<T>(y_new); |
| 263 | } |
| 264 | |
| 265 | } // namespace fputil |
| 266 | } // namespace LIBC_NAMESPACE_DECL |
| 267 | |
| 268 | #endif // LLVM_LIBC_SRC___SUPPORT_FPUTIL_HYPOT_H |
| 269 |
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