| //===-- Implementation of hypotf function ---------------------------------===// |
| // |
| // 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_SRC___SUPPORT_FPUTIL_HYPOT_H |
| #define LLVM_LIBC_SRC___SUPPORT_FPUTIL_HYPOT_H |
| |
| #include "BasicOperations.h" |
| #include "FEnvImpl.h" |
| #include "FPBits.h" |
| #include "rounding_mode.h" |
| #include "src/__support/CPP/bit.h" |
| #include "src/__support/CPP/type_traits.h" |
| #include "src/__support/common.h" |
| #include "src/__support/uint128.h" |
| |
| namespace LIBC_NAMESPACE { |
| namespace fputil { |
| |
| namespace internal { |
| |
| template <typename T> |
| LIBC_INLINE T find_leading_one(T mant, int &shift_length) { |
| shift_length = 0; |
| if (mant > 0) { |
| shift_length = (sizeof(mant) * 8) - 1 - cpp::countl_zero(mant); |
| } |
| return T(1) << shift_length; |
| } |
| |
| } // namespace internal |
| |
| template <typename T> struct DoubleLength; |
| |
| template <> struct DoubleLength<uint16_t> { |
| using Type = uint32_t; |
| }; |
| |
| template <> struct DoubleLength<uint32_t> { |
| using Type = uint64_t; |
| }; |
| |
| template <> struct DoubleLength<uint64_t> { |
| using Type = UInt128; |
| }; |
| |
| // Correctly rounded IEEE 754 HYPOT(x, y) with round to nearest, ties to even. |
| // |
| // Algorithm: |
| // - Let a = max(|x|, |y|), b = min(|x|, |y|), then we have that: |
| // a <= sqrt(a^2 + b^2) <= min(a + b, a*sqrt(2)) |
| // 1. So if b < eps(a)/2, then HYPOT(x, y) = a. |
| // |
| // - Moreover, the exponent part of HYPOT(x, y) is either the same or 1 more |
| // than the exponent part of a. |
| // |
| // 2. For the remaining cases, we will use the digit-by-digit (shift-and-add) |
| // algorithm to compute SQRT(Z): |
| // |
| // - For Y = y0.y1...yn... = SQRT(Z), |
| // let Y(n) = y0.y1...yn be the first n fractional digits of Y. |
| // |
| // - The nth scaled residual R(n) is defined to be: |
| // R(n) = 2^n * (Z - Y(n)^2) |
| // |
| // - Since Y(n) = Y(n - 1) + yn * 2^(-n), the scaled residual |
| // satisfies the following recurrence formula: |
| // R(n) = 2*R(n - 1) - yn*(2*Y(n - 1) + 2^(-n)), |
| // with the initial conditions: |
| // Y(0) = y0, and R(0) = Z - y0. |
| // |
| // - So the nth fractional digit of Y = SQRT(Z) can be decided by: |
| // yn = 1 if 2*R(n - 1) >= 2*Y(n - 1) + 2^(-n), |
| // 0 otherwise. |
| // |
| // 3. Precision analysis: |
| // |
| // - Notice that in the decision function: |
| // 2*R(n - 1) >= 2*Y(n - 1) + 2^(-n), |
| // the right hand side only uses up to the 2^(-n)-bit, and both sides are |
| // non-negative, so R(n - 1) can be truncated at the 2^(-(n + 1))-bit, so |
| // that 2*R(n - 1) is corrected up to the 2^(-n)-bit. |
| // |
| // - Thus, in order to round SQRT(a^2 + b^2) correctly up to n-fractional |
| // bits, we need to perform the summation (a^2 + b^2) correctly up to (2n + |
| // 2)-fractional bits, and the remaining bits are sticky bits (i.e. we only |
| // care if they are 0 or > 0), and the comparisons, additions/subtractions |
| // can be done in n-fractional bits precision. |
| // |
| // - For single precision (float), we can use uint64_t to store the sum a^2 + |
| // b^2 exact up to (2n + 2)-fractional bits. |
| // |
| // - Then we can feed this sum into the digit-by-digit algorithm for SQRT(Z) |
| // described above. |
| // |
| // |
| // Special cases: |
| // - HYPOT(x, y) is +Inf if x or y is +Inf or -Inf; else |
| // - HYPOT(x, y) is NaN if x or y is NaN. |
| // |
| template <typename T, cpp::enable_if_t<cpp::is_floating_point_v<T>, int> = 0> |
| LIBC_INLINE T hypot(T x, T y) { |
| using FPBits_t = FPBits<T>; |
| using StorageType = typename FPBits<T>::StorageType; |
| using DStorageType = typename DoubleLength<StorageType>::Type; |
| |
| FPBits_t x_bits(x), y_bits(y); |
| |
| if (x_bits.is_inf() || y_bits.is_inf()) { |
| return FPBits_t::inf().get_val(); |
| } |
| if (x_bits.is_nan()) { |
| return x; |
| } |
| if (y_bits.is_nan()) { |
| return y; |
| } |
| |
| uint16_t x_exp = x_bits.get_biased_exponent(); |
| uint16_t y_exp = y_bits.get_biased_exponent(); |
| uint16_t exp_diff = (x_exp > y_exp) ? (x_exp - y_exp) : (y_exp - x_exp); |
| |
| if ((exp_diff >= FPBits_t::FRACTION_LEN + 2) || (x == 0) || (y == 0)) { |
| return abs(x) + abs(y); |
| } |
| |
| uint16_t a_exp, b_exp, out_exp; |
| StorageType a_mant, b_mant; |
| DStorageType a_mant_sq, b_mant_sq; |
| bool sticky_bits; |
| |
| if (abs(x) >= abs(y)) { |
| a_exp = x_exp; |
| a_mant = x_bits.get_mantissa(); |
| b_exp = y_exp; |
| b_mant = y_bits.get_mantissa(); |
| } else { |
| a_exp = y_exp; |
| a_mant = y_bits.get_mantissa(); |
| b_exp = x_exp; |
| b_mant = x_bits.get_mantissa(); |
| } |
| |
| out_exp = a_exp; |
| |
| // Add an extra bit to simplify the final rounding bit computation. |
| constexpr StorageType ONE = StorageType(1) << (FPBits_t::FRACTION_LEN + 1); |
| |
| a_mant <<= 1; |
| b_mant <<= 1; |
| |
| StorageType leading_one; |
| int y_mant_width; |
| if (a_exp != 0) { |
| leading_one = ONE; |
| a_mant |= ONE; |
| y_mant_width = FPBits_t::FRACTION_LEN + 1; |
| } else { |
| leading_one = internal::find_leading_one(a_mant, y_mant_width); |
| a_exp = 1; |
| } |
| |
| if (b_exp != 0) { |
| b_mant |= ONE; |
| } else { |
| b_exp = 1; |
| } |
| |
| a_mant_sq = static_cast<DStorageType>(a_mant) * a_mant; |
| b_mant_sq = static_cast<DStorageType>(b_mant) * b_mant; |
| |
| // At this point, a_exp >= b_exp > a_exp - 25, so in order to line up aSqMant |
| // and bSqMant, we need to shift bSqMant to the right by (a_exp - b_exp) bits. |
| // But before that, remember to store the losing bits to sticky. |
| // The shift length is for a^2 and b^2, so it's double of the exponent |
| // difference between a and b. |
| uint16_t shift_length = static_cast<uint16_t>(2 * (a_exp - b_exp)); |
| sticky_bits = |
| ((b_mant_sq & ((DStorageType(1) << shift_length) - DStorageType(1))) != |
| DStorageType(0)); |
| b_mant_sq >>= shift_length; |
| |
| DStorageType sum = a_mant_sq + b_mant_sq; |
| if (sum >= (DStorageType(1) << (2 * y_mant_width + 2))) { |
| // a^2 + b^2 >= 4* leading_one^2, so we will need an extra bit to the left. |
| if (leading_one == ONE) { |
| // For normal result, we discard the last 2 bits of the sum and increase |
| // the exponent. |
| sticky_bits = sticky_bits || ((sum & 0x3U) != 0); |
| sum >>= 2; |
| ++out_exp; |
| if (out_exp >= FPBits_t::MAX_BIASED_EXPONENT) { |
| if (int round_mode = quick_get_round(); |
| round_mode == FE_TONEAREST || round_mode == FE_UPWARD) |
| return FPBits_t::inf().get_val(); |
| return FPBits_t::max_normal().get_val(); |
| } |
| } else { |
| // For denormal result, we simply move the leading bit of the result to |
| // the left by 1. |
| leading_one <<= 1; |
| ++y_mant_width; |
| } |
| } |
| |
| StorageType y_new = leading_one; |
| StorageType r = static_cast<StorageType>(sum >> y_mant_width) - leading_one; |
| StorageType tail_bits = static_cast<StorageType>(sum) & (leading_one - 1); |
| |
| for (StorageType current_bit = leading_one >> 1; current_bit; |
| current_bit >>= 1) { |
| r = (r << 1) + ((tail_bits & current_bit) ? 1 : 0); |
| StorageType tmp = (y_new << 1) + current_bit; // 2*y_new(n - 1) + 2^(-n) |
| if (r >= tmp) { |
| r -= tmp; |
| y_new += current_bit; |
| } |
| } |
| |
| bool round_bit = y_new & StorageType(1); |
| bool lsb = y_new & StorageType(2); |
| |
| if (y_new >= ONE) { |
| y_new -= ONE; |
| |
| if (out_exp == 0) { |
| out_exp = 1; |
| } |
| } |
| |
| y_new >>= 1; |
| |
| // Round to the nearest, tie to even. |
| int round_mode = quick_get_round(); |
| switch (round_mode) { |
| case FE_TONEAREST: |
| // Round to nearest, ties to even |
| if (round_bit && (lsb || sticky_bits || (r != 0))) |
| ++y_new; |
| break; |
| case FE_UPWARD: |
| if (round_bit || sticky_bits || (r != 0)) |
| ++y_new; |
| break; |
| } |
| |
| if (y_new >= (ONE >> 1)) { |
| y_new -= ONE >> 1; |
| ++out_exp; |
| if (out_exp >= FPBits_t::MAX_BIASED_EXPONENT) { |
| if (round_mode == FE_TONEAREST || round_mode == FE_UPWARD) |
| return FPBits_t::inf().get_val(); |
| return FPBits_t::max_normal().get_val(); |
| } |
| } |
| |
| y_new |= static_cast<StorageType>(out_exp) << FPBits_t::FRACTION_LEN; |
| return cpp::bit_cast<T>(y_new); |
| } |
| |
| } // namespace fputil |
| } // namespace LIBC_NAMESPACE |
| |
| #endif // LLVM_LIBC_SRC___SUPPORT_FPUTIL_HYPOT_H |