| //===-- 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 "FPBits.h" |
| #include "utils/CPP/TypeTraits.h" |
| |
| namespace __llvm_libc { |
| namespace fputil { |
| |
| namespace internal { |
| |
| template <typename T> static inline T findLeadingOne(T mant, int &shift_length); |
| |
| template <> |
| inline uint32_t findLeadingOne<uint32_t>(uint32_t mant, int &shift_length) { |
| shift_length = 0; |
| constexpr int nsteps = 5; |
| constexpr uint32_t bounds[nsteps] = {1 << 16, 1 << 8, 1 << 4, 1 << 2, 1 << 1}; |
| constexpr int shifts[nsteps] = {16, 8, 4, 2, 1}; |
| for (int i = 0; i < nsteps; ++i) { |
| if (mant >= bounds[i]) { |
| shift_length += shifts[i]; |
| mant >>= shifts[i]; |
| } |
| } |
| return 1U << shift_length; |
| } |
| |
| template <> |
| inline uint64_t findLeadingOne<uint64_t>(uint64_t mant, int &shift_length) { |
| shift_length = 0; |
| constexpr int nsteps = 6; |
| constexpr uint64_t bounds[nsteps] = {1ULL << 32, 1ULL << 16, 1ULL << 8, |
| 1ULL << 4, 1ULL << 2, 1ULL << 1}; |
| constexpr int shifts[nsteps] = {32, 16, 8, 4, 2, 1}; |
| for (int i = 0; i < nsteps; ++i) { |
| if (mant >= bounds[i]) { |
| shift_length += shifts[i]; |
| mant >>= shifts[i]; |
| } |
| } |
| return 1ULL << 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_t; }; |
| |
| // 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::EnableIfType<cpp::IsFloatingPointType<T>::Value, int> = 0> |
| static inline T hypot(T x, T y) { |
| using FPBits_t = FPBits<T>; |
| using UIntType = typename FPBits<T>::UIntType; |
| using DUIntType = typename DoubleLength<UIntType>::Type; |
| |
| FPBits_t x_bits(x), y_bits(y); |
| |
| if (x_bits.isInf() || y_bits.isInf()) { |
| return T(FPBits_t::inf()); |
| } |
| if (x_bits.isNaN()) { |
| return x; |
| } |
| if (y_bits.isNaN()) { |
| return y; |
| } |
| |
| uint16_t a_exp, b_exp, out_exp; |
| UIntType a_mant, b_mant; |
| DUIntType a_mant_sq, b_mant_sq; |
| bool sticky_bits; |
| |
| if ((x_bits.getUnbiasedExponent() >= |
| y_bits.getUnbiasedExponent() + MantissaWidth<T>::value + 2) || |
| (y == 0)) { |
| return abs(x); |
| } else if ((y_bits.getUnbiasedExponent() >= |
| x_bits.getUnbiasedExponent() + MantissaWidth<T>::value + 2) || |
| (x == 0)) { |
| y_bits.setSign(0); |
| return abs(y); |
| } |
| |
| if (x >= y) { |
| a_exp = x_bits.getUnbiasedExponent(); |
| a_mant = x_bits.getMantissa(); |
| b_exp = y_bits.getUnbiasedExponent(); |
| b_mant = y_bits.getMantissa(); |
| } else { |
| a_exp = y_bits.getUnbiasedExponent(); |
| a_mant = y_bits.getMantissa(); |
| b_exp = x_bits.getUnbiasedExponent(); |
| b_mant = x_bits.getMantissa(); |
| } |
| |
| out_exp = a_exp; |
| |
| // Add an extra bit to simplify the final rounding bit computation. |
| constexpr UIntType one = UIntType(1) << (MantissaWidth<T>::value + 1); |
| |
| a_mant <<= 1; |
| b_mant <<= 1; |
| |
| UIntType leading_one; |
| int y_mant_width; |
| if (a_exp != 0) { |
| leading_one = one; |
| a_mant |= one; |
| y_mant_width = MantissaWidth<T>::value + 1; |
| } else { |
| leading_one = internal::findLeadingOne(a_mant, y_mant_width); |
| } |
| |
| if (b_exp != 0) { |
| b_mant |= one; |
| } |
| |
| a_mant_sq = static_cast<DUIntType>(a_mant) * a_mant; |
| b_mant_sq = static_cast<DUIntType>(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 = 2 * (a_exp - b_exp); |
| sticky_bits = |
| ((b_mant_sq & ((DUIntType(1) << shift_length) - DUIntType(1))) != |
| DUIntType(0)); |
| b_mant_sq >>= shift_length; |
| |
| DUIntType sum = a_mant_sq + b_mant_sq; |
| if (sum >= (DUIntType(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::maxExponent) { |
| return T(FPBits_t::inf()); |
| } |
| } else { |
| // For denormal result, we simply move the leading bit of the result to |
| // the left by 1. |
| leading_one <<= 1; |
| ++y_mant_width; |
| } |
| } |
| |
| UIntType Y = leading_one; |
| UIntType R = static_cast<UIntType>(sum >> y_mant_width) - leading_one; |
| UIntType tailBits = static_cast<UIntType>(sum) & (leading_one - 1); |
| |
| for (UIntType current_bit = leading_one >> 1; current_bit; |
| current_bit >>= 1) { |
| R = (R << 1) + ((tailBits & current_bit) ? 1 : 0); |
| UIntType tmp = (Y << 1) + current_bit; // 2*y(n - 1) + 2^(-n) |
| if (R >= tmp) { |
| R -= tmp; |
| Y += current_bit; |
| } |
| } |
| |
| bool round_bit = Y & UIntType(1); |
| bool lsb = Y & UIntType(2); |
| |
| if (Y >= one) { |
| Y -= one; |
| |
| if (out_exp == 0) { |
| out_exp = 1; |
| } |
| } |
| |
| Y >>= 1; |
| |
| // Round to the nearest, tie to even. |
| if (round_bit && (lsb || sticky_bits || (R != 0))) { |
| ++Y; |
| } |
| |
| if (Y >= (one >> 1)) { |
| Y -= one >> 1; |
| ++out_exp; |
| if (out_exp >= FPBits_t::maxExponent) { |
| return T(FPBits_t::inf()); |
| } |
| } |
| |
| Y |= static_cast<UIntType>(out_exp) << MantissaWidth<T>::value; |
| return *reinterpret_cast<T *>(&Y); |
| } |
| |
| } // namespace fputil |
| } // namespace __llvm_libc |
| |
| #endif // LLVM_LIBC_SRC_SUPPORT_FPUTIL_HYPOT_H |