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 //===-- llvm/Support/MathExtras.h - Useful math functions -------*- 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
 //
 //===----------------------------------------------------------------------===//
 //
 // This file contains some functions that are useful for math stuff.
 //
 //===----------------------------------------------------------------------===//
 
 #ifndef LLVM_SUPPORT_MATHEXTRAS_H
 #define LLVM_SUPPORT_MATHEXTRAS_H
 
 #include "llvm/ADT/bit.h"
 #include "llvm/Support/Compiler.h"
 #include <cassert>
 #include <climits>
 #include <cstdint>
 #include <cstring>
 #include <limits>
 #include <type_traits>
 
 namespace llvm {
 /// Some template parameter helpers to optimize for bitwidth, for functions that
 /// take multiple arguments.
 
 // We can't verify signedness, since callers rely on implicit coercions to
 // signed/unsigned.
 template <typename T, typename U>
 using enableif_int =
     std::enable_if_t<std::is_integral_v<T> && std::is_integral_v<U>>;
 
 // Use std::common_type_t to widen only up to the widest argument.
 template <typename T, typename U, typename = enableif_int<T, U>>
 using common_uint =
     std::common_type_t<std::make_unsigned_t<T>, std::make_unsigned_t<U>>;
 template <typename T, typename U, typename = enableif_int<T, U>>
 using common_sint =
     std::common_type_t<std::make_signed_t<T>, std::make_signed_t<U>>;
 
 /// Mathematical constants.
 namespace numbers {
 // TODO: Track C++20 std::numbers.
 // clang-format off
 constexpr double e          = 0x1.5bf0a8b145769P+1, // (2.7182818284590452354) https://oeis.org/A001113
                  egamma     = 0x1.2788cfc6fb619P-1, // (.57721566490153286061) https://oeis.org/A001620
                  ln2        = 0x1.62e42fefa39efP-1, // (.69314718055994530942) https://oeis.org/A002162
                  ln10       = 0x1.26bb1bbb55516P+1, // (2.3025850929940456840) https://oeis.org/A002392
                  log2e      = 0x1.71547652b82feP+0, // (1.4426950408889634074)
                  log10e     = 0x1.bcb7b1526e50eP-2, // (.43429448190325182765)
                  pi         = 0x1.921fb54442d18P+1, // (3.1415926535897932385) https://oeis.org/A000796
                  inv_pi     = 0x1.45f306dc9c883P-2, // (.31830988618379067154) https://oeis.org/A049541
                  sqrtpi     = 0x1.c5bf891b4ef6bP+0, // (1.7724538509055160273) https://oeis.org/A002161
                  inv_sqrtpi = 0x1.20dd750429b6dP-1, // (.56418958354775628695) https://oeis.org/A087197
                  sqrt2      = 0x1.6a09e667f3bcdP+0, // (1.4142135623730950488) https://oeis.org/A00219
                  inv_sqrt2  = 0x1.6a09e667f3bcdP-1, // (.70710678118654752440)
                  sqrt3      = 0x1.bb67ae8584caaP+0, // (1.7320508075688772935) https://oeis.org/A002194
                  inv_sqrt3  = 0x1.279a74590331cP-1, // (.57735026918962576451)
                  phi        = 0x1.9e3779b97f4a8P+0; // (1.6180339887498948482) https://oeis.org/A001622
 constexpr float ef          = 0x1.5bf0a8P+1F, // (2.71828183) https://oeis.org/A001113
                 egammaf     = 0x1.2788d0P-1F, // (.577215665) https://oeis.org/A001620
                 ln2f        = 0x1.62e430P-1F, // (.693147181) https://oeis.org/A002162
                 ln10f       = 0x1.26bb1cP+1F, // (2.30258509) https://oeis.org/A002392
                 log2ef      = 0x1.715476P+0F, // (1.44269504)
                 log10ef     = 0x1.bcb7b2P-2F, // (.434294482)
                 pif         = 0x1.921fb6P+1F, // (3.14159265) https://oeis.org/A000796
                 inv_pif     = 0x1.45f306P-2F, // (.318309886) https://oeis.org/A049541
                 sqrtpif     = 0x1.c5bf8aP+0F, // (1.77245385) https://oeis.org/A002161
                 inv_sqrtpif = 0x1.20dd76P-1F, // (.564189584) https://oeis.org/A087197
                 sqrt2f      = 0x1.6a09e6P+0F, // (1.41421356) https://oeis.org/A002193
                 inv_sqrt2f  = 0x1.6a09e6P-1F, // (.707106781)
                 sqrt3f      = 0x1.bb67aeP+0F, // (1.73205081) https://oeis.org/A002194
                 inv_sqrt3f  = 0x1.279a74P-1F, // (.577350269)
                 phif        = 0x1.9e377aP+0F; // (1.61803399) https://oeis.org/A001622
 // clang-format on
 } // namespace numbers
 
 /// Create a bitmask with the N right-most bits set to 1, and all other
 /// bits set to 0.  Only unsigned types are allowed.
 template <typename T> T maskTrailingOnes(unsigned N) {
   static_assert(std::is_unsigned_v<T>, "Invalid type!");
   const unsigned Bits = CHAR_BIT * sizeof(T);
   assert(N <= Bits && "Invalid bit index");
   if (N == 0)
     return 0;
   return T(-1) >> (Bits - N);
 }
 
 /// Create a bitmask with the N left-most bits set to 1, and all other
 /// bits set to 0.  Only unsigned types are allowed.
 template <typename T> T maskLeadingOnes(unsigned N) {
   return ~maskTrailingOnes<T>(CHAR_BIT * sizeof(T) - N);
 }
 
 /// Create a bitmask with the N right-most bits set to 0, and all other
 /// bits set to 1.  Only unsigned types are allowed.
 template <typename T> T maskTrailingZeros(unsigned N) {
   return maskLeadingOnes<T>(CHAR_BIT * sizeof(T) - N);
 }
 
 /// Create a bitmask with the N left-most bits set to 0, and all other
 /// bits set to 1.  Only unsigned types are allowed.
 template <typename T> T maskLeadingZeros(unsigned N) {
   return maskTrailingOnes<T>(CHAR_BIT * sizeof(T) - N);
 }
 
 /// Macro compressed bit reversal table for 256 bits.
 ///
 /// http://graphics.stanford.edu/~seander/bithacks.html#BitReverseTable
 static const unsigned char BitReverseTable256[256] = {
 #define R2(n) n, n + 2 * 64, n + 1 * 64, n + 3 * 64
 #define R4(n) R2(n), R2(n + 2 * 16), R2(n + 1 * 16), R2(n + 3 * 16)
 #define R6(n) R4(n), R4(n + 2 * 4), R4(n + 1 * 4), R4(n + 3 * 4)
   R6(0), R6(2), R6(1), R6(3)
 #undef R2
 #undef R4
 #undef R6
 };
 
 /// Reverse the bits in \p Val.
 template <typename T> T reverseBits(T Val) {
 #if __has_builtin(__builtin_bitreverse8)
   if constexpr (std::is_same_v<T, uint8_t>)
     return __builtin_bitreverse8(Val);
 #endif
 #if __has_builtin(__builtin_bitreverse16)
   if constexpr (std::is_same_v<T, uint16_t>)
     return __builtin_bitreverse16(Val);
 #endif
 #if __has_builtin(__builtin_bitreverse32)
   if constexpr (std::is_same_v<T, uint32_t>)
     return __builtin_bitreverse32(Val);
 #endif
 #if __has_builtin(__builtin_bitreverse64)
   if constexpr (std::is_same_v<T, uint64_t>)
     return __builtin_bitreverse64(Val);
 #endif
 
   unsigned char in[sizeof(Val)];
   unsigned char out[sizeof(Val)];
   std::memcpy(in, &Val, sizeof(Val));
   for (unsigned i = 0; i < sizeof(Val); ++i)
     out[(sizeof(Val) - i) - 1] = BitReverseTable256[in[i]];
   std::memcpy(&Val, out, sizeof(Val));
   return Val;
 }
 
 // NOTE: The following support functions use the _32/_64 extensions instead of
 // type overloading so that signed and unsigned integers can be used without
 // ambiguity.
 
 /// Return the high 32 bits of a 64 bit value.
 constexpr uint32_t Hi_32(uint64_t Value) {
   return static_cast<uint32_t>(Value >> 32);
 }
 
 /// Return the low 32 bits of a 64 bit value.
 constexpr uint32_t Lo_32(uint64_t Value) {
   return static_cast<uint32_t>(Value);
 }
 
 /// Make a 64-bit integer from a high / low pair of 32-bit integers.
 constexpr uint64_t Make_64(uint32_t High, uint32_t Low) {
   return ((uint64_t)High << 32) | (uint64_t)Low;
 }
 
 /// Checks if an integer fits into the given bit width.
 template <unsigned N> constexpr bool isInt(int64_t x) {
   if constexpr (N == 0)
     return 0 == x;
   if constexpr (N == 8)
     return static_cast<int8_t>(x) == x;
   if constexpr (N == 16)
     return static_cast<int16_t>(x) == x;
   if constexpr (N == 32)
     return static_cast<int32_t>(x) == x;
   if constexpr (N < 64)
     return -(INT64_C(1) << (N - 1)) <= x && x < (INT64_C(1) << (N - 1));
   (void)x; // MSVC v19.25 warns that x is unused.
   return true;
 }
 
 /// Checks if a signed integer is an N bit number shifted left by S.
 template <unsigned N, unsigned S>
 constexpr bool isShiftedInt(int64_t x) {
   static_assert(S < 64, "isShiftedInt<N, S> with S >= 64 is too much.");
   static_assert(N + S <= 64, "isShiftedInt<N, S> with N + S > 64 is too wide.");
   return isInt<N + S>(x) && (x % (UINT64_C(1) << S) == 0);
 }
 
 /// Checks if an unsigned integer fits into the given bit width.
 template <unsigned N> constexpr bool isUInt(uint64_t x) {
   if constexpr (N == 0)
     return 0 == x;
   if constexpr (N == 8)
     return static_cast<uint8_t>(x) == x;
   if constexpr (N == 16)
     return static_cast<uint16_t>(x) == x;
   if constexpr (N == 32)
     return static_cast<uint32_t>(x) == x;
   if constexpr (N < 64)
     return x < (UINT64_C(1) << (N));
   (void)x; // MSVC v19.25 warns that x is unused.
   return true;
 }
 
 /// Checks if a unsigned integer is an N bit number shifted left by S.
 template <unsigned N, unsigned S>
 constexpr bool isShiftedUInt(uint64_t x) {
   static_assert(S < 64, "isShiftedUInt<N, S> with S >= 64 is too much.");
   static_assert(N + S <= 64,
                 "isShiftedUInt<N, S> with N + S > 64 is too wide.");
   // S must be strictly less than 64. So 1 << S is not undefined behavior.
   return isUInt<N + S>(x) && (x % (UINT64_C(1) << S) == 0);
 }
 
 /// Gets the maximum value for a N-bit unsigned integer.
 inline uint64_t maxUIntN(uint64_t N) {
   assert(N <= 64 && "integer width out of range");
 
   // uint64_t(1) << 64 is undefined behavior, so we can't do
   //   (uint64_t(1) << N) - 1
   // without checking first that N != 64.  But this works and doesn't have a
   // branch for N != 0.
   // Unfortunately, shifting a uint64_t right by 64 bit is undefined
   // behavior, so the condition on N == 0 is necessary. Fortunately, most
   // optimizers do not emit branches for this check.
   if (N == 0)
     return 0;
   return UINT64_MAX >> (64 - N);
 }
 
 /// Gets the minimum value for a N-bit signed integer.
 inline int64_t minIntN(int64_t N) {
   assert(N <= 64 && "integer width out of range");
 
   if (N == 0)
     return 0;
   return UINT64_C(1) + ~(UINT64_C(1) << (N - 1));
 }
 
 /// Gets the maximum value for a N-bit signed integer.
 inline int64_t maxIntN(int64_t N) {
   assert(N <= 64 && "integer width out of range");
 
   // This relies on two's complement wraparound when N == 64, so we convert to
   // int64_t only at the very end to avoid UB.
   if (N == 0)
     return 0;
   return (UINT64_C(1) << (N - 1)) - 1;
 }
 
 /// Checks if an unsigned integer fits into the given (dynamic) bit width.
 inline bool isUIntN(unsigned N, uint64_t x) {
   return N >= 64 || x <= maxUIntN(N);
 }
 
 /// Checks if an signed integer fits into the given (dynamic) bit width.
 inline bool isIntN(unsigned N, int64_t x) {
   return N >= 64 || (minIntN(N) <= x && x <= maxIntN(N));
 }
 
 /// Return true if the argument is a non-empty sequence of ones starting at the
 /// least significant bit with the remainder zero (32 bit version).
 /// Ex. isMask_32(0x0000FFFFU) == true.
 constexpr bool isMask_32(uint32_t Value) {
   return Value && ((Value + 1) & Value) == 0;
 }
 
 /// Return true if the argument is a non-empty sequence of ones starting at the
 /// least significant bit with the remainder zero (64 bit version).
 constexpr bool isMask_64(uint64_t Value) {
   return Value && ((Value + 1) & Value) == 0;
 }
 
 /// Return true if the argument contains a non-empty sequence of ones with the
 /// remainder zero (32 bit version.) Ex. isShiftedMask_32(0x0000FF00U) == true.
 constexpr bool isShiftedMask_32(uint32_t Value) {
   return Value && isMask_32((Value - 1) | Value);
 }
 
 /// Return true if the argument contains a non-empty sequence of ones with the
 /// remainder zero (64 bit version.)
 constexpr bool isShiftedMask_64(uint64_t Value) {
   return Value && isMask_64((Value - 1) | Value);
 }
 
 /// Return true if the argument is a power of two > 0.
 /// Ex. isPowerOf2_32(0x00100000U) == true (32 bit edition.)
 constexpr bool isPowerOf2_32(uint32_t Value) {
   return llvm::has_single_bit(Value);
 }
 
 /// Return true if the argument is a power of two > 0 (64 bit edition.)
 constexpr bool isPowerOf2_64(uint64_t Value) {
   return llvm::has_single_bit(Value);
 }
 
 /// Return true if the argument contains a non-empty sequence of ones with the
 /// remainder zero (32 bit version.) Ex. isShiftedMask_32(0x0000FF00U) == true.
 /// If true, \p MaskIdx will specify the index of the lowest set bit and \p
 /// MaskLen is updated to specify the length of the mask, else neither are
 /// updated.
 inline bool isShiftedMask_32(uint32_t Value, unsigned &MaskIdx,
                              unsigned &MaskLen) {
   if (!isShiftedMask_32(Value))
     return false;
   MaskIdx = llvm::countr_zero(Value);
   MaskLen = llvm::popcount(Value);
   return true;
 }
 
 /// Return true if the argument contains a non-empty sequence of ones with the
 /// remainder zero (64 bit version.) If true, \p MaskIdx will specify the index
 /// of the lowest set bit and \p MaskLen is updated to specify the length of the
 /// mask, else neither are updated.
 inline bool isShiftedMask_64(uint64_t Value, unsigned &MaskIdx,
                              unsigned &MaskLen) {
   if (!isShiftedMask_64(Value))
     return false;
   MaskIdx = llvm::countr_zero(Value);
   MaskLen = llvm::popcount(Value);
   return true;
 }
 
 /// Compile time Log2.
 /// Valid only for positive powers of two.
 template <size_t kValue> constexpr size_t CTLog2() {
   static_assert(kValue > 0 && llvm::isPowerOf2_64(kValue),
                 "Value is not a valid power of 2");
   return 1 + CTLog2<kValue / 2>();
 }
 
 template <> constexpr size_t CTLog2<1>() { return 0; }
 
 /// Return the floor log base 2 of the specified value, -1 if the value is zero.
 /// (32 bit edition.)
 /// Ex. Log2_32(32) == 5, Log2_32(1) == 0, Log2_32(0) == -1, Log2_32(6) == 2
 inline unsigned Log2_32(uint32_t Value) {
   return 31 - llvm::countl_zero(Value);
 }
 
 /// Return the floor log base 2 of the specified value, -1 if the value is zero.
 /// (64 bit edition.)
 inline unsigned Log2_64(uint64_t Value) {
   return 63 - llvm::countl_zero(Value);
 }
 
 /// Return the ceil log base 2 of the specified value, 32 if the value is zero.
 /// (32 bit edition).
 /// Ex. Log2_32_Ceil(32) == 5, Log2_32_Ceil(1) == 0, Log2_32_Ceil(6) == 3
 inline unsigned Log2_32_Ceil(uint32_t Value) {
   return 32 - llvm::countl_zero(Value - 1);
 }
 
 /// Return the ceil log base 2 of the specified value, 64 if the value is zero.
 /// (64 bit edition.)
 inline unsigned Log2_64_Ceil(uint64_t Value) {
   return 64 - llvm::countl_zero(Value - 1);
 }
 
 /// A and B are either alignments or offsets. Return the minimum alignment that
 /// may be assumed after adding the two together.
 template <typename U, typename V, typename T = common_uint<U, V>>
 constexpr T MinAlign(U A, V B) {
   // The largest power of 2 that divides both A and B.
   //
   // Replace "-Value" by "1+~Value" in the following commented code to avoid
   // MSVC warning C4146
   //    return (A | B) & -(A | B);
   return (A | B) & (1 + ~(A | B));
 }
 
 /// Fallback when arguments aren't integral.
 constexpr uint64_t MinAlign(uint64_t A, uint64_t B) {
   return (A | B) & (1 + ~(A | B));
 }
 
 /// Returns the next power of two (in 64-bits) that is strictly greater than A.
 /// Returns zero on overflow.
 constexpr uint64_t NextPowerOf2(uint64_t A) {
   A |= (A >> 1);
   A |= (A >> 2);
   A |= (A >> 4);
   A |= (A >> 8);
   A |= (A >> 16);
   A |= (A >> 32);
   return A + 1;
 }
 
 /// Returns the power of two which is greater than or equal to the given value.
 /// Essentially, it is a ceil operation across the domain of powers of two.
 inline uint64_t PowerOf2Ceil(uint64_t A) {
   if (!A || A > UINT64_MAX / 2)
     return 0;
   return UINT64_C(1) << Log2_64_Ceil(A);
 }
 
 /// Returns the integer ceil(Numerator / Denominator). Unsigned version.
 /// Guaranteed to never overflow.
 template <typename U, typename V, typename T = common_uint<U, V>>
 constexpr T divideCeil(U Numerator, V Denominator) {
   assert(Denominator && "Division by zero");
   T Bias = (Numerator != 0);
   return (Numerator - Bias) / Denominator + Bias;
 }
 
 /// Fallback when arguments aren't integral.
 constexpr uint64_t divideCeil(uint64_t Numerator, uint64_t Denominator) {
   assert(Denominator && "Division by zero");
   uint64_t Bias = (Numerator != 0);
   return (Numerator - Bias) / Denominator + Bias;
 }
 
 // Check whether divideCeilSigned or divideFloorSigned would overflow. This
 // happens only when Numerator = INT_MIN and Denominator = -1.
 template <typename U, typename V>
 constexpr bool divideSignedWouldOverflow(U Numerator, V Denominator) {
   return Numerator == std::numeric_limits<U>::min() && Denominator == -1;
 }
 
 /// Returns the integer ceil(Numerator / Denominator). Signed version.
 /// Overflow is explicitly forbidden with an assert.
 template <typename U, typename V, typename T = common_sint<U, V>>
 constexpr T divideCeilSigned(U Numerator, V Denominator) {
   assert(Denominator && "Division by zero");
   assert(!divideSignedWouldOverflow(Numerator, Denominator) &&
          "Divide would overflow");
   if (!Numerator)
     return 0;
   // C's integer division rounds towards 0.
   T Bias = Denominator >= 0 ? 1 : -1;
   bool SameSign = (Numerator >= 0) == (Denominator >= 0);
   return SameSign ? (Numerator - Bias) / Denominator + 1
                   : Numerator / Denominator;
 }
 
 /// Returns the integer floor(Numerator / Denominator). Signed version.
 /// Overflow is explicitly forbidden with an assert.
 template <typename U, typename V, typename T = common_sint<U, V>>
 constexpr T divideFloorSigned(U Numerator, V Denominator) {
   assert(Denominator && "Division by zero");
   assert(!divideSignedWouldOverflow(Numerator, Denominator) &&
          "Divide would overflow");
   if (!Numerator)
     return 0;
   // C's integer division rounds towards 0.
   T Bias = Denominator >= 0 ? -1 : 1;
   bool SameSign = (Numerator >= 0) == (Denominator >= 0);
   return SameSign ? Numerator / Denominator
                   : (Numerator - Bias) / Denominator - 1;
 }
 
 /// Returns the remainder of the Euclidean division of LHS by RHS. Result is
 /// always non-negative.
 template <typename U, typename V, typename T = common_sint<U, V>>
 constexpr T mod(U Numerator, V Denominator) {
   assert(Denominator >= 1 && "Mod by non-positive number");
   T Mod = Numerator % Denominator;
   return Mod < 0 ? Mod + Denominator : Mod;
 }
 
 /// Returns (Numerator / Denominator) rounded by round-half-up. Guaranteed to
 /// never overflow.
 template <typename U, typename V, typename T = common_uint<U, V>>
 constexpr T divideNearest(U Numerator, V Denominator) {
   assert(Denominator && "Division by zero");
   T Mod = Numerator % Denominator;
   return (Numerator / Denominator) +
          (Mod > (static_cast<T>(Denominator) - 1) / 2);
 }
 
 /// Returns the next integer (mod 2**nbits) that is greater than or equal to
 /// \p Value and is a multiple of \p Align. \p Align must be non-zero.
 ///
 /// Examples:
 /// \code
 ///   alignTo(5, 8) = 8
 ///   alignTo(17, 8) = 24
 ///   alignTo(~0LL, 8) = 0
 ///   alignTo(321, 255) = 510
 /// \endcode
 ///
 /// Will overflow only if result is not representable in T.
 template <typename U, typename V, typename T = common_uint<U, V>>
 constexpr T alignTo(U Value, V Align) {
   assert(Align != 0u && "Align can't be 0.");
   T CeilDiv = divideCeil(Value, Align);
   return CeilDiv * Align;
 }
 
 /// Fallback when arguments aren't integral.
 constexpr uint64_t alignTo(uint64_t Value, uint64_t Align) {
   assert(Align != 0u && "Align can't be 0.");
   uint64_t CeilDiv = divideCeil(Value, Align);
   return CeilDiv * Align;
 }
 
 /// Will overflow only if result is not representable in T.
 template <typename U, typename V, typename T = common_uint<U, V>>
 constexpr T alignToPowerOf2(U Value, V Align) {
   assert(Align != 0 && (Align & (Align - 1)) == 0 &&
          "Align must be a power of 2");
   T NegAlign = static_cast<T>(0) - Align;
   return (Value + (Align - 1)) & NegAlign;
 }
 
 /// Fallback when arguments aren't integral.
 constexpr uint64_t alignToPowerOf2(uint64_t Value, uint64_t Align) {
   assert(Align != 0 && (Align & (Align - 1)) == 0 &&
          "Align must be a power of 2");
   uint64_t NegAlign = 0 - Align;
   return (Value + (Align - 1)) & NegAlign;
 }
 
 /// If non-zero \p Skew is specified, the return value will be a minimal integer
 /// that is greater than or equal to \p Size and equal to \p A * N + \p Skew for
 /// some integer N. If \p Skew is larger than \p A, its value is adjusted to '\p
 /// Skew mod \p A'. \p Align must be non-zero.
 ///
 /// Examples:
 /// \code
 ///   alignTo(5, 8, 7) = 7
 ///   alignTo(17, 8, 1) = 17
 ///   alignTo(~0LL, 8, 3) = 3
 ///   alignTo(321, 255, 42) = 552
 /// \endcode
 ///
 /// May overflow.
 template <typename U, typename V, typename W,
           typename T = common_uint<common_uint<U, V>, W>>
 constexpr T alignTo(U Value, V Align, W Skew) {
   assert(Align != 0u && "Align can't be 0.");
   Skew %= Align;
   return alignTo(Value - Skew, Align) + Skew;
 }
 
 /// Returns the next integer (mod 2**nbits) that is greater than or equal to
 /// \p Value and is a multiple of \c Align. \c Align must be non-zero.
 ///
 /// Will overflow only if result is not representable in T.
 template <auto Align, typename V, typename T = common_uint<decltype(Align), V>>
 constexpr T alignTo(V Value) {
   static_assert(Align != 0u, "Align must be non-zero");
   T CeilDiv = divideCeil(Value, Align);
   return CeilDiv * Align;
 }
 
 /// Returns the largest unsigned integer less than or equal to \p Value and is
 /// \p Skew mod \p Align. \p Align must be non-zero. Guaranteed to never
 /// overflow.
 template <typename U, typename V, typename W = uint8_t,
           typename T = common_uint<common_uint<U, V>, W>>
 constexpr T alignDown(U Value, V Align, W Skew = 0) {
   assert(Align != 0u && "Align can't be 0.");
   Skew %= Align;
   return (Value - Skew) / Align * Align + Skew;
 }
 
 /// Sign-extend the number in the bottom B bits of X to a 32-bit integer.
 /// Requires B <= 32.
 template <unsigned B> constexpr int32_t SignExtend32(uint32_t X) {
   static_assert(B <= 32, "Bit width out of range.");
   if constexpr (B == 0)
     return 0;
   return int32_t(X << (32 - B)) >> (32 - B);
 }
 
 /// Sign-extend the number in the bottom B bits of X to a 32-bit integer.
 /// Requires B <= 32.
 inline int32_t SignExtend32(uint32_t X, unsigned B) {
   assert(B <= 32 && "Bit width out of range.");
   if (B == 0)
     return 0;
   return int32_t(X << (32 - B)) >> (32 - B);
 }
 
 /// Sign-extend the number in the bottom B bits of X to a 64-bit integer.
 /// Requires B <= 64.
 template <unsigned B> constexpr int64_t SignExtend64(uint64_t x) {
   static_assert(B <= 64, "Bit width out of range.");
   if constexpr (B == 0)
     return 0;
   return int64_t(x << (64 - B)) >> (64 - B);
 }
 
 /// Sign-extend the number in the bottom B bits of X to a 64-bit integer.
 /// Requires B <= 64.
 inline int64_t SignExtend64(uint64_t X, unsigned B) {
   assert(B <= 64 && "Bit width out of range.");
   if (B == 0)
     return 0;
   return int64_t(X << (64 - B)) >> (64 - B);
 }
 
 /// Subtract two unsigned integers, X and Y, of type T and return the absolute
 /// value of the result.
 template <typename U, typename V, typename T = common_uint<U, V>>
 constexpr T AbsoluteDifference(U X, V Y) {
   return X > Y ? (X - Y) : (Y - X);
 }
 
 /// Add two unsigned integers, X and Y, of type T.  Clamp the result to the
 /// maximum representable value of T on overflow.  ResultOverflowed indicates if
 /// the result is larger than the maximum representable value of type T.
 template <typename T>
 std::enable_if_t<std::is_unsigned_v<T>, T>
 SaturatingAdd(T X, T Y, bool *ResultOverflowed = nullptr) {
   bool Dummy;
   bool &Overflowed = ResultOverflowed ? *ResultOverflowed : Dummy;
   // Hacker's Delight, p. 29
   T Z = X + Y;
   Overflowed = (Z < X || Z < Y);
   if (Overflowed)
     return std::numeric_limits<T>::max();
   else
     return Z;
 }
 
 /// Add multiple unsigned integers of type T.  Clamp the result to the
 /// maximum representable value of T on overflow.
 template <class T, class... Ts>
 std::enable_if_t<std::is_unsigned_v<T>, T> SaturatingAdd(T X, T Y, T Z,
                                                          Ts... Args) {
   bool Overflowed = false;
   T XY = SaturatingAdd(X, Y, &Overflowed);
   if (Overflowed)
     return SaturatingAdd(std::numeric_limits<T>::max(), T(1), Args...);
   return SaturatingAdd(XY, Z, Args...);
 }
 
 /// Multiply two unsigned integers, X and Y, of type T.  Clamp the result to the
 /// maximum representable value of T on overflow.  ResultOverflowed indicates if
 /// the result is larger than the maximum representable value of type T.
 template <typename T>
 std::enable_if_t<std::is_unsigned_v<T>, T>
 SaturatingMultiply(T X, T Y, bool *ResultOverflowed = nullptr) {
   bool Dummy;
   bool &Overflowed = ResultOverflowed ? *ResultOverflowed : Dummy;
 
   // Hacker's Delight, p. 30 has a different algorithm, but we don't use that
   // because it fails for uint16_t (where multiplication can have undefined
   // behavior due to promotion to int), and requires a division in addition
   // to the multiplication.
 
   Overflowed = false;
 
   // Log2(Z) would be either Log2Z or Log2Z + 1.
   // Special case: if X or Y is 0, Log2_64 gives -1, and Log2Z
   // will necessarily be less than Log2Max as desired.
   int Log2Z = Log2_64(X) + Log2_64(Y);
   const T Max = std::numeric_limits<T>::max();
   int Log2Max = Log2_64(Max);
   if (Log2Z < Log2Max) {
     return X * Y;
   }
   if (Log2Z > Log2Max) {
     Overflowed = true;
     return Max;
   }
 
   // We're going to use the top bit, and maybe overflow one
   // bit past it. Multiply all but the bottom bit then add
   // that on at the end.
   T Z = (X >> 1) * Y;
   if (Z & ~(Max >> 1)) {
     Overflowed = true;
     return Max;
   }
   Z <<= 1;
   if (X & 1)
     return SaturatingAdd(Z, Y, ResultOverflowed);
 
   return Z;
 }
 
 /// Multiply two unsigned integers, X and Y, and add the unsigned integer, A to
 /// the product. Clamp the result to the maximum representable value of T on
 /// overflow. ResultOverflowed indicates if the result is larger than the
 /// maximum representable value of type T.
 template <typename T>
 std::enable_if_t<std::is_unsigned_v<T>, T>
 SaturatingMultiplyAdd(T X, T Y, T A, bool *ResultOverflowed = nullptr) {
   bool Dummy;
   bool &Overflowed = ResultOverflowed ? *ResultOverflowed : Dummy;
 
   T Product = SaturatingMultiply(X, Y, &Overflowed);
   if (Overflowed)
     return Product;
 
   return SaturatingAdd(A, Product, &Overflowed);
 }
 
 /// Use this rather than HUGE_VALF; the latter causes warnings on MSVC.
 extern const float huge_valf;
 
 /// Add two signed integers, computing the two's complement truncated result,
 /// returning true if overflow occurred.
 template <typename T>
 std::enable_if_t<std::is_signed_v<T>, T> AddOverflow(T X, T Y, T &Result) {
 #if __has_builtin(__builtin_add_overflow)
   return __builtin_add_overflow(X, Y, &Result);
 #else
   // Perform the unsigned addition.
   using U = std::make_unsigned_t<T>;
   const U UX = static_cast<U>(X);
   const U UY = static_cast<U>(Y);
   const U UResult = UX + UY;
 
   // Convert to signed.
   Result = static_cast<T>(UResult);
 
   // Adding two positive numbers should result in a positive number.
   if (X > 0 && Y > 0)
     return Result <= 0;
   // Adding two negatives should result in a negative number.
   if (X < 0 && Y < 0)
     return Result >= 0;
   return false;
 #endif
 }
 
 /// Subtract two signed integers, computing the two's complement truncated
 /// result, returning true if an overflow ocurred.
 template <typename T>
 std::enable_if_t<std::is_signed_v<T>, T> SubOverflow(T X, T Y, T &Result) {
 #if __has_builtin(__builtin_sub_overflow)
   return __builtin_sub_overflow(X, Y, &Result);
 #else
   // Perform the unsigned addition.
   using U = std::make_unsigned_t<T>;
   const U UX = static_cast<U>(X);
   const U UY = static_cast<U>(Y);
   const U UResult = UX - UY;
 
   // Convert to signed.
   Result = static_cast<T>(UResult);
 
   // Subtracting a positive number from a negative results in a negative number.
   if (X <= 0 && Y > 0)
     return Result >= 0;
   // Subtracting a negative number from a positive results in a positive number.
   if (X >= 0 && Y < 0)
     return Result <= 0;
   return false;
 #endif
 }
 
 /// Multiply two signed integers, computing the two's complement truncated
 /// result, returning true if an overflow ocurred.
 template <typename T>
 std::enable_if_t<std::is_signed_v<T>, T> MulOverflow(T X, T Y, T &Result) {
 #if __has_builtin(__builtin_mul_overflow)
   return __builtin_mul_overflow(X, Y, &Result);
 #else
   // Perform the unsigned multiplication on absolute values.
   using U = std::make_unsigned_t<T>;
   const U UX = X < 0 ? (0 - static_cast<U>(X)) : static_cast<U>(X);
   const U UY = Y < 0 ? (0 - static_cast<U>(Y)) : static_cast<U>(Y);
   const U UResult = UX * UY;
 
   // Convert to signed.
   const bool IsNegative = (X < 0) ^ (Y < 0);
   Result = IsNegative ? (0 - UResult) : UResult;
 
   // If any of the args was 0, result is 0 and no overflow occurs.
   if (UX == 0 || UY == 0)
     return false;
 
   // UX and UY are in [1, 2^n], where n is the number of digits.
   // Check how the max allowed absolute value (2^n for negative, 2^(n-1) for
   // positive) divided by an argument compares to the other.
   if (IsNegative)
     return UX > (static_cast<U>(std::numeric_limits<T>::max()) + U(1)) / UY;
   else
     return UX > (static_cast<U>(std::numeric_limits<T>::max())) / UY;
 #endif
 }
 
 /// Type to force float point values onto the stack, so that x86 doesn't add
 /// hidden precision, avoiding rounding differences on various platforms.
 #if defined(__i386__) || defined(_M_IX86)
 using stack_float_t = volatile float;
 #else
 using stack_float_t = float;
 #endif
 
 } // namespace llvm
 
 #endif

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