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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 {

/// The behavior an operation has on an input of 0.

enum ZeroBehavior {

  /// The returned value is undefined.

  ZB_Undefined,

  /// The returned value is numeric_limits<T>::max()

  ZB_Max

};

/// Mathematical constants.

namespace numbers {

// TODO: Track C++20 std::numbers.

// TODO: Favor using the hexadecimal FP constants (requires C++17).

constexpr double e          = 2.7182818284590452354, // (0x1.5bf0a8b145749P+1) https://oeis.org/A001113

                 egamma     = .57721566490153286061, // (0x1.2788cfc6fb619P-1) https://oeis.org/A001620

                 ln2        = .69314718055994530942, // (0x1.62e42fefa39efP-1) https://oeis.org/A002162

                 ln10       = 2.3025850929940456840, // (0x1.24bb1bbb55516P+1) https://oeis.org/A002392

                 log2e      = 1.4426950408889634074, // (0x1.71547652b82feP+0)

                 log10e     = .43429448190325182765, // (0x1.bcb7b1526e50eP-2)

                 pi         = 3.1415926535897932385, // (0x1.921fb54442d18P+1) https://oeis.org/A000796

                 inv_pi     = .31830988618379067154, // (0x1.45f306bc9c883P-2) https://oeis.org/A049541

                 sqrtpi     = 1.7724538509055160273, // (0x1.c5bf891b4ef6bP+0) https://oeis.org/A002161

                 inv_sqrtpi = .56418958354775628695, // (0x1.20dd750429b6dP-1) https://oeis.org/A087197

                 sqrt2      = 1.4142135623730950488, // (0x1.6a09e667f3bcdP+0) https://oeis.org/A00219

                 inv_sqrt2  = .70710678118654752440, // (0x1.6a09e667f3bcdP-1)

                 sqrt3      = 1.7320508075688772935, // (0x1.bb67ae8584caaP+0) https://oeis.org/A002194

                 inv_sqrt3  = .57735026918962576451, // (0x1.279a74590331cP-1)

                 phi        = 1.6180339887498948482; // (0x1.9e3779b97f4a8P+0) https://oeis.org/A001622

constexpr float ef          = 2.71828183F, // (0x1.5bf0a8P+1) https://oeis.org/A001113

                egammaf     = .577215665F, // (0x1.2788d0P-1) https://oeis.org/A001620

                ln2f        = .693147181F, // (0x1.62e430P-1) https://oeis.org/A002162

                ln10f       = 2.30258509F, // (0x1.26bb1cP+1) https://oeis.org/A002392

                log2ef      = 1.44269504F, // (0x1.715476P+0)

                log10ef     = .434294482F, // (0x1.bcb7b2P-2)

                pif         = 3.14159265F, // (0x1.921fb6P+1) https://oeis.org/A000796

                inv_pif     = .318309886F, // (0x1.45f306P-2) https://oeis.org/A049541

                sqrtpif     = 1.77245385F, // (0x1.c5bf8aP+0) https://oeis.org/A002161

                inv_sqrtpif = .564189584F, // (0x1.20dd76P-1) https://oeis.org/A087197

                sqrt2f      = 1.41421356F, // (0x1.6a09e6P+0) https://oeis.org/A002193

                inv_sqrt2f  = .707106781F, // (0x1.6a09e6P-1)

                sqrt3f      = 1.73205081F, // (0x1.bb67aeP+0) https://oeis.org/A002194

                inv_sqrt3f  = .577350269F, // (0x1.279a74P-1)

                phif        = 1.61803399F; // (0x1.9e377aP+0) https://oeis.org/A001622

} // namespace numbers

/// Count number of 0's from the least significant bit to the most

///   stopping at the first 1.

///

/// Only unsigned integral types are allowed.

///

/// Returns std::numeric_limits<T>::digits on an input of 0.

template <typename T> unsigned countTrailingZeros(T Val) {

  static_assert(std::is_unsigned_v<T>,

                "Only unsigned integral types are allowed.");

  return llvm::countr_zero(Val);

}

/// Count number of 0's from the most significant bit to the least

///   stopping at the first 1.

///

/// Only unsigned integral types are allowed.

///

/// Returns std::numeric_limits<T>::digits on an input of 0.

template <typename T> unsigned countLeadingZeros(T Val) {

  static_assert(std::is_unsigned_v<T>,

                "Only unsigned integral types are allowed.");

  return llvm::countl_zero(Val);

}

/// Get the index of the first set bit starting from the least

///   significant bit.

///

/// Only unsigned integral types are allowed.

///

/// \param ZB the behavior on an input of 0.

template <typename T> T findFirstSet(T Val, ZeroBehavior ZB = ZB_Max) {

  if (ZB == ZB_Max && Val == 0)

    return std::numeric_limits<T>::max();

  return llvm::countr_zero(Val);

}

/// 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<T>::value, "Invalid type!");

  const unsigned Bits = CHAR_BIT * sizeof(T);

  assert(N <= Bits && "Invalid bit index");

  return N == 0 ? 0 : (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);

}

/// Get the index of the last set bit starting from the least

///   significant bit.

///

/// Only unsigned integral types are allowed.

///

/// \param ZB the behavior on an input of 0.

template <typename T> T findLastSet(T Val, ZeroBehavior ZB = ZB_Max) {

  if (ZB == ZB_Max && Val == 0)

    return std::numeric_limits<T>::max();

  // Use ^ instead of - because both gcc and llvm can remove the associated ^

  // in the __builtin_clz intrinsic on x86.

  return llvm::countl_zero(Val) ^ (std::numeric_limits<T>::digits - 1);

}

/// 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 inline 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 inline 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 inline 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 inline bool isInt(int64_t 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 inline bool isShiftedInt(int64_t x) {

  static_assert(

      N > 0, "isShiftedInt<0> doesn't make sense (refers to a 0-bit number.");

  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 inline bool isUInt(uint64_t x) {

  static_assert(N > 0, "isUInt<0> doesn't make sense");

  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 inline bool isShiftedUInt(uint64_t x) {

  static_assert(

      N > 0, "isShiftedUInt<0> doesn't make sense (refers to a 0-bit number)");

  static_assert(N + S <= 64,

                "isShiftedUInt<N, S> with N + S > 64 is too wide.");

  // Per the two static_asserts above, 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 > 0 && 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.

  return UINT64_MAX >> (64 - N);

}

/// Gets the minimum value for a N-bit signed integer.

inline int64_t minIntN(int64_t N) {

  assert(N > 0 && N <= 64 && "integer width out of range");

  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 > 0 && 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.

  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 inline 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 inline 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 inline 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 inline 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 inline 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 inline bool isPowerOf2_64(uint64_t Value) {

  return llvm::has_single_bit(Value);

}

/// Count the number of ones from the most significant bit to the first

/// zero bit.

///

/// Ex. countLeadingOnes(0xFF0FFF00) == 8.

/// Only unsigned integral types are allowed.

///

/// Returns std::numeric_limits<T>::digits on an input of all ones.

template <typename T> unsigned countLeadingOnes(T Value) {

  static_assert(std::is_unsigned_v<T>,

                "Only unsigned integral types are allowed.");

  return llvm::countl_one<T>(Value);

}

/// Count the number of ones from the least significant bit to the first

/// zero bit.

///

/// Ex. countTrailingOnes(0x00FF00FF) == 8.

/// Only unsigned integral types are allowed.

///

/// Returns std::numeric_limits<T>::digits on an input of all ones.

template <typename T> unsigned countTrailingOnes(T Value) {

  static_assert(std::is_unsigned_v<T>,

                "Only unsigned integral types are allowed.");

  return llvm::countr_one<T>(Value);

}

/// Count the number of set bits in a value.

/// Ex. countPopulation(0xF000F000) = 8

/// Returns 0 if the word is zero.

template <typename T>

inline unsigned countPopulation(T Value) {

  static_assert(std::is_unsigned_v<T>,

                "Only unsigned integral types are allowed.");

  return (unsigned)llvm::popcount(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 inline 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 inline 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);

}

/// This function takes a 64-bit integer and returns the bit equivalent double.

inline double BitsToDouble(uint64_t Bits) {

  static_assert(sizeof(uint64_t) == sizeof(double), "Unexpected type sizes");

  return llvm::bit_cast<double>(Bits);

}

/// This function takes a 32-bit integer and returns the bit equivalent float.

inline float BitsToFloat(uint32_t Bits) {

  static_assert(sizeof(uint32_t) == sizeof(float), "Unexpected type sizes");

  return llvm::bit_cast<float>(Bits);

}

/// This function takes a double and returns the bit equivalent 64-bit integer.

/// Note that copying doubles around changes the bits of NaNs on some hosts,

/// notably x86, so this routine cannot be used if these bits are needed.

inline uint64_t DoubleToBits(double Double) {

  static_assert(sizeof(uint64_t) == sizeof(double), "Unexpected type sizes");

  return llvm::bit_cast<uint64_t>(Double);

}

/// This function takes a float and returns the bit equivalent 32-bit integer.

/// Note that copying floats around changes the bits of NaNs on some hosts,

/// notably x86, so this routine cannot be used if these bits are needed.

inline uint32_t FloatToBits(float Float) {

  static_assert(sizeof(uint32_t) == sizeof(float), "Unexpected type sizes");

  return llvm::bit_cast<uint32_t>(Float);

}

/// A and B are either alignments or offsets. Return the minimum alignment that

/// may be assumed after adding the two together.

constexpr inline uint64_t MinAlign(uint64_t A, uint64_t 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));

}

/// Returns the next power of two (in 64-bits) that is strictly greater than A.

/// Returns zero on overflow.

constexpr inline 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 less than or equal to the given value.

/// Essentially, it is a floor operation across the domain of powers of two.

inline uint64_t PowerOf2Floor(uint64_t A) {

  return llvm::bit_floor(A);

}

/// 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)

    return 0;

  return NextPowerOf2(A - 1);

}

/// Returns the next integer (mod 2**64) 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

inline uint64_t alignTo(uint64_t Value, uint64_t Align) {

  assert(Align != 0u && "Align can't be 0.");

  return (Value + Align - 1) / Align * Align;

}

inline uint64_t alignToPowerOf2(uint64_t Value, uint64_t Align) {

  assert(Align != 0 && (Align & (Align - 1)) == 0 &&

         "Align must be a power of 2");

  return (Value + Align - 1) & -Align;

}

/// 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

inline uint64_t alignTo(uint64_t Value, uint64_t Align, uint64_t Skew) {

  assert(Align != 0u && "Align can't be 0.");

  Skew %= Align;

  return alignTo(Value - Skew, Align) + Skew;

}

/// Returns the next integer (mod 2**64) that is greater than or equal to

/// \p Value and is a multiple of \c Align. \c Align must be non-zero.

template <uint64_t Align> constexpr inline uint64_t alignTo(uint64_t Value) {

  static_assert(Align != 0u, "Align must be non-zero");

  return (Value + Align - 1) / Align * Align;

}

/// Returns the integer ceil(Numerator / Denominator).

inline uint64_t divideCeil(uint64_t Numerator, uint64_t Denominator) {

  return alignTo(Numerator, Denominator) / Denominator;

}

/// Returns the integer nearest(Numerator / Denominator).

inline uint64_t divideNearest(uint64_t Numerator, uint64_t Denominator) {

  return (Numerator + (Denominator / 2)) / Denominator;

}

/// Returns the largest uint64_t less than or equal to \p Value and is

/// \p Skew mod \p Align. \p Align must be non-zero

inline uint64_t alignDown(uint64_t Value, uint64_t Align, uint64_t 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 0 < B <= 32.

template <unsigned B> constexpr inline int32_t SignExtend32(uint32_t X) {

  static_assert(B > 0, "Bit width can't be 0.");

  static_assert(B <= 32, "Bit width out of range.");

  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 0 < B <= 32.

inline int32_t SignExtend32(uint32_t X, unsigned B) {

  assert(B > 0 && "Bit width can't be 0.");

  assert(B <= 32 && "Bit width out of range.");

  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 0 < B <= 64.

template <unsigned B> constexpr inline int64_t SignExtend64(uint64_t x) {

  static_assert(B > 0, "Bit width can't be 0.");

  static_assert(B <= 64, "Bit width out of range.");

  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 0 < B <= 64.

inline int64_t SignExtend64(uint64_t X, unsigned B) {

  assert(B > 0 && "Bit width can't be 0.");

  assert(B <= 64 && "Bit width out of range.");

  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 T>

std::enable_if_t<std::is_unsigned<T>::value, T> AbsoluteDifference(T X, T 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<T>::value, 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<T>::value, 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<T>::value, 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<T>::value, 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<T>::value, 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<T>::value, T> MulOverflow(T X, T Y, T &Result) {

  // 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);