```1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 ``` ```------------------------------------------------------------------------------ --- A collection of common operations on integer numbers. --- Most operations make no assumption on the precision of integers. --- Operation `bitNot` is necessarily an exception. --- --- @author Sergio Antoy --- @version July 2014 --- @category general ------------------------------------------------------------------------------ module Integer((^), pow, ilog, isqrt, factorial, binomial, abs, max3, min3, maxlist, minlist, bitTrunc, bitAnd, bitOr, bitNot, bitXor, even, odd) where infixr 8 ^ ------------------------------------------------------------------ -- Public Operations ------------------------------------------------------------------ --- The value of `a ^ b` is `a` raised to the power of `b`. --- Fails if `b < 0`. --- Executes in `O(log b)` steps. --- --- @param a - The base. --- @param b - The exponent. --- @return `a` raised to the power of `b`. (^) :: Int -> Int -> Int a ^ b = pow a b --- The value of `pow a b` is `a` --- raised to the power of `b`. --- Fails if `b < 0`. --- Executes in `O(log b)` steps. --- --- @param a - The base. --- @param b - The exponent. --- @return `a` raised to the power of `b`. pow :: Int -> Int -> Int pow a b | b>= 0 = powaux 1 a b where powaux n x y = if y == 0 then n else powaux (n * if (y `mod` 2 == 1) then x else 1) (x * x) (y `div` 2) --- The value of `ilog n` is the floor of the logarithm --- in the base 10 of `n`. --- Fails if `n <= 0`. --- For positive integers, the returned value is --- 1 less the number of digits in the decimal representation of `n`. --- --- @param n - The argument. --- @return the floor of the logarithm in the base 10 of `n`. ilog :: Int -> Int ilog n | n>0 = if n<10 then 0 else 1 + ilog (n `div` 10) --- The value of `isqrt n` is the floor --- of the square root of `n`. --- Fails if `n < 0`. --- Executes in `O(log n)` steps, but there must be a better way. --- --- @param n - The argument. --- @return the floor of the square root of `n`. isqrt :: Int -> Int isqrt n | n >= 0 = if n == 0 then 0 else if n < 4 then 1 else aux 2 n where aux low past = -- invariant low <= result < past if past == low+1 then low else let cand = (past + low) `div` 2 in if cand*cand > n then aux low cand else aux cand past --- The value of `factorial n` is the factorial of `n`. --- Fails if `n < 0`. --- --- @param n - The argument. --- @return the factorial of `n`. factorial :: Int -> Int factorial n | n >= 0 = if n == 0 then 1 else n * factorial (n-1) --- The value of `binomial n m` is --- n*(n-1)*...*(n-m+1)/m*(m-1)*...1 --- Fails if `m <= 0` or `n < m`. --- --- @param n - Argument. --- @param m - Argument. --- @return the binomial coefficient of `n` over `m`. binomial :: Int -> Int -> Int binomial n m | m > 0 && n >= m = aux m n `div` factorial m where aux x y = if x == 0 then 1 else y * aux (x-1) (y-1) --- The value of `abs n` is the absolute value of `n`. --- --- @param n - The argument. --- @return the absolute value of `n`. abs :: Int -> Int abs n = if n<0 then -n else n --- Returns the maximum of the three arguments. --- --- @param n - Argument. --- @param m - Argument. --- @param p - Argument. --- @return the maximum among `n`, `m` and `p`. max3 :: a -> a -> a -> a max3 n m p = max n (max m p) --- Returns the minimum of the three arguments. --- --- @param n - Argument. --- @param m - Argument. --- @param p - Argument. --- @return the minimum among `n`, `m` and `p`. min3 :: a -> a -> a -> a min3 n m p = min n (min m p) --- Returns the maximum of a list of integer values. --- Fails if the list is empty. --- --- @param l - The list of values. --- @return the maximum element of `l`. maxlist :: [a] -> a maxlist [n] = n maxlist (n:m:ns) = max n (maxlist (m:ns)) --- Returns the minimum of a list of integer values. --- Fails if the list is empty. --- --- @param l - The list of values. --- @return the minimum element of `l`. minlist :: [a] -> a minlist [n] = n minlist (n:m:ns) = min n (minlist (m:ns)) --- The value of `bitTrunc n m` is the value of the `n` --- least significant bits of `m`. --- --- @param n - Argument. --- @param m - Argument. --- @return `m` truncated to the `n` least significant bits. bitTrunc :: Int -> Int -> Int bitTrunc n m = bitAnd (pow 2 n - 1) m --- Returns the bitwise AND of the two arguments. --- --- @param n - Argument. --- @param m - Argument. --- @return the bitwise and of `n` and `m`. bitAnd :: Int -> Int -> Int bitAnd n m = if m == 0 then 0 else let p = 2 * bitAnd (n `div` 2) (m `div` 2) q = if m `mod` 2 == 0 then 0 else n `mod` 2 in p + q --- Returns the bitwise inclusive OR of the two arguments. --- --- @param n - Argument. --- @param m - Argument. --- @return the bitwise inclusive or of `n` and `m`. bitOr :: Int -> Int -> Int bitOr n m = if m == 0 then n else let p = 2 * bitOr (n `div` 2) (m `div` 2) q = if m `mod` 2 == 1 then 1 else n `mod` 2 in p + q --- Returns the bitwise NOT of the argument. --- Since integers have unlimited precision, --- only the 32 least significant bits are computed. --- --- @param n - Argument. --- @return the bitwise negation of `n` truncated to 32 bits. bitNot :: Int -> Int bitNot n = aux 32 n where aux c m = if c==0 then 0 else let p = 2 * aux (c-1) (m `div` 2) q = 1 - m `mod` 2 in p + q --- Returns the bitwise exclusive OR of the two arguments. --- --- @param n - Argument. --- @param m - Argument. --- @return the bitwise exclusive of `n` and `m`. bitXor :: Int -> Int -> Int bitXor n m = if m == 0 then n else let p = 2 * bitXor (n `div` 2) (m `div` 2) q = if m `mod` 2 == n `mod` 2 then 0 else 1 in p + q --- Returns whether an integer is even --- --- @param n - Argument. --- @return whether `n` is even. even :: Int -> Bool even n = n `mod` 2 == 0 --- Returns whether an integer is odd --- --- @param n - Argument. --- @return whether `n` is odd. odd :: Int -> Bool odd n = n `mod` 2 /= 0 ```