Integer.curry 6.24 KB
 Michael Hanus committed Nov 27, 2017 1 2 3 4 5 6 7 8 9 ``````------------------------------------------------------------------------------ --- 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 October 2016 --- @category general ------------------------------------------------------------------------------ `````` Michael Hanus committed Aug 31, 2020 10 ``````{-# OPTIONS_CYMAKE -Wno-incomplete-patterns #-} `````` Michael Hanus committed Nov 27, 2017 11 `````` `````` Michael Hanus committed Oct 22, 2019 12 13 14 15 16 ``````module Integer ( (^), pow, ilog, isqrt, factorial, binomial , max3, min3, maxlist, minlist , bitTrunc, bitAnd, bitOr, bitNot, bitXor , even, odd ) where `````` Michael Hanus committed Nov 27, 2017 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 `````` 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 `````` Michael Hanus committed Oct 22, 2019 46 47 48 `````` where powaux n x y = if y == 0 then n `````` Michael Hanus committed Nov 27, 2017 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 `````` 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 `````` Michael Hanus committed Oct 22, 2019 74 75 76 77 78 79 80 81 82 ``````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 `````` Michael Hanus committed Nov 27, 2017 83 84 85 86 87 88 89 90 91 92 `````` --- 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) `````` Michael Hanus committed Oct 22, 2019 93 ``````--- The value of `binomial n m` is `n*(n-1)*...*(n-m+1)/m*(m-1)*...1`. `````` Michael Hanus committed Nov 27, 2017 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 ``````--- 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) --- 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 :: Ord a => 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 :: Ord a => 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 :: Ord a => [a] -> a `````` Michael Hanus committed Oct 22, 2019 131 ``````maxlist [n] = n `````` Michael Hanus committed Nov 27, 2017 132 133 134 135 136 137 138 139 140 ``````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 :: Ord a => [a] -> a `````` Michael Hanus committed Oct 22, 2019 141 ``````minlist [n] = n `````` Michael Hanus committed Nov 27, 2017 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 ``````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 `````` Michael Hanus committed Oct 22, 2019 161 162 163 164 165 ``````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 `````` Michael Hanus committed Nov 27, 2017 166 167 168 169 170 171 172 173 `````` --- 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 `````` Michael Hanus committed Oct 22, 2019 174 175 176 177 178 ``````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 `````` Michael Hanus committed Nov 27, 2017 179 180 181 182 183 184 185 186 187 188 `````` --- 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 `````` Michael Hanus committed Oct 22, 2019 189 190 191 192 193 194 `````` 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 `````` Michael Hanus committed Nov 27, 2017 195 196 197 198 199 200 201 202 `````` --- 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 `````` Michael Hanus committed Oct 22, 2019 203 204 205 206 207 ``````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 `````` Michael Hanus committed Nov 27, 2017 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 `````` --- 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``````