src/FunctionValues.hs

   1 {-# LANGUAGE BangPatterns #-}
   2
   3 -- | The FunctionValues module contains the 'FunctionValues' type and
   4 --   the functions used to manipulate it.
   5 --
   6 module FunctionValues (
   7   FunctionValues(..),
   8   empty_values,
   9   eval,
  10   make_values,
  11   rotate,
  12   function_values_tests,
  13   function_values_properties,
  14   value_at )
  15 where
  16
  17 import Prelude hiding ( LT )
  18 import Test.Tasty ( TestTree, testGroup )
  19 import Test.Tasty.HUnit ( Assertion, testCase )
  20 import Test.Tasty.QuickCheck ( Arbitrary( arbitrary ), choose, testProperty )
  21
  22 import Assertions ( assertTrue )
  23 import Cardinal ( Cardinal(..), cwx, cwy, cwz )
  24 import Examples ( trilinear )
  25 import Values ( Values3D, dims, idx )
  26
  27 -- | The FunctionValues type represents the value of our function f at
  28 --   the 27 points surrounding (and including) the center of a
  29 --   cube. Each value of f can be accessed by the name of its
  30 --   direction.
  31 --
  32 data FunctionValues =
  33     FunctionValues { front              :: !Double,
  34                      back               :: !Double,
  35                      left               :: !Double,
  36                      right              :: !Double,
  37                      top                :: !Double,
  38                      down               :: !Double,
  39                      front_left         :: !Double,
  40                      front_right        :: !Double,
  41                      front_down         :: !Double,
  42                      front_top          :: !Double,
  43                      back_left          :: !Double,
  44                      back_right         :: !Double,
  45                      back_down          :: !Double,
  46                      back_top           :: !Double,
  47                      left_down          :: !Double,
  48                      left_top           :: !Double,
  49                      right_down         :: !Double,
  50                      right_top          :: !Double,
  51                      front_left_down    :: !Double,
  52                      front_left_top     :: !Double,
  53                      front_right_down   :: !Double,
  54                      front_right_top    :: !Double,
  55                      back_left_down     :: !Double,
  56                      back_left_top      :: !Double,
  57                      back_right_down    :: !Double,
  58                      back_right_top     :: !Double,
  59                      interior           :: !Double }
  60       deriving (Eq, Show)
  61
  62
  63 instance Arbitrary FunctionValues where
  64     arbitrary = do
  65       front'  <- choose (min_double, max_double)
  66       back'   <- choose (min_double, max_double)
  67       left'   <- choose (min_double, max_double)
  68       right'  <- choose (min_double, max_double)
  69       top'    <- choose (min_double, max_double)
  70       down'   <- choose (min_double, max_double)
  71       front_left' <- choose (min_double, max_double)
  72       front_right' <- choose (min_double, max_double)
  73       front_top' <- choose (min_double, max_double)
  74       front_down' <- choose (min_double, max_double)
  75       back_left' <- choose (min_double, max_double)
  76       back_right' <- choose (min_double, max_double)
  77       back_top' <- choose (min_double, max_double)
  78       back_down' <- choose (min_double, max_double)
  79       left_top' <- choose (min_double, max_double)
  80       left_down' <- choose (min_double, max_double)
  81       right_top' <- choose (min_double, max_double)
  82       right_down' <- choose (min_double, max_double)
  83       front_left_top' <- choose (min_double, max_double)
  84       front_left_down' <- choose (min_double, max_double)
  85       front_right_top' <- choose (min_double, max_double)
  86       front_right_down' <- choose (min_double, max_double)
  87       back_left_top' <- choose (min_double, max_double)
  88       back_left_down' <- choose (min_double, max_double)
  89       back_right_top' <- choose (min_double, max_double)
  90       back_right_down' <- choose (min_double, max_double)
  91       interior' <- choose (min_double, max_double)
  92
  93       return empty_values { front = front',
  94                             back  = back',
  95                             left  = left',
  96                             right = right',
  97                             top   = top',
  98                             down  = down',
  99                             front_left = front_left',
 100                             front_right = front_right',
 101                             front_top = front_top',
 102                             front_down = front_down',
 103                             back_left = back_left',
 104                             back_right = back_right',
 105                             back_top = back_top',
 106                             back_down = back_down',
 107                             left_top = left_top',
 108                             left_down = left_down',
 109                             right_top = right_top',
 110                             right_down = right_down',
 111                             front_left_top = front_left_top',
 112                             front_left_down = front_left_down',
 113                             front_right_top = front_right_top',
 114                             front_right_down = front_right_down',
 115                             back_left_top = back_left_top',
 116                             back_left_down = back_left_down',
 117                             back_right_top = back_right_top',
 118                             back_right_down = back_right_down',
 119                             interior = interior' }
 120       where
 121         -- | We perform addition with the function values contained in a
 122         --   FunctionValues object. If we choose random doubles near the machine
 123         --   min/max, we risk overflowing or underflowing the 'Double'. This
 124         --   places a reasonably safe limit on the maximum size of our generated
 125         --   'Double' members.
 126         max_double :: Double
 127         max_double = 10000.0
 128
 129         -- | See 'max_double'.
 130         min_double :: Double
 131         min_double = (-1) * max_double
 132
 133
 134 -- | Return a 'FunctionValues' with a bunch of zeros for data points.
 135 empty_values :: FunctionValues
 136 empty_values =
 137     FunctionValues 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 138
 139
 140 -- | The eval function is where the magic happens for the
 141 --   FunctionValues type. Given a 'Cardinal' direction and a
 142 --   'FunctionValues' object, eval will return the value of the
 143 --   function f in that 'Cardinal' direction. Note that 'Cardinal' can
 144 --   be a composite type; eval is what performs the \"arithmetic\" on
 145 --   'Cardinal' directions.
 146 eval :: FunctionValues -> Cardinal -> Double
 147 eval f F = front f
 148 eval f B = back f
 149 eval f L = left f
 150 eval f R = right f
 151 eval f T = top f
 152 eval f D = down f
 153 eval f FL = front_left f
 154 eval f FR = front_right f
 155 eval f FD = front_down f
 156 eval f FT = front_top f
 157 eval f BL = back_left f
 158 eval f BR = back_right f
 159 eval f BD = back_down f
 160 eval f BT = back_top f
 161 eval f LD = left_down f
 162 eval f LT = left_top f
 163 eval f RD = right_down f
 164 eval f RT = right_top f
 165 eval f FLD = front_left_down f
 166 eval f FLT = front_left_top f
 167 eval f FRD = front_right_down f
 168 eval f FRT = front_right_top f
 169 eval f BLD = back_left_down f
 170 eval f BLT = back_left_top f
 171 eval f BRD = back_right_down f
 172 eval f BRT = back_right_top f
 173 eval f I   = interior f
 174 eval _ (Scalar x) = x
 175 eval f (Sum x y) = (eval f x) + (eval f y)
 176 eval f (Difference x y) = (eval f x) - (eval f y)
 177 eval f (Product x y) = (eval f x) * (eval f y)
 178 eval f (Quotient x y) = (eval f x) / (eval f y)
 179
 180
 181 -- | Takes a three-dimensional list of 'Double' and a set of 3D
 182 --   coordinates (i,j,k), and returns the value at (i,j,k) in the
 183 --   supplied list. If there is no such value, we calculate one
 184 --   according to Sorokina and Zeilfelder, remark 7.3, p. 99.
 185 --
 186 --   We specifically do not consider values more than one unit away
 187 --   from our grid.
 188 --
 189 --   Examples:
 190 --
 191 --   >>> value_at Examples.trilinear 0 0 0
 192 --   1.0
 193 --
 194 --   >>> value_at Examples.trilinear (-1) 0 0
 195 --   0.0
 196 --
 197 --   >>> value_at Examples.trilinear 0 0 4
 198 --   1.0
 199 --
 200 --   >>> value_at Examples.trilinear 1 3 0
 201 --   5.0
 202 --
 203 value_at :: Values3D -> Int -> Int -> Int -> Double
 204 value_at v3d !i !j !k
 205   -- Put the most common case first!
 206   | (valid_i i) && (valid_j j) && (valid_k k) =
 207       idx v3d i j k
 208
 209   -- The next three are from the first line in (7.3). Analogous cases
 210   -- have been added where the indices are one-too-big. These are the
 211   -- "one index is bad" cases.
 212   | not (valid_i i) =
 213       if (dim_i == 1)
 214       then
 215         -- We're one-dimensional in our first coordinate, so just
 216         -- return the data point that we do have. If we try to use
 217         -- the formula from remark 7.3, we go into an infinite loop.
 218         value_at v3d 0 j k
 219       else
 220         if (i == -1)
 221         then
 222           2*(value_at v3d 0 j k) - (value_at v3d 1 j k)
 223         else
 224           2*(value_at v3d (i-1) j k) - (value_at v3d (i-2) j k)
 225
 226   | not (valid_j j) =
 227       if (dim_j == 1)
 228       then
 229         -- We're one-dimensional in our second coordinate, so just
 230         -- return the data point that we do have. If we try to use
 231         -- the formula from remark 7.3, we go into an infinite loop.
 232         value_at v3d i 0 k
 233       else
 234         if (j == -1)
 235         then
 236           2*(value_at v3d i 0 k) - (value_at v3d i 1 k)
 237         else
 238           2*(value_at v3d i (j-1) k) - (value_at v3d i (j-2) k)
 239
 240   | not (valid_k k) =
 241       if (dim_k == 1)
 242       then
 243         -- We're one-dimensional in our third coordinate, so just
 244         -- return the data point that we do have. If we try to use
 245         -- the formula from remark 7.3, we go into an infinite loop.
 246         value_at v3d i j 0
 247       else
 248         if (k == -1)
 249         then
 250           2*(value_at v3d i j 0) - (value_at v3d i j 1)
 251         else
 252           2*(value_at v3d i j (k-1)) - (value_at v3d i j (k-2))
 253   where
 254     (dim_i, dim_j, dim_k) = dims v3d
 255
 256     valid_i :: Int -> Bool
 257     valid_i i' = (i' >= 0) && (i' < dim_i)
 258
 259     valid_j :: Int -> Bool
 260     valid_j j' = (j' >= 0) && (j' < dim_j)
 261
 262     valid_k :: Int -> Bool
 263     valid_k k' = (k' >= 0) && (k' < dim_k)
 264
 265
 266
 267 -- | Given a three-dimensional list of 'Double' and a set of 3D
 268 --   coordinates (i,j,k), constructs and returns the 'FunctionValues'
 269 --   object centered at (i,j,k)
 270 make_values :: Values3D -> Int -> Int -> Int -> FunctionValues
 271 make_values values !i !j !k =
 272     empty_values { front  = value_at values (i-1) j k,
 273                    back   = value_at values (i+1) j k,
 274                    left   = value_at values i (j-1) k,
 275                    right  = value_at values i (j+1) k,
 276                    down   = value_at values i j (k-1),
 277                    top    = value_at values i j (k+1),
 278                    front_left = value_at values (i-1) (j-1) k,
 279                    front_right = value_at values (i-1) (j+1) k,
 280                    front_down =value_at values (i-1) j (k-1),
 281                    front_top = value_at values (i-1) j (k+1),
 282                    back_left = value_at values (i+1) (j-1) k,
 283                    back_right = value_at values (i+1) (j+1) k,
 284                    back_down = value_at values (i+1) j (k-1),
 285                    back_top = value_at values (i+1) j (k+1),
 286                    left_down = value_at values i (j-1) (k-1),
 287                    left_top = value_at values i (j-1) (k+1),
 288                    right_down = value_at values i (j+1) (k-1),
 289                    right_top = value_at values i (j+1) (k+1),
 290                    front_left_down = value_at values (i-1) (j-1) (k-1),
 291                    front_left_top = value_at values (i-1) (j-1) (k+1),
 292                    front_right_down = value_at values (i-1) (j+1) (k-1),
 293                    front_right_top = value_at values (i-1) (j+1) (k+1),
 294                    back_left_down = value_at values (i+1) (j-1) (k-1),
 295                    back_left_top = value_at values (i+1) (j-1) (k+1),
 296                    back_right_down = value_at values (i+1) (j+1) (k-1),
 297                    back_right_top = value_at values (i+1) (j+1) (k+1),
 298                    interior = value_at values i j k }
 299
 300 -- | Takes a 'FunctionValues' and a function that transforms one
 301 --   'Cardinal' to another (called a rotation). Then it applies the
 302 --   rotation to each element of the 'FunctionValues' object, and
 303 --   returns the result.
 304 rotate :: (Cardinal -> Cardinal) -> FunctionValues -> FunctionValues
 305 rotate rotation fv =
 306     FunctionValues { front  = eval fv (rotation F),
 307                      back   = eval fv (rotation B),
 308                      left   = eval fv (rotation L),
 309                      right  = eval fv (rotation R),
 310                      down   = eval fv (rotation D),
 311                      top    = eval fv (rotation T),
 312                      front_left = eval fv (rotation FL),
 313                      front_right = eval fv (rotation FR),
 314                      front_down = eval fv (rotation FD),
 315                      front_top = eval fv (rotation FT),
 316                      back_left = eval fv (rotation BL),
 317                      back_right = eval fv (rotation BR),
 318                      back_down = eval fv (rotation BD),
 319                      back_top = eval fv (rotation BT),
 320                      left_down = eval fv (rotation LD),
 321                      left_top = eval fv (rotation LT),
 322                      right_down = eval fv (rotation RD),
 323                      right_top = eval fv (rotation RT),
 324                      front_left_down = eval fv (rotation FLD),
 325                      front_left_top = eval fv (rotation FLT),
 326                      front_right_down = eval fv (rotation FRD),
 327                      front_right_top = eval fv (rotation FRT),
 328                      back_left_down = eval fv (rotation BLD),
 329                      back_left_top = eval fv (rotation BLT),
 330                      back_right_down = eval fv (rotation BRD),
 331                      back_right_top = eval fv (rotation BRT),
 332                      interior = interior fv }
 333
 334
 335
 336 -- | Ensure that the trilinear values wind up where we think they
 337 --   should.
 338 test_directions :: Assertion
 339 test_directions =
 340     assertTrue "all direction functions work" (and equalities)
 341         where
 342           fvs = make_values trilinear 1 1 1
 343           equalities = [ interior fvs == 4,
 344                          front fvs == 1,
 345                          back  fvs == 7,
 346                          left  fvs == 2,
 347                          right fvs == 6,
 348                          down  fvs == 3,
 349                          top   fvs == 5,
 350                          front_left fvs == 1,
 351                          front_right fvs == 1,
 352                          front_down fvs == 1,
 353                          front_top fvs == 1,
 354                          back_left fvs == 3,
 355                          back_right fvs == 11,
 356                          back_down fvs == 5,
 357                          back_top fvs == 9,
 358                          left_down fvs == 2,
 359                          left_top fvs == 2,
 360                          right_down fvs == 4,
 361                          right_top fvs == 8,
 362                          front_left_down fvs == 1,
 363                          front_left_top fvs == 1,
 364                          front_right_down fvs == 1,
 365                          front_right_top fvs == 1,
 366                          back_left_down fvs == 3,
 367                          back_left_top fvs == 3,
 368                          back_right_down fvs == 7,
 369                          back_right_top fvs == 15]
 370
 371
 372 function_values_tests :: TestTree
 373 function_values_tests =
 374     testGroup "FunctionValues tests"
 375                   [ testCase "test directions" test_directions ]
 376
 377
 378 prop_x_rotation_doesnt_affect_front :: FunctionValues -> Bool
 379 prop_x_rotation_doesnt_affect_front fv0 =
 380     expr1 == expr2
 381     where
 382       fv1 = rotate cwx fv0
 383       expr1 = front fv0
 384       expr2 = front fv1
 385
 386 prop_x_rotation_doesnt_affect_back :: FunctionValues -> Bool
 387 prop_x_rotation_doesnt_affect_back fv0 =
 388     expr1 == expr2
 389     where
 390       fv1 = rotate cwx fv0
 391       expr1 = back fv0
 392       expr2 = back fv1
 393
 394
 395 prop_y_rotation_doesnt_affect_left :: FunctionValues -> Bool
 396 prop_y_rotation_doesnt_affect_left fv0 =
 397     expr1 == expr2
 398     where
 399       fv1 = rotate cwy fv0
 400       expr1 = left fv0
 401       expr2 = left fv1
 402
 403 prop_y_rotation_doesnt_affect_right :: FunctionValues -> Bool
 404 prop_y_rotation_doesnt_affect_right fv0 =
 405     expr1 == expr2
 406     where
 407       fv1 = rotate cwy fv0
 408       expr1 = right fv0
 409       expr2 = right fv1
 410
 411
 412 prop_z_rotation_doesnt_affect_down :: FunctionValues -> Bool
 413 prop_z_rotation_doesnt_affect_down fv0 =
 414     expr1 == expr2
 415     where
 416       fv1 = rotate cwz fv0
 417       expr1 = down fv0
 418       expr2 = down fv1
 419
 420
 421 prop_z_rotation_doesnt_affect_top :: FunctionValues -> Bool
 422 prop_z_rotation_doesnt_affect_top fv0 =
 423     expr1 == expr2
 424     where
 425       fv1 = rotate cwz fv0
 426       expr1 = top fv0
 427       expr2 = top fv1
 428
 429
 430 function_values_properties :: TestTree
 431 function_values_properties =
 432   testGroup "FunctionValues properties" [
 433     testProperty
 434       "x rotation doesn't affect front"
 435       prop_x_rotation_doesnt_affect_front,
 436     testProperty
 437       "x rotation doesn't affect back"
 438       prop_x_rotation_doesnt_affect_back,
 439     testProperty
 440       "y rotation doesn't affect left"
 441       prop_y_rotation_doesnt_affect_left,
 442     testProperty
 443       "y rotation doesn't affect right"
 444       prop_y_rotation_doesnt_affect_right,
 445     testProperty
 446       "z rotation doesn't affect top"
 447       prop_z_rotation_doesnt_affect_top,
 448     testProperty
 449       "z rotation doesn't affect down"
 450       prop_z_rotation_doesnt_affect_down ]