X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FGrid.hs;h=dc52482b0eb3cc2bc88656df24681759b8a4d346;hb=715be016934300f596a11e4fc5b8ca2ec42d6c34;hp=1a436acdc00d9276ceb1e25b7080355577822bf3;hpb=30bc407abc86ffa9a0eb4b26bdb4890f3ea423d1;p=spline3.git

diff --git a/src/Grid.hs b/src/Grid.hs
index 1a436ac..dc52482 100644
--- a/src/Grid.hs
+++ b/src/Grid.hs
@@ -1,14 +1,40 @@
+{-# LANGUAGE BangPatterns #-}
 -- | The Grid module just contains the Grid type and two constructors
 --   for it. We hide the main Grid constructor because we don't want
 --   to allow instantiation of a grid with h <= 0.
-module Grid
+module Grid (
+  cube_at,
+  grid_tests,
+  make_grid,
+  slow_tests,
+  zoom
+  )
 where
 
-import Cube (Cube(Cube))
-import FunctionValues
-import Misc (flatten)
-import Point (Point)
-import ThreeDimensional (contains_point)
+import qualified Data.Array.Repa as R
+import Test.HUnit (Assertion, assertEqual)
+import Test.Framework (Test, testGroup)
+import Test.Framework.Providers.HUnit (testCase)
+import Test.Framework.Providers.QuickCheck2 (testProperty)
+import Test.QuickCheck ((==>),
+                        Arbitrary(..),
+                        Gen,
+                        Positive(..),
+                        Property,
+                        choose)
+import Assertions (assertAlmostEqual, assertTrue)
+import Comparisons ((~=))
+import Cube (Cube(Cube),
+             find_containing_tetrahedron,
+             tetrahedra,
+             tetrahedron)
+import Examples (trilinear, trilinear9x9x9, zeros, naturals_1d)
+import FunctionValues (make_values, value_at)
+import Point (Point(..))
+import ScaleFactor (ScaleFactor)
+import Tetrahedron (Tetrahedron, c, polynomial, v0, v1, v2, v3)
+import ThreeDimensional (ThreeDimensional(..))
+import Values (Values3D, dims, empty3d, zoom_shape)
 
 
 -- | Our problem is defined on a Grid. The grid size is given by the
@@ -16,59 +42,471 @@ import ThreeDimensional (contains_point)
 --   function at the grid points, which are distance h from one
 --   another in each direction (x,y,z).
 data Grid = Grid { h :: Double, -- MUST BE GREATER THAN ZERO!
-                   function_values :: [[[Double]]] }
-          deriving (Eq, Show)
+                   function_values :: Values3D }
+          deriving (Show)
+
+
+instance Arbitrary Grid where
+    arbitrary = do
+      (Positive h') <- arbitrary :: Gen (Positive Double)
+      fvs <- arbitrary :: Gen Values3D
+      return (make_grid h' fvs)
 
 
 -- | The constructor that we want people to use. If we're passed a
 --   non-positive grid size, we throw an error.
-make_grid :: Double -> [[[Double]]] -> Grid
+make_grid :: Double -> Values3D -> Grid
 make_grid grid_size values
     | grid_size <= 0 = error "grid size must be positive"
     | otherwise = Grid grid_size values
 
 
--- | Creates an empty grid with grid size 1.
-empty_grid :: Grid
-empty_grid = Grid 1 [[[]]]
+
+-- | Takes a grid and a position as an argument and returns the cube
+--   centered on that position. If there is no cube there (i.e. the
+--   position is outside of the grid), it will throw an error.
+cube_at :: Grid -> Int -> Int -> Int -> Cube
+cube_at !g !i !j !k
+    | i < 0      = error "i < 0 in cube_at"
+    | i >= xsize = error "i >= xsize in cube_at"
+    | j < 0      = error "j < 0 in cube_at"
+    | j >= ysize = error "j >= ysize in cube_at"
+    | k < 0      = error "k < 0 in cube_at"
+    | k >= zsize = error "k >= zsize in cube_at"
+    | otherwise = Cube delta i j k fvs' tet_vol
+      where
+        fvs = function_values g
+        (xsize, ysize, zsize) = dims fvs
+        fvs' = make_values fvs i j k
+        delta = h g
+        tet_vol = (1/24)*(delta^(3::Int))
+
+--   The first cube along any axis covers (-h/2, h/2). The second
+--   covers (h/2, 3h/2).  The third, (3h/2, 5h/2), and so on.
+--
+--   We translate the (x,y,z) coordinates forward by 'h/2' so that the
+--   first covers (0, h), the second covers (h, 2h), etc. This makes
+--   it easy to figure out which cube contains the given point.
+calculate_containing_cube_coordinate :: Grid -> Double -> Int
+calculate_containing_cube_coordinate g coord
+    -- Don't use a cube on the boundary if we can help it. This
+    -- returns cube #1 if we would have returned cube #0 and cube #1
+    -- exists.
+    | coord < offset = 0
+    | coord == offset && (xsize > 1 && ysize > 1 && zsize > 1) = 1
+    | otherwise = (ceiling ( (coord + offset) / cube_width )) - 1
+    where
+      (xsize, ysize, zsize) = dims (function_values g)
+      cube_width = (h g)
+      offset = cube_width / 2
+
+
+-- | Takes a 'Grid', and returns a 'Cube' containing the given 'Point'.
+--   Since our grid is rectangular, we can figure this out without having
+--   to check every cube.
+find_containing_cube :: Grid -> Point -> Cube
+find_containing_cube g (Point x y z) =
+    cube_at g i j k
+    where
+      i = calculate_containing_cube_coordinate g x
+      j = calculate_containing_cube_coordinate g y
+      k = calculate_containing_cube_coordinate g z
 
 
--- | Returns a three-dimensional list of cubes centered on the grid
---   points of g with the appropriate 'FunctionValues'.
-cubes :: Grid -> [[[Cube]]]
-cubes g
-    | fvs == [[[]]] = [[[]]]
-    | head fvs == [[]] = [[[]]]
+zoom_lookup :: Values3D -> ScaleFactor -> a -> (R.DIM3 -> Double)
+zoom_lookup v3d scale_factor _ =
+    zoom_result v3d scale_factor
+
+
+zoom_result :: Values3D -> ScaleFactor -> R.DIM3 -> Double
+zoom_result v3d (sfx, sfy, sfz) (R.Z R.:. m R.:. n R.:. o) =
+  f p
+  where
+    g = make_grid 1 v3d
+    offset = (h g)/2
+    m' = (fromIntegral m) / (fromIntegral sfx) - offset
+    n' = (fromIntegral n) / (fromIntegral sfy) - offset
+    o' = (fromIntegral o) / (fromIntegral sfz) - offset
+    p  = Point m' n' o'
+    cube = find_containing_cube g p
+    t = find_containing_tetrahedron cube p
+    f = polynomial t
+
+
+zoom :: Values3D -> ScaleFactor -> Values3D
+zoom v3d scale_factor
+    | xsize == 0 || ysize == 0 || zsize == 0 = empty3d
     | otherwise =
-        [[[ Cube (h g) i j k (make_values fvs i j k) | i <- [0..xsize]]
-              | j <- [0..ysize]]
-              | k <- [0..zsize]]
+        R.compute $ R.unsafeTraverse v3d transExtent f
+          where
+            (xsize, ysize, zsize) = dims v3d
+            transExtent = zoom_shape scale_factor
+            f = zoom_lookup v3d scale_factor
+
+
+-- | Check all coefficients of tetrahedron0 belonging to the cube
+--   centered on (1,1,1) with a grid constructed from the trilinear
+--   values. See example one in the paper.
+--
+--   We also verify that the four vertices on face0 of the cube are
+--   in the correct location.
+--
+trilinear_c0_t0_tests :: Test.Framework.Test
+trilinear_c0_t0_tests =
+  testGroup "trilinear c0 t0"
+    [testGroup "coefficients"
+      [testCase "c0030 is correct" test_trilinear_c0030,
+       testCase "c0003 is correct" test_trilinear_c0003,
+       testCase "c0021 is correct" test_trilinear_c0021,
+       testCase "c0012 is correct" test_trilinear_c0012,
+       testCase "c0120 is correct" test_trilinear_c0120,
+       testCase "c0102 is correct" test_trilinear_c0102,
+       testCase "c0111 is correct" test_trilinear_c0111,
+       testCase "c0210 is correct" test_trilinear_c0210,
+       testCase "c0201 is correct" test_trilinear_c0201,
+       testCase "c0300 is correct" test_trilinear_c0300,
+       testCase "c1020 is correct" test_trilinear_c1020,
+       testCase "c1002 is correct" test_trilinear_c1002,
+       testCase "c1011 is correct" test_trilinear_c1011,
+       testCase "c1110 is correct" test_trilinear_c1110,
+       testCase "c1101 is correct" test_trilinear_c1101,
+       testCase "c1200 is correct" test_trilinear_c1200,
+       testCase "c2010 is correct" test_trilinear_c2010,
+       testCase "c2001 is correct" test_trilinear_c2001,
+       testCase "c2100 is correct" test_trilinear_c2100,
+       testCase "c3000 is correct" test_trilinear_c3000],
+
+    testGroup "face0 vertices"
+      [testCase "v0 is correct" test_trilinear_f0_t0_v0,
+       testCase "v1 is correct" test_trilinear_f0_t0_v1,
+       testCase "v2 is correct" test_trilinear_f0_t0_v2,
+       testCase "v3 is correct" test_trilinear_f0_t0_v3]
+    ]
+  where
+    g = make_grid 1 trilinear
+    cube = cube_at g 1 1 1
+    t = tetrahedron cube 0
+
+    test_trilinear_c0030 :: Assertion
+    test_trilinear_c0030 =
+      assertAlmostEqual "c0030 is correct" (c t 0 0 3 0) (17/8)
+
+    test_trilinear_c0003 :: Assertion
+    test_trilinear_c0003 =
+      assertAlmostEqual "c0003 is correct" (c t 0 0 0 3) (27/8)
+
+    test_trilinear_c0021 :: Assertion
+    test_trilinear_c0021 =
+      assertAlmostEqual "c0021 is correct" (c t 0 0 2 1) (61/24)
+
+    test_trilinear_c0012 :: Assertion
+    test_trilinear_c0012 =
+      assertAlmostEqual "c0012 is correct" (c t 0 0 1 2) (71/24)
+
+    test_trilinear_c0120 :: Assertion
+    test_trilinear_c0120 =
+      assertAlmostEqual "c0120 is correct" (c t 0 1 2 0) (55/24)
+
+    test_trilinear_c0102 :: Assertion
+    test_trilinear_c0102 =
+      assertAlmostEqual "c0102 is correct" (c t 0 1 0 2) (73/24)
+
+    test_trilinear_c0111 :: Assertion
+    test_trilinear_c0111 =
+      assertAlmostEqual "c0111 is correct" (c t 0 1 1 1) (8/3)
+
+    test_trilinear_c0210 :: Assertion
+    test_trilinear_c0210 =
+      assertAlmostEqual "c0210 is correct" (c t 0 2 1 0) (29/12)
+
+    test_trilinear_c0201 :: Assertion
+    test_trilinear_c0201 =
+      assertAlmostEqual "c0201 is correct" (c t 0 2 0 1) (11/4)
+
+    test_trilinear_c0300 :: Assertion
+    test_trilinear_c0300 =
+      assertAlmostEqual "c0300 is correct" (c t 0 3 0 0) (5/2)
+
+    test_trilinear_c1020 :: Assertion
+    test_trilinear_c1020 =
+      assertAlmostEqual "c1020 is correct" (c t 1 0 2 0) (8/3)
+
+    test_trilinear_c1002 :: Assertion
+    test_trilinear_c1002 =
+      assertAlmostEqual "c1002 is correct" (c t 1 0 0 2) (23/6)
+
+    test_trilinear_c1011 :: Assertion
+    test_trilinear_c1011 =
+      assertAlmostEqual "c1011 is correct" (c t 1 0 1 1) (13/4)
+
+    test_trilinear_c1110 :: Assertion
+    test_trilinear_c1110 =
+      assertAlmostEqual "c1110 is correct" (c t 1 1 1 0) (23/8)
+
+    test_trilinear_c1101 :: Assertion
+    test_trilinear_c1101 =
+      assertAlmostEqual "c1101 is correct" (c t 1 1 0 1) (27/8)
+
+    test_trilinear_c1200 :: Assertion
+    test_trilinear_c1200 =
+      assertAlmostEqual "c1200 is correct" (c t 1 2 0 0) 3
+
+    test_trilinear_c2010 :: Assertion
+    test_trilinear_c2010 =
+      assertAlmostEqual "c2010 is correct" (c t 2 0 1 0) (10/3)
+
+    test_trilinear_c2001 :: Assertion
+    test_trilinear_c2001 =
+      assertAlmostEqual "c2001 is correct" (c t 2 0 0 1) 4
+
+    test_trilinear_c2100 :: Assertion
+    test_trilinear_c2100 =
+      assertAlmostEqual "c2100 is correct" (c t 2 1 0 0) (7/2)
+
+    test_trilinear_c3000 :: Assertion
+    test_trilinear_c3000 =
+      assertAlmostEqual "c3000 is correct" (c t 3 0 0 0) 4
+
+    test_trilinear_f0_t0_v0 :: Assertion
+    test_trilinear_f0_t0_v0 =
+      assertEqual "v0 is correct" (v0 t) (Point 1 1 1)
+
+    test_trilinear_f0_t0_v1 :: Assertion
+    test_trilinear_f0_t0_v1 =
+      assertEqual "v1 is correct" (v1 t) (Point 0.5 1 1)
+
+    test_trilinear_f0_t0_v2 :: Assertion
+    test_trilinear_f0_t0_v2 =
+      assertEqual "v2 is correct" (v2 t) (Point 0.5 0.5 1.5)
+
+    test_trilinear_f0_t0_v3 :: Assertion
+    test_trilinear_f0_t0_v3 =
+      assertEqual "v3 is correct" (v3 t) (Point 0.5 1.5 1.5)
+
+
+test_trilinear_reproduced :: Assertion
+test_trilinear_reproduced =
+    assertTrue "trilinears are reproduced correctly" $
+             and [p (Point i' j' k') ~= value_at trilinear i j k
+                    | i <- [0..2],
+                      j <- [0..2],
+                      k <- [0..2],
+                      c0 <- cs,
+                      t <- tetrahedra c0,
+                      let p = polynomial t,
+                      let i' = fromIntegral i,
+                      let j' = fromIntegral j,
+                      let k' = fromIntegral k]
     where
-      fvs = function_values g
-      zsize = (length fvs) - 1
-      ysize = length (head fvs) - 1
-      xsize = length (head $ head fvs) - 1
+      g = make_grid 1 trilinear
+      cs = [ cube_at g ci cj ck | ci <- [0..2], cj <- [0..2], ck <- [0..2] ]
 
 
--- | Takes a grid and a position as an argument and returns the cube
---   centered on that position. If there is no cube there (i.e. the
---   position is outside of the grid), it will return 'Nothing'.
-cube_at :: Grid -> Int -> Int -> Int -> Maybe Cube
-cube_at g i j k
-    | i < 0 = Nothing
-    | j < 0 = Nothing
-    | k < 0 = Nothing
-    | i >= length (cubes g) = Nothing
-    | j >= length ((cubes g) !! i) = Nothing
-    | k >= length (((cubes g) !! i) !! j) = Nothing
-    | otherwise = Just $ (((cubes g) !! i) !! j) !! k
-
-
--- | Takes a 'Grid', and returns all 'Cube's belonging to it that
---   contain the given 'Point'.
-find_containing_cubes :: Grid -> Point -> [Cube]
-find_containing_cubes g p =
-    filter contains_our_point all_cubes
+test_zeros_reproduced :: Assertion
+test_zeros_reproduced =
+    assertTrue "the zero function is reproduced correctly" $
+             and [p (Point i' j' k') ~= value_at zeros i j k
+                    | i <- [0..2],
+                      j <- [0..2],
+                      k <- [0..2],
+                      let i' = fromIntegral i,
+                      let j' = fromIntegral j,
+                      let k' = fromIntegral k,
+                      c0 <- cs,
+                      t0 <- tetrahedra c0,
+                      let p = polynomial t0 ]
     where
-      all_cubes = flatten $ cubes g
-      contains_our_point = flip contains_point p
+      g = make_grid 1 zeros
+      cs = [ cube_at g ci cj ck | ci <- [0..2], cj <- [0..2], ck <- [0..2] ]
+
+
+-- | Make sure we can reproduce a 9x9x9 trilinear from the 3x3x3 one.
+test_trilinear9x9x9_reproduced :: Assertion
+test_trilinear9x9x9_reproduced =
+    assertTrue "trilinear 9x9x9 is reproduced correctly" $
+      and [p (Point i' j' k') ~= value_at trilinear9x9x9 i j k
+            | i <- [0..8],
+              j <- [0..8],
+              k <- [0..8],
+              t <- tetrahedra c0,
+              let p = polynomial t,
+              let i' = (fromIntegral i) * 0.5,
+              let j' = (fromIntegral j) * 0.5,
+              let k' = (fromIntegral k) * 0.5]
+    where
+      g = make_grid 1 trilinear
+      c0 = cube_at g 1 1 1
+
+
+-- | The point 'p' in this test lies on the boundary of tetrahedra 12 and 15.
+--   However, the 'contains_point' test fails due to some numerical innacuracy.
+--   This bug should have been fixed by setting a positive tolerance level.
+--
+--   Example from before the fix:
+--
+--   b1 (tetrahedron c 20) (0, 17.5, 0.5)
+--   -0.0
+--
+test_tetrahedra_collision_sensitivity :: Assertion
+test_tetrahedra_collision_sensitivity =
+  assertTrue "tetrahedron collision tests isn't too sensitive" $
+             contains_point t20 p
+  where
+    g = make_grid 1 naturals_1d
+    cube = cube_at g 0 18 0
+    p = Point 0 17.5 0.5
+    t20 = tetrahedron cube 20
+
+
+prop_cube_indices_never_go_out_of_bounds :: Grid -> Gen Bool
+prop_cube_indices_never_go_out_of_bounds g =
+  do
+    let delta = Grid.h g
+    let coordmin = negate (delta/2)
+
+    let (xsize, ysize, zsize) = dims $ function_values g
+    let xmax = delta*(fromIntegral xsize) - (delta/2)
+    let ymax = delta*(fromIntegral ysize) - (delta/2)
+    let zmax = delta*(fromIntegral zsize) - (delta/2)
+
+    x <- choose (coordmin, xmax)
+    y <- choose (coordmin, ymax)
+    z <- choose (coordmin, zmax)
+
+    let idx_x = calculate_containing_cube_coordinate g x
+    let idx_y = calculate_containing_cube_coordinate g y
+    let idx_z = calculate_containing_cube_coordinate g z
+
+    return $
+      idx_x >= 0 &&
+      idx_x <= xsize - 1 &&
+      idx_y >= 0 &&
+      idx_y <= ysize - 1 &&
+      idx_z >= 0 &&
+      idx_z <= zsize - 1
+
+
+-- | Given in Sorokina and Zeilfelder, p. 80, (2.9). Note that the
+--   third and fourth indices of c-t10 have been switched. This is
+--   because we store the triangles oriented such that their volume is
+--   positive. If T and T-tilde share \<v1,v2,v3\> and v0,v0-tilde point
+--   in opposite directions, one of them has to have negative volume!
+prop_c0120_identity :: Grid -> Property
+prop_c0120_identity g =
+  and [xsize >= 3, ysize >= 3, zsize >= 3] ==>
+    c t0 0 1 2 0 ~= (c t0 1 0 2 0 + c t10 1 0 0 2) / 2
+  where
+    fvs = function_values g
+    (xsize, ysize, zsize) = dims fvs
+    cube0 = cube_at g 1 1 1
+    cube1 = cube_at g 0 1 1
+    t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
+    t10 = tetrahedron cube1 10
+
+
+-- | Given in Sorokina and Zeilfelder, p. 80, (2.9). See
+--   'prop_c0120_identity'.
+prop_c0111_identity :: Grid -> Property
+prop_c0111_identity g =
+  and [xsize >= 3, ysize >= 3, zsize >= 3] ==>
+    c t0 0 1 1 1 ~= (c t0 1 0 1 1 + c t10 1 0 1 1) / 2
+  where
+    fvs = function_values g
+    (xsize, ysize, zsize) = dims fvs
+    cube0 = cube_at g 1 1 1
+    cube1 = cube_at g 0 1 1
+    t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
+    t10 = tetrahedron cube1 10
+
+
+-- | Given in Sorokina and Zeilfelder, p. 80, (2.9). See
+--   'prop_c0120_identity'.
+prop_c0201_identity :: Grid -> Property
+prop_c0201_identity g =
+  and [xsize >= 3, ysize >= 3, zsize >= 3] ==>
+    c t0 0 2 0 1 ~= (c t0 1 1 0 1 + c t10 1 1 1 0) / 2
+  where
+    fvs = function_values g
+    (xsize, ysize, zsize) = dims fvs
+    cube0 = cube_at g 1 1 1
+    cube1 = cube_at g 0 1 1
+    t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
+    t10 = tetrahedron cube1 10
+
+
+-- | Given in Sorokina and Zeilfelder, p. 80, (2.9). See
+--   'prop_c0120_identity'.
+prop_c0102_identity :: Grid -> Property
+prop_c0102_identity g =
+  and [xsize >= 3, ysize >= 3, zsize >= 3] ==>
+    c t0 0 1 0 2 ~= (c t0 1 0 0 2 + c t10 1 0 2 0) / 2
+  where
+    fvs = function_values g
+    (xsize, ysize, zsize) = dims fvs
+    cube0 = cube_at g 1 1 1
+    cube1 = cube_at g 0 1 1
+    t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
+    t10 = tetrahedron cube1 10
+
+
+-- | Given in Sorokina and Zeilfelder, p. 80, (2.9). See
+--   'prop_c0120_identity'.
+prop_c0210_identity :: Grid -> Property
+prop_c0210_identity g =
+  and [xsize >= 3, ysize >= 3, zsize >= 3] ==>
+    c t0 0 2 1 0 ~= (c t0 1 1 1 0 + c t10 1 1 0 1) / 2
+  where
+    fvs = function_values g
+    (xsize, ysize, zsize) = dims fvs
+    cube0 = cube_at g 1 1 1
+    cube1 = cube_at g 0 1 1
+    t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
+    t10 = tetrahedron cube1 10
+
+
+-- | Given in Sorokina and Zeilfelder, p. 80, (2.9). See
+--   'prop_c0120_identity'.
+prop_c0300_identity :: Grid -> Property
+prop_c0300_identity g =
+  and [xsize >= 3, ysize >= 3, zsize >= 3] ==>
+    c t0 0 3 0 0 ~= (c t0 1 2 0 0 + c t10 1 2 0 0) / 2
+  where
+    fvs = function_values g
+    (xsize, ysize, zsize) = dims fvs
+    cube0 = cube_at g 1 1 1
+    cube1 = cube_at g 0 1 1
+    t0 = tetrahedron cube0 0 -- These two tetrahedra share a face.
+    t10 = tetrahedron cube1 10
+
+
+-- | All of the properties from Section (2.9), p. 80. These require a
+--   grid since they refer to two adjacent cubes.
+p80_29_properties :: Test.Framework.Test
+p80_29_properties =
+  testGroup "p. 80, Section (2.9) Properties" [
+    testProperty "c0120 identity" prop_c0120_identity,
+    testProperty "c0111 identity" prop_c0111_identity,
+    testProperty "c0201 identity" prop_c0201_identity,
+    testProperty "c0102 identity" prop_c0102_identity,
+    testProperty "c0210 identity" prop_c0210_identity,
+    testProperty "c0300 identity" prop_c0300_identity ]
+
+
+grid_tests :: Test.Framework.Test
+grid_tests =
+    testGroup "Grid Tests" [
+      trilinear_c0_t0_tests,
+      p80_29_properties,
+      testCase "tetrahedra collision test isn't too sensitive"
+        test_tetrahedra_collision_sensitivity,
+      testProperty "cube indices within bounds"
+        prop_cube_indices_never_go_out_of_bounds ]
+
+
+-- Do the slow tests last so we can stop paying attention.
+slow_tests :: Test.Framework.Test
+slow_tests =
+    testGroup "Slow Tests" [
+      testCase "trilinear reproduced" test_trilinear_reproduced,
+      testCase "trilinear9x9x9 reproduced" test_trilinear9x9x9_reproduced,
+      testCase "zeros reproduced" test_zeros_reproduced ]