optimise root-finding functions

* use PrimArray to represent polynomials
  * add some strictness annotations
  * turn on some optimisation flags
  * use quadratic formula for quadratic polynomials

MetaBrush.cabal

@@ -40,6 +40,8 @@ common common
           >= 1.2.0.1 && < 2.0
       , groups
           ^>= 0.4.1.0
+      , primitive
+          ^>= 0.7.1.0
       , transformers
           ^>= 0.5.6.2
 
@@ -49,7 +51,11 @@ common common
     ghc-options:
         -O1
         -fexpose-all-unfoldings
+        -funfolding-use-threshold=16
+        -fexcess-precision
         -fspecialise-aggressively
+        -optc-O3
+        -optc-ffast-math
         -Wall
         -Wcompat
         -fwarn-missing-local-signatures
@@ -85,6 +91,8 @@ library
           ^>= 0.20.0.0
       , monad-par
           ^>= 0.3.5
+      , prim-instances
+          ^>= 0.2
       , vector
           ^>= 0.12.1.2
 

app/Main.hs

@@ -214,7 +214,7 @@ main = do
   maxHistorySizeTVar <- STM.newTVarIO @Int                                         1000
   fitParametersTVar  <- STM.newTVarIO @FitParameters
     ( FitParameters
-      { maxSubdiv  = 10
+      { maxSubdiv  = 6
       , nbSegments = 12
       , dist_tol   = 5e-3
       , t_tol      = 1e-4

src/app/MetaBrush/Unique.hs

@@ -1,3 +1,4 @@
+{-# LANGUAGE BangPatterns               #-}
 {-# LANGUAGE DerivingStrategies         #-}
 {-# LANGUAGE FlexibleContexts           #-}
 {-# LANGUAGE GeneralizedNewtypeDeriving #-}
@@ -72,7 +73,7 @@ newtype UniqueSupply = UniqueSupply { uniqueSupplyTVar :: STM.TVar Unique }
 
 freshUnique :: UniqueSupply -> STM Unique
 freshUnique ( UniqueSupply { uniqueSupplyTVar } ) = do
-  uniq@( Unique i ) <- STM.readTVar uniqueSupplyTVar
+  uniq@( Unique !i ) <- STM.readTVar uniqueSupplyTVar
   STM.writeTVar uniqueSupplyTVar ( Unique ( succ i ) )
   pure uniq
 

src/lib/Math/Bezier/Cubic.hs

@@ -1,4 +1,5 @@
 {-# LANGUAGE AllowAmbiguousTypes   #-}
+{-# LANGUAGE BangPatterns          #-}
 {-# LANGUAGE DeriveAnyClass        #-}
 {-# LANGUAGE DeriveGeneric         #-}
 {-# LANGUAGE DeriveTraversable     #-}
@@ -57,6 +58,10 @@ import Data.Group
 import Data.Group.Generics
   ()
 
+-- primitive
+import Data.Primitive.Types
+  ( Prim )
+
 -- MetaBrush
 import qualified Math.Bezier.Quadratic as Quadratic
   ( Bezier(..), bezier )
@@ -154,28 +159,30 @@ subdivide ( Bezier {..} ) t = ( Bezier p0 q1 q2 pt, Bezier pt r1 r2 p3 )
 
 -- | Polynomial coefficients of the derivative of the distance to a cubic Bézier curve.
 ddist :: forall v r p. ( Torsor v p, Inner r v, RealFloat r ) => Bezier p -> p -> [ r ]
-ddist ( Bezier {..} ) c = [ a0, a1, a2, a3, a4, a5 ]
+ddist ( Bezier {..} ) c = [ a5, a4, a3, a2, a1, a0 ]
   where
     v, v', v'', v''' :: v
-    v    = c  --> p0
+    !v    = c  --> p0
-    v'   = p0 --> p1
+    !v'   = p0 --> p1
-    v''  = p1 --> p0 ^+^ p1 --> p2
+    !v''  = p1 --> p0 ^+^ p1 --> p2
-    v''' = p0 --> p3 ^+^ 3 *^ ( p2 --> p1 )
+    !v''' = p0 --> p3 ^+^ 3 *^ ( p2 --> p1 )
 
     a0, a1, a2, a3, a4, a5 :: r
-    a0 = v ^.^ v'
+    !a0 = v ^.^ v'
-    a1 = 3 * squaredNorm v' + 2 * v ^.^ v''
+    !a1 = 3 * squaredNorm v' + 2 * v ^.^ v''
-    a2 = 9 * v' ^.^ v'' + v ^.^ v'''
+    !a2 = 9 * v' ^.^ v'' + v ^.^ v'''
-    a3 = 6 * squaredNorm v'' + 4 * v' ^.^ v'''
+    !a3 = 6 * squaredNorm v'' + 4 * v' ^.^ v'''
-    a4 = 5 * v'' ^.^ v'''
+    !a4 = 5 * v'' ^.^ v'''
-    a5 = squaredNorm v'''
+    !a5 = squaredNorm v'''
 
 -- | Finds the closest point to a given point on a cubic Bézier curve.
-closestPoint :: forall v r p. ( Torsor v p, Inner r v, RealFloat r ) => Bezier p -> p -> ArgMin r ( r, p )
+closestPoint
+  :: forall v r p. ( Torsor v p, Inner r v, RealFloat r, Prim r )
+  => Bezier p -> p -> ArgMin r ( r, p )
 closestPoint pts@( Bezier {..} ) c = pickClosest ( 0 :| 1 : roots ) -- todo: also include the self-intersection point if one exists
   where
     roots :: [ r ]
-    roots = filter ( \ r -> r > 0 && r < 1 ) ( realRoots $ ddist @v pts c )
+    roots = filter ( \ r -> r > 0 && r < 1 ) ( realRoots 2000 $ ddist @v pts c )
 
     pickClosest :: NonEmpty r -> ArgMin r ( r, p )
     pickClosest ( s :| ss ) = go s q nm0 ss

src/lib/Math/Bezier/Cubic/Fit.hs

@@ -46,6 +46,10 @@ import qualified Data.Sequence as Seq
 import Control.DeepSeq
   ( NFData )
 
+-- primitive
+import Data.Primitive.PrimArray
+  ( primArrayFromListN, unsafeThawPrimArray )
+
 -- transformers
 import Control.Monad.Trans.State.Strict
   ( execStateT, modify' )
@@ -241,9 +245,12 @@ fitPiece dist_tol t_tol maxIters p tp qs r tr =
       ( dts_changed, argmax_sq_dist ) <- ( `execStateT` ( False, Max ( Arg 0 0 ) ) ) $ for_ ( zip qs [ 0 .. ] ) \( q, i ) -> do
         ti <- lift ( Unboxed.MVector.unsafeRead ts i )
         let
-          poly :: [ Complex Double ]
+          laguerreStepResult :: Complex Double
-          poly = map (:+ 0) $ Cubic.ddist @( Vector2D Double ) bez q
+          laguerreStepResult = runST do
-        ti' <- case laguerre epsilon 1 poly ( ti :+ 0 ) of
+            coeffs <- unsafeThawPrimArray . primArrayFromListN 6 . map (:+ 0)
+                    $ Cubic.ddist @( Vector2D Double ) bez q
+            laguerre epsilon 1 coeffs ( ti :+ 0 )
+        ti' <- case laguerreStepResult of
           x :+ y
             |  isNaN x
             || isNaN y

src/lib/Math/Bezier/Quadratic.hs

@@ -1,4 +1,5 @@
 {-# LANGUAGE AllowAmbiguousTypes   #-}
+{-# LANGUAGE BangPatterns          #-}
 {-# LANGUAGE DeriveAnyClass        #-}
 {-# LANGUAGE DeriveGeneric         #-}
 {-# LANGUAGE DeriveTraversable     #-}
@@ -54,6 +55,10 @@ import Data.Group
 import Data.Group.Generics
   ()
 
+-- primitive
+import Data.Primitive.Types
+  ( Prim )
+
 -- MetaBrush
 import Math.Epsilon
   ( epsilon )
@@ -125,25 +130,27 @@ subdivide ( Bezier {..} ) t = ( Bezier p0 q1 pt, Bezier pt r1 p2 )
 
 -- | Polynomial coefficients of the derivative of the distance to a quadratic Bézier curve.
 ddist :: forall v r p. ( Torsor v p, Inner r v, RealFloat r ) => Bezier p -> p -> [ r ]
-ddist ( Bezier {..} ) c = [ a0, a1, a2, a3 ]
+ddist ( Bezier {..} ) c = [ a3, a2, a1, a0 ]
   where
     v, v', v'' :: v
-    v   = c  --> p0
+    !v   = c  --> p0
-    v'  = p0 --> p1
+    !v'  = p0 --> p1
-    v'' = p1 --> p0 ^+^ p1 --> p2
+    !v'' = p1 --> p0 ^+^ p1 --> p2
 
     a0, a1, a2, a3 :: r
-    a0 = v ^.^ v'
+    !a0 = v ^.^ v'
-    a1 = v ^.^ v'' + 2 * squaredNorm v'
+    !a1 = v ^.^ v'' + 2 * squaredNorm v'
-    a2 = 3 * v' ^.^ v''
+    !a2 = 3 * v' ^.^ v''
-    a3 = squaredNorm v''
+    !a3 = squaredNorm v''
 
 -- | Finds the closest point to a given point on a quadratic Bézier curve.
-closestPoint :: forall v r p. ( Torsor v p, Inner r v, RealFloat r ) => Bezier p -> p -> ArgMin r ( r, p )
+closestPoint
+  :: forall v r p. ( Torsor v p, Inner r v, RealFloat r, Prim r )
+  => Bezier p -> p -> ArgMin r ( r, p )
 closestPoint pts@( Bezier {..} ) c = pickClosest ( 0 :| 1 : roots )
   where
     roots :: [ r ]
-    roots = filter ( \ r -> r > 0 && r < 1 ) ( realRoots $ ddist @v pts c ) 
+    roots = filter ( \ r -> r > 0 && r < 1 ) ( realRoots 2000 $ ddist @v pts c ) 
 
     pickClosest :: NonEmpty r -> ArgMin r ( r, p )
     pickClosest ( s :| ss ) = go s q nm0 ss

src/lib/Math/Bezier/Stroke.hs

@@ -78,7 +78,7 @@ import Math.Module
   , lerp, squaredNorm
   )
 import Math.Roots
-  ( realRoots )
+  ( solveQuadratic )
 import Math.Vector2D
   ( Point2D(..), Vector2D(..), cross )
 
@@ -580,12 +580,12 @@ withTangent tgt ( spt0 :<| spt1 :<| spts ) =
           | otherwise
           = Nothing
       in
-        case mapMaybe correctTangentParam $ realRoots [ c01, 2 * ( c12 - c01 ), c01 + c23 - 2 * c12 ] of
+        case mapMaybe correctTangentParam $ solveQuadratic c01 ( 2 * ( c12 - c01 ) ) ( c01 + c23 - 2 * c12 ) of
           ( t : _ )
             -> Offset i ( Just t ) ( MkVector2D $ Cubic.bezier @( Vector2D Double ) bez t )
-          -- Fallback in case we couldn't solve the quadratic for some reason.
           _
             | Just s <- between tgt tgt0 tgt2
+            -- Fallback in case we couldn't solve the quadratic for some reason.
             -> Offset i ( Just s ) ( MkVector2D $ Cubic.bezier @( Vector2D Double ) bez s )
             -- Otherwise: go to next piece of the curve.
             | otherwise

src/lib/Math/Epsilon.hs

@@ -1,7 +1,7 @@
 {-# LANGUAGE ScopedTypeVariables #-}
 
 module Math.Epsilon
-  ( epsilon )
+  ( epsilon, nearZero )
   where
 
 --------------------------------------------------------------------------------
@@ -10,3 +10,6 @@ module Math.Epsilon
 {-# SPECIALISE epsilon :: Double #-}
 epsilon :: forall r. RealFloat r => r
 epsilon = encodeFloat 1 ( 5 - floatDigits ( 0 :: r ) )
+
+nearZero :: RealFloat r => r -> Bool
+nearZero x = abs x < epsilon

src/lib/Math/Roots.hs

@@ -1,108 +1,172 @@
+{-# LANGUAGE BangPatterns          #-}
+{-# LANGUAGE BlockArguments        #-}
+{-# LANGUAGE GADTs                 #-}
+{-# LANGUAGE NamedWildCards        #-}
+{-# LANGUAGE PartialTypeSignatures #-}
 {-# LANGUAGE ScopedTypeVariables   #-}
+{-# LANGUAGE TypeApplications      #-}
 
-module Math.Roots
+{-# OPTIONS_GHC -fno-warn-partial-type-signatures #-}
 
-  where
+module Math.Roots where
 
 -- base
+import Control.Monad
+  ( unless )
+import Control.Monad.ST
+  ( ST, runST )
 import Data.Complex
   ( Complex(..), magnitude )
-import Data.List.NonEmpty
-  ( NonEmpty(..), toList )
 import Data.Maybe
   ( mapMaybe )
 
+-- primitive
+import Control.Monad.Primitive
+  ( PrimMonad(PrimState) )
+import Data.Primitive.PrimArray
+  ( PrimArray, MutablePrimArray
+  , primArrayFromList
+  , getSizeofMutablePrimArray, sizeofPrimArray
+  , unsafeThawPrimArray, cloneMutablePrimArray
+  , shrinkMutablePrimArray, readPrimArray, writePrimArray
+  )
+import Data.Primitive.Types
+  ( Prim )
+
+-- prim-instances
+import Data.Primitive.Instances
+  () -- instance Prim a => Prim ( Complex a )
+
 -- MetaBrush
 import Math.Epsilon
-  ( epsilon )
+  ( epsilon, nearZero )
 
 --------------------------------------------------------------------------------
 
--- | Find real roots of a polynomial.
+-- | Real solutions to a quadratic equation.
-
+solveQuadratic :: forall a. RealFloat a => a -> a -> a -> [ a ]
--- Coefficients are given in order of increasing degree, e.g.:
+solveQuadratic a0 a1 a2
---  x² + 7 is given by [ 7, 0, 1 ].
+  | nearZero a1 && nearZero a2
-realRoots :: forall r. RealFloat r => [ r ] -> [ r ]
+  = if   nearZero a0
-realRoots p = mapMaybe isReal ( roots epsilon 10000 ( map (:+ 0) p ) )
+    then [ 0, 0.5, 1 ] -- convention
+    else []
+  | nearZero ( a0 * a0 * a2 / ( a1 * a1 ) )
+  = [ - a0 / a1 ]
+  | disc < 0
+  = [] -- non-real solutions
+  | otherwise
+  = let
+      r :: a
+      r =
+        if a1 >= 0
+        then 2 * a0 / ( - a1 - sqrt disc )
+        else 0.5 * ( - a1 + sqrt disc) / a2
+    in [ r, -r - a1 ]
   where
-    isReal :: Complex r -> Maybe r
+    disc :: a  
+    disc = a1 * a1 - 4 * a0 * a2
+
+-- | Find real roots of a polynomial.
+--
+-- Coefficients are given in order of decreasing degree, e.g.:
+--  x² + 7 is given by [ 1, 0, 7 ].
+realRoots :: forall a. ( RealFloat a, Prim a ) => Int -> [ a ] -> [ a ]
+realRoots maxIters coeffs = mapMaybe isReal ( roots epsilon maxIters ( map (:+ 0) coeffs ) )
+  where
+    isReal :: Complex a -> Maybe a
     isReal ( a :+ b )
       | abs b < epsilon = Just a
       | otherwise       = Nothing
 
 -- | Compute all roots of a polynomial using Laguerre's method and (forward) deflation.
 --
--- Polynomial coefficients are given in order of ascending degree (e.g. constant coefficient first).
+-- Polynomial coefficients are given in order of descending degree (e.g. constant coefficient last).
 --
 -- N.B. The forward deflation process is only guaranteed to be numerically stable
 -- if Laguerre's method finds roots in increasing order of magnitude.
-roots :: forall a. RealFloat a => a -> Int -> [ Complex a ] -> [ Complex a ]
+roots :: forall a. ( RealFloat a, Prim a ) => a -> Int -> [ Complex a ] -> [ Complex a ]
-roots eps maxIters p = go p []
+roots eps maxIters coeffs = runST do
-  where
+  let
-    go :: [ Complex a ] -> [ Complex a ] -> [ Complex a ]
+    coeffPrimArray :: PrimArray ( Complex a )
-    go q rs
+    coeffPrimArray = primArrayFromList coeffs
-      | length q <= 2 = r : rs
+    sz :: Int
-      | otherwise     = go ( deflate r q ) ( r : rs )
+    sz = sizeofPrimArray coeffPrimArray
-      where
+  p <- unsafeThawPrimArray coeffPrimArray
-        r :: Complex a
+  let
-        r = laguerre eps maxIters q 0
+    go :: Int -> [ Complex a ] -> ST _s [ Complex a ]
-        -- Start the iteration at 0 for best chance of numerical stability.
+    go i rs = do
+      !r <- laguerre eps maxIters p 0 -- Start Laguerre's method at 0 for best chance of numerical stability.
+      if i <= 2
+      then pure ( r : rs )
+      else do
+        deflate r p
+        go ( i - 1 ) ( r : rs )
+  go sz []
 
 -- | Deflate a polynomial: factor out a root of the polynomial.
 -- 
 -- The polynomial must have degree at least 2.
-deflate :: forall a. Num a => a -> [ a ] -> [ a ]
+deflate :: forall a m s. ( Num a, Prim a, PrimMonad m, s ~ PrimState m ) => a -> MutablePrimArray s a -> m ()
-deflate r ( _ : c : cs ) = toList $ go ( c :| cs )
+deflate r p = do
-  where
+  deg <- subtract 1 <$> getSizeofMutablePrimArray p
-    go :: NonEmpty a -> NonEmpty a
+  case compare deg 2 of
-    go ( a :| [] ) = a :| []
+    LT -> pure ()
-    go ( a :| a' : as ) = case go ( a' :| as ) of
+    EQ -> shrinkMutablePrimArray p deg
-      ( b' :| bs ) -> ( a + r * b' ) :| ( b' : bs )
+    GT -> do
-deflate _ _ = error "deflate: polynomial of degree < 2"
+      shrinkMutablePrimArray p deg
+      let
+        go :: a -> Int -> m ()
+        go b i = unless ( i >= deg ) do
+          ai <- readPrimArray p i
+          writePrimArray p i ( ai + r * b )
+          go ai ( i + 1 )
+      a0 <- readPrimArray p 0
+      go a0 1
 
 -- | Laguerre's method.
 laguerre
-  :: forall a. RealFloat a
+  :: forall a m s
+  .  ( RealFloat a, Prim a, PrimMonad m, s ~ PrimState m )
   => a                                -- ^ error tolerance
   -> Int                              -- ^ max number of iterations
-  -> [ Complex a ] -- ^ polynomial
+  -> MutablePrimArray s ( Complex a ) -- ^ polynomial
   -> Complex a                        -- ^ initial point
-  -> Complex a
+  -> m ( Complex a )
-laguerre eps maxIters p = go maxIters
+laguerre eps maxIters p x0 = do
-  where
+  p'  <- derivative p
-    p', p'' :: [ Complex a ]
+  p'' <- derivative p'
-    p'  = derivative p
+  let
-    p'' = derivative p'
+    go :: Int -> Complex a -> m ( Complex a )
-    go :: Int -> Complex a -> Complex a
+    go iterationsLeft x = do
-    go iterationsLeft x
+      x' <- laguerreStep eps p p' p'' x
-      | iterationsLeft <= 1
+      if iterationsLeft <= 1 || magnitude ( x' - x ) < eps
-      || magnitude ( x' - x ) < eps = x'
+      then pure x'
-      | otherwise                   = go ( iterationsLeft - 1 ) x'
+      else go ( iterationsLeft - 1 ) x'
-      where
+  go maxIters x0
-        x' :: Complex a
-        x' = laguerreStep eps p p' p'' x
 
 -- | Take a single step in Laguerre's method.
 laguerreStep
-  :: forall a. RealFloat a
+  :: forall a m s
+  .  ( RealFloat a, Prim a, PrimMonad m, s ~ PrimState m )
   => a                                -- ^ error tolerance
-  -> [ Complex a ] -- ^ polynomial
+  -> MutablePrimArray s ( Complex a ) -- ^ polynomial
-  -> [ Complex a ] -- ^ first derivative of polynomial
+  -> MutablePrimArray s ( Complex a ) -- ^ first derivative of polynomial
-  -> [ Complex a ] -- ^ second derivative of polynomial
+  -> MutablePrimArray s ( Complex a ) -- ^ second derivative of polynomial
   -> Complex a                        -- ^ initial point
-  -> Complex a
+  -> m ( Complex a )
-laguerreStep eps p p' p'' x 
+laguerreStep eps p p' p'' x = do
-  | magnitude px < eps = x
+  n  <- fromIntegral @_ @a <$> getSizeofMutablePrimArray p
-  | otherwise          = x - n / denom
+  px <- eval p x
-  where
+  if magnitude px < eps
-    n     = fromIntegral ( length p )
+  then pure x
-    px    = eval p   x
+  else do
-    p'x   = eval p'  x
+    p'x  <- eval p'  x
-    p''x  = eval p'' x
+    p''x <- eval p'' x
+    let
       g     = p'x / px
       g²    = g * g
       h     = g² - p''x / px
-    delta = sqrt $ ( n - 1 ) * ( n * h - g² )
+      delta = sqrt $ ( n - 1 ) *: ( n *: h - g² )
       gp    = g + delta
       gm    = g - delta
       denom
@@ -110,11 +174,42 @@ laguerreStep eps p p' p'' x
         = gm
         | otherwise
         = gp
+    pure $ x - n *: ( recip denom )
+
+  where
+    (*:) :: a -> Complex a -> Complex a
+    r *: (u :+ v) = ( r * u ) :+ ( r * v )
 
 -- | Evaluate a polynomial.
-eval :: Num a => [ a ] -> a -> a
+eval
-eval as x = foldr ( \ a b -> a + x * b ) 0 as
+  :: forall a m s
+  . ( Num a, Prim a, PrimMonad m, s ~ PrimState m )
+  => MutablePrimArray s a -> a -> m a
+eval p x = do
+  n <- getSizeofMutablePrimArray p
+  let
+    go :: a -> Int -> m a
+    go !a i =
+      if i >= n
+      then pure a
+      else do
+        !b <- readPrimArray p i
+        go ( b + x * a ) ( i + 1 )
+  an <- readPrimArray p 0
+  go an 1
 
 -- | Derivative of a polynomial.
-derivative :: Num a => [ a ] -> [ a ]
+derivative
-derivative as = zipWith ( \ i a -> fromIntegral i * a ) [ ( 1 :: Int ) .. ] ( tail as )
+  :: forall a m s
+  . ( Num a, Prim a, PrimMonad m, s ~ PrimState m )
+  => MutablePrimArray s a -> m ( MutablePrimArray s a )
+derivative p = do
+  deg <- subtract 1 <$> getSizeofMutablePrimArray p
+  p' <- cloneMutablePrimArray p 0 deg
+  let
+    go :: Int -> m ()
+    go i = unless ( i >= deg ) do
+      a <- readPrimArray p' i
+      writePrimArray p' i ( a * fromIntegral ( deg - i ) )
+  go 0
+  pure p'