Speed, Correctness, or Simplicity: Choose 3

Transcript

1. 1.

   Speed,  Correctness,  or  Simplicity:     Choose  3   Tom

    Switzer   @9xxit     h;ps://github.com/9xxit/fpfilter-­‐talk  
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   Overview     Floa9ng  point  is  “good  enough”…    

   most  of  the  9me.  
3. 3.

   Op9ons   Use  Double,  live  with  the  errors.    

   Use  higher  precision  type,   live  with  performance  loss.     But,  there  is  a  3rd  op9on…  
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   Floa9ng  Point  Filters     Use  floa9ng  point  when  you

    can.     Use  higher  precision  when  you  can’t.  
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   Err…  Not  So  Simple   Solve  problem  using  floa9ng  point

     approxima9on…     Maintain  an  error  bound  on  approxima9on.     Re-­‐evaluate  with  exact  type  if  error  too  large.  
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   The  Catch  

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       What  is  the  determinant  of  my  matrix?  

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   Not  Good  For:  Minimizing  Errors   in  Floa9ng  Point  Arithme9c

    
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       What  is  the  sign  of  the  determinant  of

    my   matrix?  
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    Good  For:  Making  a  Decision  

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    FpFilter[A]   Simple  wrapper:  FpFilter[Rational] Standard  Opera2ons   +,  -­‐,

     *,  /,  .sqrt,  etc     Fast  predictes   signum,  compare,  isWhole,  etc.  
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    FpFilter[A]   class FpFilter[A]( apx: Double, mes: Double, ind: Int,

    exact: => A ) { … } floa9ng  point  approxima9on   error  bounds  
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    FpFilter[A]   class FpFilter[A]( apx: Double, mes: Double, ind: Int,

    exact: => A ) { … } error  bounds   “Exact  Geometric  Computa2on   Using  Cascading.”   Burnikel,  Funke  &  Seel.  
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    FpFilter[A]   class FpFilter[A]( apx: Double, mes: Double, ind: Int,

    exact: => A ) { … } error  bounds   thunk  for  higher  precision   Welcome  to  …  
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    …  Macro  City   def abs(implicit ev: Signed[A]): FpFilter[A] =

    macro FpFilter.absImpl[A] def unary_- (implicit ev: Rng[A]) : FpFilter[A] = macro FpFilter.negateImpl[A] def +(rhs: FpFilter[A])(implicit ev: Semiring[A]): FpFilter[A] = macro FpFilter.plusImpl[A] def -(rhs: FpFilter[A])(implicit ev: Rng[A]): FpFilter[A] = macro FpFilter.minusImpl[A] def *(rhs: FpFilter[A])(implicit ev: Semiring[A]): FpFilter[A] = macro FpFilter.timesImpl[A] def /(rhs: FpFilter[A])(implicit ev: Field[A]): FpFilter[A] = macro FpFilter.divideImpl[A] def sqrt(implicit ev: NRoot[A]): FpFilter[A] = macro FpFilter.sqrtImpl[A] def <(rhs: FpFilter[A])(implicit ev0: Signed[A], ev1: Rng[A]): Boolean = macro FpFilter.ltImpl[A] def >(rhs: FpFilter[A])(implicit ev0: Signed[A], ev1: Rng[A]): Boolean = macro FpFilter.gtImpl[A] def <=(rhs: FpFilter[A])(implicit ev0: Signed[A], ev1: Rng[A]): Boolean = macro FpFilter.ltEqImpl[A] def >=(rhs: FpFilter[A])(implicit ev0: Signed[A], ev1: Rng[A]): Boolean = macro FpFilter.gtEqImpl[A] def ===(rhs: FpFilter[A])(implicit ev0: Signed[A], ev1: Rng[A]): Boolean = macro FpFilter.eqImpl[A] def signum(implicit ev: Signed[A]): Int = macro FpFilter.signImpl[A]
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    …  Macro  City   •  Operator  fusion   – No  intermediate

     alloca9ons   •  In-­‐line  approxima9on  +  error  bounds   – Fast,  Double  arithme9c   •  Thunk  becomes  inner  defs   – Compile  down  to  private  methods  
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    Turn  this…   (x + y).signum

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    …  into  this.   val fpf$tmp$macro$38 = x.value; val fpf$apx$macro$39

    = fpf$tmp$macro$38; val fpf$mes$macro$40 = java.lang.Math.abs(fpf$tmp$macro$38); def fpf$exact$macro$42 = spire.algebra.Field.apply[spire.math.Algebraic](Algebraic.AlgebraicAlgebra).fromDouble(fpf $tmp$macro$38); val fpf$tmp$macro$43 = y.value; val fpf$apx$macro$44 = fpf$tmp$macro$43; val fpf$mes$macro$45 = java.lang.Math.abs(fpf$tmp$macro$43); def fpf$exact$macro$47 = spire.algebra.Field.apply[Algebraic](Algebraic.AlgebraicAlgebra).fromDouble(fpf$tmp$macro $43); val fpf$apx$macro$48 = fpf$apx$macro$39.+(fpf$apx$macro$44); val fpf$mes$macro$49 = fpf$mes$macro$40.+(fpf$mes$macro$45); def fpf$exact$macro$51 = Algebraic.AlgebraicAlgebra.plus( fpf$exact$macro$42, fpf$exact$macro$47); val fpf$err$macro$52 = fpf$mes$macro$49.$times(1).$times(2.220446049250313E-16); if (fpf$apx$macro$48 > fpf$err$macro$52 && fpf$apx$macro$48 < Double.POSITIVE_INFINITY) 1 else if (fpf$apx$macro$48 < fpf$err$macro$52.unary_$minus && fpf$apx$macro$48 > Double.NEGATIVE_INFINITY) -1 else if (fpf$err$macro$52 == 0.0) 0 else Algebraic.AlgebraicAlgebra.signum(fpf$exact$macro$51)
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    Examples  

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    2D  Orienta2on  

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    p   q   r  

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    p   q   r  

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    p   q   r   RIGHT  

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    p   r   q  

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    p   r   q   LEFT  

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    p   r   q  

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    p   r   q   NO  TURN  

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    trait Turn[@spec A] { def apply( px: A, py: A,

    qx: A, qy: A, rx: A, ry: A ): Int }
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    object FastTurn extends Turn[Double] { def apply( px: Double, py:

    Double, qx: Double, qy: Double, rx: Double, ry: Double ): Int = signum { (qx - px) * (ry - py) - (rx - px) * (qy - py) } }
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    Accuracy  of  Fast  Turn  

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    object ExactTurn extends Turn[Double] { def apply( px: Double, py:

    Double, qx: Double, qy: Double, rx: Double, ry: Double ): Int = { val pxa = Algebraic(px) val pya = Algebraic(py) val qxa = Algebraic(qx) val qya = Algebraic(qy) val rxa = Algebraic(rx) val rya = Algebraic(ry) ((qxa - pxa) * (rya - pya) – (rxa - pxa) * (qya - pya)).signum } }
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    10,000x  Slower!  

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    Let’s  try  again…  

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    object FilteredTurn extends Turn[Double] { def apply( px: Double, py:

    Double, qx: Double, qy: Double, rx: Double, ry: Double ): Int = { val pxf = FpFilter.exact[Algebraic](px) val pyf = FpFilter.exact[Algebraic](py) val qxf = FpFilter.exact[Algebraic](qx) val qyf = FpFilter.exact[Algebraic](qy) val rxf = FpFilter.exact[Algebraic](rx) val ryf = FpFilter.exact[Algebraic](ry) ((qxf - pxf) * (ryf - pyf) – (rxf - pxf) * (qyf - pyf)).signum } }
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    FilteredTurn  Speed   Rela9ve  to  FastTurn  

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    Polynomial  Root  Finding  

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    Polynomial[A]  

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    Interval[A]   Root  

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    “Quadra2c  Interval  Refinement  for   Real  Roots.”  John  AbboT.  

    QIR  for  short.  
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    QIR  (N=8)  

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    QIR  (N=8)  

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    QIR  (N=8)  

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    QIR  (N=8)  

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    QIR  (N=8)  

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    QIR  (N=8)  

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    QIR  (N=8)  

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    QIR  (N=8)  

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    QIR  (N=16)  

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    QIR  (N=16)  

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    QIR  (N=16)  

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    QIR   •  Requires  2  polynomial  evalua9ons   – High  precision

     generally  required   •  Very  fast  convergence  (quadra9c)   – Under  some  assump9ons   •  Occasionally  fails  when  assump9ons  not  met   – Fallsback  to  bisec9on!  
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    QIR  (N=8)  

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    QIR  (N=8)  

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    QIR  (N=8)   FAILED!  

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    Falls  back  to  bisec9on…  

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    Bisec9on  

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    Bisec9on  

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    Bisec9on  

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    Bisec9on  

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    Bisec9on  

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    Bisec9on  

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    Bisec9on  

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    Bisec9on     Requires  only  sign  tests     Converges

     slowly,  1  bit  at  a  9me!  
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    trait SignTest[A] { def apply( poly: Polynomial[A], x: A ):

    Sign }
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    final class FilteredSignTest[@sp A: Semiring]( implicit A: IsAlgebraic[A] ) extends

    SignTest[A] { def apply(poly: Polynomial[A], x: A): Sign = { val x0 = FpFilter(A.toDouble(x), A.toAlgebraic(x)) @tailrec def loop(acc: FpFilter[Algebraic], i: Int): Sign = if (i >= 0) { val c = poly.nth(i) val cftr = FpFilter(A.toDouble(c), A.toAlgebraic(c)) loop(cftr + acc * x0, i - 1) } else { Sign(acc.signum) } loop(FpFilter.approx(Algebraic.Zero), poly.degree) } }
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    Accuracy  Using  Double  

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    Speed  Up  from  Exact  Sign  Test   Fast  (d=8)  

    Fast  (d=16)   Fast  (d=32)   Filtered  (d=8)   Filtered  (d=16)   Filtered  (d=16)  
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    Summary   •  Works  like  any  other  number  type  

    – Operator  fusion  +  inlining  within  expressions   •  Speeds  up  predicates   – Sign  tests,  comparisons,  etc.   •  Near-­‐Double  performance   – 2-­‐4x  in  most  cases   h;p://github.com/9xxit/fpfilter-­‐talk  
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    Thanks!   h;p://github.com/non/spire     Tom  Switzer  @9xxit