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products/Sources/formale Sprachen/PVS/float/ (Beweissystem der NASA Version 6.0.9^©) Datei vom 28.9.2014 mit Größe 5 kB

Quelle axpy.pvs Sprache: PVS

axpy[radix:above(1), (IMPORTING float[radix]) b:Format]: THEORY
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  This theory defines a faithful computation of  a*x + y, where
%  a,x, and y are floating point numbers.
%  Author:
%  Sylvie Boldo (ENS-Lyon)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
BEGIN

FcanonicBounded2: JUDGEMENT {x:float | Fcanonic?(b)(x)} SUBTYPE_OF {x:float | Fbounded?(b)(x)}

MinOrMax?(r:real,f:(Fbounded?(b))):bool=isMin?(b)(r,f) OR isMax?(b)(r,f)
Fless(f:float):float= if    0 < FtoR(f) then Fpred(b)(f)
        elsif FtoR(f) < 0 then Fsucc(b)(f)
                      else  f endif

% Variables and lemmas

f,p,q       : VAR float
a,x,y,t,u   : VAR {x:float | Fcanonic?(b)(x)}
a1,x1,y1,z  : VAR real
n           : VAR nat

% MinOrMax properties
MinOrMax_Rlt : lemma Fbounded?(b)(f) => MinOrMax?(z,f) => abs(FtoR(f)-z) < Fulp(b)(f)

MinOrMax_Fopp : lemma Fbounded?(b)(f) =>MinOrMax?(z,f) => MinOrMax?(-z,Fopp(f))

MinOrMax1 : lemma Fcanonic?(b)(f) => 0 < FtoR(f)
     => abs(FtoR(f)-z) < Fulp(b)(Fpred(b)(f)) => MinOrMax?(z,f)

MinOrMax2 : lemma Fcanonic?(b)(f) => 0 < FtoR(f)
     => abs(FtoR(f)-z) < Fulp(b)(f) => FtoR(f) <= z => MinOrMax?(z,f)

MinOrMax3 : lemma Fcanonic?(b)(f) => 0 = FtoR(f)
     => abs(FtoR(f)-z) < Fulp(b)(f) => MinOrMax?(z,f)

% Links between a real and its rounding
RoundLe : lemma Fcanonic?(b)(f) => NOT FtoR(f)=0 => Closest?(b)(z,f)
     => abs(FtoR(f)) <= abs(z)/(1-1/(2*abs(Fnum(f))))

RoundGe : lemma Fcanonic?(b)(f) => NOT FtoR(f)=0 => Closest?(b)(z,f)
     =>  abs(z)/(1+1/(2*abs(Fnum(f)))) <= abs(FtoR(f))

% Various lemmas
ExactSum_Near : lemma
  Fbounded?(b)(p) => Fbounded?(b)(q) => Fbounded?(b)(f) => Closest?(b)(FtoR(p)+FtoR(q),f)
    => abs(FtoR(f)-(FtoR(p)+FtoR(q))) < radix^(-dExp(b)) => FtoR(f)=FtoR(p)+FtoR(q)

Normal_iff : lemma
  Fcanonic?(b)(f) =>
    (Fnormal?(b)(f) IFF radix^(Prec(b)-1-dExp(b)) <= abs(FtoR(f)))

% Let us go !
Axpy_aux1 : lemma Closest?(b)(FtoR(a)*FtoR(x),t) => Closest?(b)(FtoR(t)+FtoR(y),u) => 0 < FtoR(u)
   => abs(FtoR(t)-FtoR(a)*FtoR(x))          <=  Fulp(b)(Fpred(b)(u))/4
   => abs(y1-FtoR(y)+a1*x1-FtoR(a)*FtoR(x)) <   Fulp(b)(Fpred(b)(u))/4
   => MinOrMax?(y1+a1*x1,u)

Axpy_aux1_aux1 : lemma Closest?(b)(FtoR(a)*FtoR(x),t) => Closest?(b)(FtoR(t)+FtoR(y),u) => 0 < FtoR(u)
  => Fnormal?(b)(t) => radix*abs(FtoR(t)) <= FtoR(Fpred(b)(u))
  => abs(FtoR(t)-FtoR(a)*FtoR(x))  <=  Fulp(b)(Fpred(b)(u))/4

Axpy_aux1_aux2 : lemma Closest?(b)(FtoR(a)*FtoR(x),t) => Closest?(b)(FtoR(t)+FtoR(y),u) => 0 < FtoR(u)
  => Fsubnormal?(b)(t) => 1-dExp(b) <= Fexp(Fpred(b)(u))
  => abs(FtoR(t)-FtoR(a)*FtoR(x)) <=  Fulp(b)(Fpred(b)(u))/4

Axpy_aux2 : lemma Closest?(b)(FtoR(a)*FtoR(x),t) => Closest?(b)(FtoR(t)+FtoR(y),u) => 0 < FtoR(u)
  => Fsubnormal?(b)(t) => FtoR(u)=FtoR(t)+FtoR(y)
  => abs(y1-FtoR(y)+a1*x1-FtoR(a)*FtoR(x)) < Fulp(b)(Fpred(b)(u))/4
  => MinOrMax?(y1+a1*x1,u)

Axpy_aux3 : lemma Closest?(b)(FtoR(a)*FtoR(x),t) => Closest?(b)(FtoR(t)+FtoR(y),u) => 0 < FtoR(u)
  => Fsubnormal?(b)(t)
  => -dExp(b) = Fexp(Fpred(b)(u)) =>  1-dExp(b) <= Fexp(u)
  => abs(y1-FtoR(y)+a1*x1-FtoR(a)*FtoR(x)) < Fulp(b)(Fpred(b)(u))/4
  => MinOrMax?(y1+a1*x1,u)

AxpyPos : lemma Closest?(b)(FtoR(a)*FtoR(x),t) => Closest?(b)(FtoR(t)+FtoR(y),u) => 0 < FtoR(u)
  => (Fnormal?(b)(t) => radix*abs(FtoR(t)) <= FtoR(Fpred(b)(u)))
  => abs(y1-FtoR(y)+a1*x1-FtoR(a)*FtoR(x)) < Fulp(b)(Fpred(b)(u))/4
  => MinOrMax?(y1+a1*x1,u)

Axpy_opt_aux1_aux1 : lemma Fnormal?(b)(t) => Fnormal?(b)(u) => 0 < FtoR(u)
    => Prec(b) >= 3 =>
   (1+radix*(1+1/(2*abs(Fnum(u))))*(1+1/abs(Fnum(Fpred(b)(u)))))/(1-1/(2*abs(Fnum(t))))
      <= 1+radix+radix^(4-Prec(b))

Axpy_opt_aux1 : lemma Closest?(b)(FtoR(a)*FtoR(x),t) => Closest?(b)(FtoR(t)+FtoR(y),u) => 0 < FtoR(u)
  =>  Prec(b) >= 3 => Fnormal?(b)(t)
  => (radix+1+radix^(4-Prec(b)))*abs(FtoR(a)*FtoR(x)) <= abs(FtoR(y))
  =>  radix*abs(FtoR(t)) <= FtoR(Fpred(b)(u))

Axpy_opt_aux2 : lemma  Closest?(b)(FtoR(a)*FtoR(x),t) => Closest?(b)(FtoR(t)+FtoR(y),u) => 0 < FtoR(u)
  =>  Prec(b) >= 6 =>  Fnormal?(b)(t)
  => (radix+1+radix^(4-Prec(b)))*abs(FtoR(a)*FtoR(x)) <= abs(FtoR(y))
  =>  abs(FtoR(y))*radix^(1-Prec(b))/(radix+1) < Fulp(b)(Fpred(b)(u))

Axpy_opt_aux3 : lemma  Closest?(b)(FtoR(a)*FtoR(x),t) => Closest?(b)(FtoR(t)+FtoR(y),u) => 0 < FtoR(u)
  =>  Prec(b) >= 3  =>  Fsubnormal?(b)(t)
  => (radix+1+radix^(4-Prec(b)))*abs(FtoR(a)*FtoR(x)) <= abs(FtoR(y))
  =>  abs(FtoR(y))*radix^(1-Prec(b))/(radix+radix/2) <= Fulp(b)(Fpred(b)(u))

Axpy_optPos : lemma Closest?(b)(FtoR(a)*FtoR(x),t) => Closest?(b)(FtoR(t)+FtoR(y),u) => 0 < FtoR(u)
  => Prec(b) >= 6
  => (radix+1+radix^(4-Prec(b)))*abs(FtoR(a)*FtoR(x)) <= abs(FtoR(y))
  => abs(y1-FtoR(y)+a1*x1-FtoR(a)*FtoR(x)) < abs(FtoR(y))*radix^(1-Prec(b))/(6*radix)
  => MinOrMax?(y1+a1*x1,u)

Axpy_optZero : lemma Closest?(b)(FtoR(a)*FtoR(x),t) => Closest?(b)(FtoR(t)+FtoR(y),u) => 0 = FtoR(u)
  => (radix+1+radix^(4-Prec(b)))*abs(FtoR(a)*FtoR(x)) <= abs(FtoR(y))
  => abs(y1-FtoR(y)+a1*x1-FtoR(a)*FtoR(x)) < abs(FtoR(y))*radix^(1-Prec(b))/(6*radix)
  => MinOrMax?(y1+a1*x1,u)

Axpy_opt : lemma Closest?(b)(FtoR(a)*FtoR(x),t) => Closest?(b)(FtoR(t)+FtoR(y),u)
  => Prec(b) >= 6
  => (radix+1+radix^(4-Prec(b)))*abs(FtoR(a)*FtoR(x)) <= abs(FtoR(y))
  => abs(y1-FtoR(y)+a1*x1-FtoR(a)*FtoR(x)) < abs(FtoR(y))*radix^(1-Prec(b))/(6*radix)
  => MinOrMax?(y1+a1*x1,u)

Axpy_simpl : lemma Closest?(b)(FtoR(a)*FtoR(x),t) => Closest?(b)(FtoR(t)+FtoR(y),u)
  => Prec(b) >= 24 => radix=2
  => (3+1/100000)*abs(FtoR(a)*FtoR(x)) <= abs(FtoR(y))
  => abs(y1-FtoR(y)+a1*x1-FtoR(a)*FtoR(x)) < abs(FtoR(y))*2^(1-Prec(b))/12
  => MinOrMax?(y1+a1*x1,u)

END axpy

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