ClosestMult

# Library ClosestMult

`Require Export FroundMult.
Require Export ClosestProp.
Section FRoundP.
Variable b : Fbound.
Variable radix : Z.
Variable precision : nat.
Coercion Local FtoRradix := FtoR radix.
Hypothesis radixMoreThanOne : (1 < radix)%Z.
Hypothesis precisionGreaterThanOne : 1 < precision.
Hypothesis pGivesBound : Zpos (vNum b) = Zpower_nat radix precision.
Theorem closestLessMultPos :
forall (p : float) (r : R),
Closest b radix r p -> (0 <= r)%R -> (p <= 2%nat * r)%R.
Theorem closestLessMultNeg :
forall (p : float) (r : R),
Closest b radix r p -> (r <= 0)%R -> (2%nat * r <= p)%R.
Theorem closestLessMultAbs :
forall (p : float) (r : R),
Closest b radix r p -> (Rabs p <= 2%nat * Rabs r)%R.
Theorem errorBoundedMultClosest_aux :
forall p q pq : float,
Fbounded b p ->
Fbounded b q ->
Closest b radix (p * q) pq ->
(- dExp b <= Fexp p + Fexp q)%Z ->
(p * q - pq)%R <> 0%R :>R ->
ex
(fun r : float =>
ex
(fun s : float =>
Fcanonic radix b r /\
Fbounded b r /\
Fbounded b s /\
r = pq :>R /\
s = (p * q - r)%R :>R /\
Fexp s = (Fexp p + Fexp q)%Z :>Z /\
(Fexp s <= Fexp r)%Z /\ (Fexp r <= precision + (Fexp p + Fexp q))%Z)).
Theorem errorBoundedMultClosest :
forall p q pq : float,
Fbounded b p ->
Fbounded b q ->
Closest b radix (p * q) pq ->
(- dExp b <= Fexp p + Fexp q)%Z ->
(- dExp b <= Fexp (Fnormalize radix b precision pq) - precision)%Z ->
ex
(fun r : float =>
ex
(fun s : float =>
Fcanonic radix b r /\
Fbounded b r /\
Fbounded b s /\
r = pq :>R /\
s = (p * q - r)%R :>R /\ Fexp s = (Fexp r - precision)%Z :>Z)).
End FRoundP.
`