FroundDivSqrt

# Library FroundDivSqrt

`Require Export AllFloat.
Require Export Classical.
Section FroundDiv.
Variable b : Fbound.
Variable radix : Z.
Variable precision : nat.
Let FtoRradix := FtoR radix.
Coercion FtoRradix : float >-> R.
Hypothesis radixMoreThanOne : (1 < radix)%Z.
Let radixMoreThanZERO := Zlt_1_O _ (Zlt_le_weak _ _ radixMoreThanOne).
Hint Resolve radixMoreThanZERO: zarith.
Hypothesis precisionGreaterThanOne : 1 < precision.
Hypothesis pGivesBound : Zpos (vNum b) = Zpower_nat radix precision.
Theorem NearestInteger :
forall (r : R) (n : Z),
(forall z : Z, (Rabs (n - r) <= Rabs (z - r))%R) ->
(Rabs (n - r) <= / 2)%R.
Theorem errorBoundedModulo_aux :
forall (x y : float) (n : Z),
Fbounded b x ->
Fcanonic radix b x ->
Fbounded b y ->
Fcanonic radix b y ->
FtoRradix y <> 0%R ->
(0 < y)%R ->
(0 <= x)%R ->
(forall z : Z, (Rabs (n - x / y) <= Rabs (z - x / y))%R) ->
Fbounded b (Fminus radix x (Fmult (Float n 0) y)).
Theorem errorBoundedModulo_aux_y :
forall (x y : float) (n : Z),
Fbounded b x ->
Fcanonic radix b x ->
Fbounded b y ->
Fcanonic radix b y ->
FtoRradix y <> 0%R ->
(0 <= x)%R ->
(forall z : Z, (Rabs (n - x / y) <= Rabs (z - x / y))%R) ->
Fbounded b (Fminus radix x (Fmult (Float n 0) y)).
Theorem errorBoundedModuloCan :
forall (x y : float) (n : Z),
Fbounded b x ->
Fcanonic radix b x ->
Fbounded b y ->
Fcanonic radix b y ->
FtoRradix y <> 0%R ->
(forall z : Z, (Rabs (n - x / y) <= Rabs (z - x / y))%R) ->
Fbounded b (Fminus radix x (Fmult (Float n 0) y)).
Theorem errorBoundedRem :
forall (x y : float) (n : Z),
(Fbounded b x) -> (Fbounded b y) ->
y <> 0 :>R ->
(forall z : Z, (Rabs (n - x / y) <= Rabs (z - x / y))%R) ->
(Fbounded b
(Fminus radix (Fnormalize radix b precision x)
(Fmult (Float n 0) (Fnormalize radix b precision y)))).
Theorem errorBoundedDiv :
forall (x y q : float) P,
(RoundedModeP b radix P) ->
(Fbounded b x) -> (Fbounded b y) -> (Fbounded b q) ->
y <> 0 :>R -> (P (x / y)%R q) ->
(- dExp b <= Fexp q + Fexp y)%Z ->
(Rabs q) <> (powerRZ radix (- dExp b)) \/
((powerRZ radix (- dExp b)) / 2%nat <= Rabs (x / y))%R ->
(Fbounded b (Fminus radix x (Fmult q y))).
Theorem errorBoundedDivSimplHyp :
forall (x y q : float) P,
RoundedModeP b radix P ->
Fbounded b x ->
Fbounded b y ->
Fbounded b q ->
FtoRradix y <> 0%R ->
P (x / y)%R q ->
(- dExp b <= Fexp (Fnormalize radix b precision x) - precision)%Z ->
(- dExp b <= Fexp q + Fexp y)%Z.
Theorem errorBoundedDivClosest :
forall x y q : float,
Fbounded b x ->
Fbounded b y ->
Fbounded b q ->
FtoRradix y <> 0%R ->
Closest b radix (x / y) q ->
(- dExp b <= Fexp q + Fexp y)%Z -> Fbounded b (Fminus radix x (Fmult q y)).
Theorem errorBoundedDivToZero :
forall x y q : float,
Fbounded b x ->
Fbounded b y ->
Fbounded b q ->
FtoRradix y <> 0%R ->
ToZeroP b radix (x / y) q ->
(- dExp b <= Fexp q + Fexp y)%Z -> Fbounded b (Fminus radix x (Fmult q y)).
Theorem errorBoundedSqrt :
forall x q : float,
(Fbounded b x) -> (Fbounded b q) ->
(0 <= x)%R -> (Closest b radix (sqrt x) q) ->
(- dExp b <= Fexp q + Fexp q)%Z ->
(Fbounded b (Fminus radix x (Fmult q q))).
End FroundDiv.
`