FArgReduct3

# Library FArgReduct3

``
FArgReduct3 file
Sylvie Boldo
This file explains an improvement of Cody & Waite argument reduction technique using the FMA (fused-multiply-and-add).
`Require Export FArgReduct2.
Section Total.
Let radix := 2%Z.
Let FtoRradix := FtoR radix.
Coercion FtoRradix : float >-> R.
`
Variables
`
Variable b : Fbound.
Variables prec : nat.
Variable N : Z.
Variables alpha gamma x zH1 u : float.
Variable zL : R.
`
Various bounds
`
Let bmoinsq :=
Bound
(P_of_succ_nat (pred (Zabs_nat (Zpower_nat radix (prec - 2)%nat))))
(dExp b).
`
All the hypotheses
`
Hypothesis pGivesBound : Zpos (vNum b) = Zpower_nat radix prec.
Hypothesis Fboundedx : Fbounded b x.
Hypothesis xCanonic : Fcanonic radix b x.
Hypothesis xPos : (0 <= x)%R.
`
alpha (the constant, such as pi) and gamma (its inverse)
`Hypothesis alphaNormal : Fnormal radix b alpha.
Hypothesis gammaNormal : Fnormal radix bmoinsq gamma.
Hypothesis alphaPos : (0 < alpha)%R.
Hypothesis gammaPos : (0 < gamma)%R.
Hypothesis
gamma_not_pow_2 : forall e : Z, FtoRradix gamma <> powerRZ radix e.
Hypothesis gammaDef : Closest bmoinsq radix (/ alpha) gamma.
`
About the computation of z
`Hypothesis
uDef :
Closest b radix
(3%nat * powerRZ radix (Zpred (Zpred (prec - N))) + x * alpha) u.
Hypothesis
zH1Def :
Closest b radix (u - 3%nat * powerRZ radix (Zpred (Zpred (prec - N))))
zH1.
Hypothesis precMoreThanThree : 3 < prec.
Hypothesis N_not_too_big : (N <= dExp b)%Z.
`
x not too big
`Hypothesis
xalpha_small :
(x * alpha <=
powerRZ radix (Zpred (Zpred (prec - N))) - powerRZ radix (- N))%R.
`
No underflow
`Hypothesis gamma_ge: (powerRZ radix (-dExp b+prec+(Zmax (-1) N))<= gamma)%R.
`
A few lemmas
`Theorem gamma_ge2: (powerRZ radix (-dExp b+prec -2 +(Zmax 1 (N-1)))<= gamma)%R.
Theorem exp_gamma_enough2: (- dExp b <= Zpred (Zpred (Fexp gamma)))%Z.
Theorem exp_alpha_le: (- dExp b <= 3 + - (Fexp alpha + (prec + prec)))%Z.
Theorem exp_gamma_enough3: (- dExp b <= Fexp gamma - N-3)%Z.
Theorem gamma_p:
ex (fun gam2 : float => FtoRradix gam2 = gamma /\ Fnormal radix b gam2 /\ (Fexp gam2=Zpred (Zpred (Fexp gamma))) /\
((powerRZ radix (Zpred prec))+4 <=Fnum gam2)%R /\ (Fnum gam2 <= (powerRZ radix prec)-4)%R).
`
Main result: q=2
`
Theorem Fmac_arg_reduct_correct3 :
ex (fun u : float => FtoRradix u = (x - zH1 * gamma)%R /\ Fbounded b u)
/\ (Rabs (x-zH1*gamma) < powerRZ radix (prec-N+Fexp gamma))%R.
End Total.
`