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(** {1 Floating-Point Computations with overflow check}
The following functions could be used to encode floating-point
operations in programs, in the case where one wants to forbids
overflows.
Each function comes with two pre-conditions and two post-conditions,
one using interpretation in real numbers, one using the predicate
`t'isFinite` from the `ieee_float` library. The second form should be
better with a prover that has built-in support for floating-point
numbers.
*)
(** {2 Single-precision floats} *)
module Single
use real.RealInfix
use export ieee_float.Float32
predicate add_pre_ieee (x:t) (y:t) =
t'isFinite (x .+ y)
predicate add_post_ieee (x:t) (y:t) (r:t) =
r = x .+ y
predicate add_pre_real (x:t) (y:t) =
no_overflow RNE (t'real x +. t'real y)
predicate add_post_real (x:t) (y:t) (r:t) =
t'real r = round RNE (t'real x +. t'real y)
val safe_add (x y: t) : t
requires { [@expl:finite floating-point number] t'isFinite x }
requires { [@expl:finite floating-point number] t'isFinite y }
requires { [@expl:floating-point overflow]
add_pre_ieee x y \/ add_pre_real x y }
ensures { t'isFinite result }
ensures { add_post_ieee x y result /\ add_post_real x y result }
predicate sub_pre_ieee (x:t) (y:t) =
t'isFinite (x .- y)
predicate sub_post_ieee (x:t) (y:t) (r:t) =
r = x .- y
predicate sub_pre_real (x:t) (y:t) =
no_overflow RNE (t'real x -. t'real y)
predicate sub_post_real (x:t) (y:t) (r:t) =
t'real r = round RNE (t'real x -. t'real y)
val safe_sub (x y: t) : t
requires { [@expl:finite floating-point number] t'isFinite x }
requires { [@expl:finite floating-point number] t'isFinite y }
requires { [@expl:floating-point overflow]
sub_pre_ieee x y \/ sub_pre_real x y }
ensures { t'isFinite result }
ensures { sub_post_ieee x y result /\ sub_post_real x y result }
predicate mul_pre_ieee (x:t) (y:t) =
t'isFinite (x .* y)
predicate mul_post_ieee (x:t) (y:t) (r:t) =
r = x .* y
predicate mul_pre_real (x:t) (y:t) =
no_overflow RNE (t'real x *. t'real y)
predicate mul_post_real (x:t) (y:t) (r:t) =
t'real r = round RNE (t'real x *. t'real y)
val safe_mul (x y: t) : t
requires { [@expl:finite floating-point number] t'isFinite x }
requires { [@expl:finite floating-point number] t'isFinite y }
requires { [@expl:floating-point overflow]
mul_pre_ieee x y \/ mul_pre_real x y }
ensures { t'isFinite result }
ensures { mul_post_ieee x y result /\ mul_post_real x y result }
predicate div_pre_ieee (x:t) (y:t) =
t'isFinite (x ./ y)
predicate div_post_ieee (x:t) (y:t) (r:t) =
r = x ./ y
predicate div_pre_real (x:t) (y:t) =
no_overflow RNE (t'real x /. t'real y)
predicate div_post_real (x:t) (y:t) (r:t) =
t'real r = round RNE (t'real x /. t'real y)
val safe_div (x y: t) : t
requires { [@expl:finite floating-point number] t'isFinite x }
requires { [@expl:finite floating-point number] t'isFinite y }
requires { [@expl:non-zero floating-point number] not (is_zero y) }
requires { [@expl:floating-point overflow]
div_pre_ieee x y \/ div_pre_real x y }
ensures { t'isFinite result }
ensures { div_post_ieee x y result /\ div_post_real x y result }
end
(** {2 Double-precision floats} *)
module Double
use real.RealInfix
use export ieee_float.Float64
predicate add_pre_ieee (x:t) (y:t) =
t'isFinite (x .+ y)
predicate add_post_ieee (x:t) (y:t) (r:t) =
r = x .+ y
predicate add_pre_real (x:t) (y:t) =
no_overflow RNE (t'real x +. t'real y)
predicate add_post_real (x:t) (y:t) (r:t) =
t'real r = round RNE (t'real x +. t'real y)
val safe_add (x y: t) : t
requires { [@expl:finite floating-point number] t'isFinite x }
requires { [@expl:finite floating-point number] t'isFinite y }
requires { [@expl:floating-point overflow]
add_pre_ieee x y \/ add_pre_real x y }
ensures { t'isFinite result }
ensures { add_post_ieee x y result /\ add_post_real x y result }
predicate sub_pre_ieee (x:t) (y:t) =
t'isFinite (x .- y)
predicate sub_post_ieee (x:t) (y:t) (r:t) =
r = x .- y
predicate sub_pre_real (x:t) (y:t) =
no_overflow RNE (t'real x -. t'real y)
predicate sub_post_real (x:t) (y:t) (r:t) =
t'real r = round RNE (t'real x -. t'real y)
val safe_sub (x y: t) : t
requires { [@expl:finite floating-point number] t'isFinite x }
requires { [@expl:finite floating-point number] t'isFinite y }
requires { [@expl:floating-point overflow]
sub_pre_ieee x y \/ sub_pre_real x y }
ensures { t'isFinite result }
ensures { sub_post_ieee x y result /\ sub_post_real x y result }
predicate mul_pre_ieee (x:t) (y:t) =
t'isFinite (x .* y)
predicate mul_post_ieee (x:t) (y:t) (r:t) =
r = x .* y
predicate mul_pre_real (x:t) (y:t) =
no_overflow RNE (t'real x *. t'real y)
predicate mul_post_real (x:t) (y:t) (r:t) =
t'real r = round RNE (t'real x *. t'real y)
val safe_mul (x y: t) : t
requires { [@expl:finite floating-point number] t'isFinite x }
requires { [@expl:finite floating-point number] t'isFinite y }
requires { [@expl:floating-point overflow]
mul_pre_ieee x y \/ mul_pre_real x y }
ensures { t'isFinite result }
ensures { mul_post_ieee x y result /\ mul_post_real x y result }
predicate div_pre_ieee (x:t) (y:t) =
t'isFinite (x ./ y)
predicate div_post_ieee (x:t) (y:t) (r:t) =
r = x ./ y
predicate div_pre_real (x:t) (y:t) =
no_overflow RNE (t'real x /. t'real y)
predicate div_post_real (x:t) (y:t) (r:t) =
t'real r = round RNE (t'real x /. t'real y)
val safe_div (x y: t) : t
requires { [@expl:finite floating-point number] t'isFinite x }
requires { [@expl:finite floating-point number] t'isFinite y }
requires { [@expl:non-zero floating-point number] not (is_zero y) }
requires { [@expl:floating-point overflow]
div_pre_ieee x y \/ div_pre_real x y }
ensures { t'isFinite result }
ensures { div_post_ieee x y result /\ div_post_real x y result }
end
|
