## Approximated Cosine in Why3

This is a Why3 version of the Approximated Cosine example

Authors: Claude Marché

Tools: Why3 / Gappa / Coq

see also the index (by topic, by tool, by reference, by year)

```(* Approximated cosine

module M

use real.RealInfix
use real.Abs
use real.Trigonometry
use floating_point.Rounding
use floating_point.Single

val single_of_real_exact (x: real) : single
ensures { value result = x }

val add (x y: single) : single
requires { no_overflow NearestTiesToEven ((value x) +. (value y)) }
ensures  { value result = round NearestTiesToEven ((value x) +. (value y))}

val sub (x y: single) : single
requires { no_overflow NearestTiesToEven ((value x) -. (value y)) }
ensures  { value result = round NearestTiesToEven ((value x) -. (value y))}

val mul (x y: single) : single
requires { no_overflow NearestTiesToEven ((value x) *. (value y)) }
ensures  { value result = round NearestTiesToEven ((value x) *. (value y))}

let my_cosine (x:single) : single
requires { abs (value x) <=. 0x1p-5 }
ensures { abs (value result -. cos (value x)) <=. 0x1p-23 }
= assert {
abs (1.0 -. (value x) *. (value x) *. 0.5 -. cos (value x)) <=. 0x1p-24
};
sub (single_of_real_exact 1.0) (mul (mul x x) (single_of_real_exact 0.5))

end
```