Re: [Caml-list] Approximations when converting from string to float

[Jacques-Henri Jourdan <jacques-henri.jourdan@normalesup.org>]

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Ok.

But really, you should use ZArith, which is much more modern, faster and 
well-maintained than Big_int.

On 10/25/2016 06:58 PM, Tung Vu Xuan wrote:
> Hi,
>
> Thanks for suggestions. Since I am using a library for interval 
> arithmetic [1], your idea becomes easier for me.
>
> let intv_of_big_int bi =
>     let n = Big_int.num_bits_big_int bi in
>     if n <= 53 then
>       let bi_float = Big_int.float_of_big_int bi in
>       {low=bi_float;high=bi_float}
>     else begin
>       let n = n - 53 in
>       (* Extract top 53 bits of x *)
>       let top = Big_int.shift_right_big_int bi n in
>
>       let top_float = Big_int.float_of_big_int top in
>       (* interval [2, 2] *)
>       let two_I = {low=2.;high=2.} in
>       (* interval: [2, 2]**n *)
>       let two_I_to_n = pow_I_i two_I n in
>
>       (* compute upper bound of mantissa, represented as an intv *)
>       (* interval arithmetic is needed because rounding error might 
> occur *)
>       let mantissa_upper_bound_intv = {low=top_float;high=top_float} 
> +$ one_I in
>
>       (* intv of mantissa *)
>       let mantissa_intv = {low=top_float; 
> high=mantissa_upper_bound_intv.high} in
>       (* compute final interval *)
>       mantissa_intv *$ two_I_to_n
>     end
>
> [1] Alliot, J.M., Gotteland, J.B., Vanaret, C., Durand, N., Gianazza, 
> D.: Implementing an interval computation library for OCaml on 
> x86/amd64 architectures. In: ICFP. ACM (2012)
>
> Thanks again,
> Tung.
>
> On Tue, Oct 25, 2016 at 5:28 PM, Soegtrop, Michael 
> <michael.soegtrop@intel.com <mailto:michael.soegtrop@intel.com>> wrote:
>
>     Dear Jacques-Henri,
>
>     if the number is cut off after n digits, the upper and lower
>     bounds I suggested are the (truncated
>     integer)*10^(#cut-off-digits) and (truncated
>     integer+1)*10^(#cut-off-digits). The true number is obviously
>     between these two. Since the integer has higher precision than the
>     mantissa in the case of cut off, this is only a fraction of a
>     mantissa bit. The errors you get by multiplying with the powers of
>     10 is likely larger in most cases.
>
>     In the case you mentioned, 2^70=1180591620717411303424 and 32 bit
>     one would truncate after
>
>     1180591620 * 10^12
>
>     Maxima gives
>
>     is(1180591620 * 10^12 <= 2^70);
>
>     true
>
>     is(1180591621 * 10^12 >= 2^70);
>
>     true
>
>     As I said, this method doesn’t give the tightest bounds possible,
>     but it gives you true upper and lower bounds without multi
>     precision arithmetic. It gives 0 intervals e.g. for integers which
>     fit in the mantissa, but not for large exact powers of 2.
>
>     > Moreover, it is not possible to implement interval arithmetic
>     with OCaml, since you cannot change the rounding mode without a
>     bit of C...
>
>     It should be possible to make the powers of 10 tables such that
>     this works even with round to nearest, but it would likely be
>     easier to make a C library for this.
>
>     Of cause it is a good question if I/O is the right place to save
>     performance and waste precision. On the other hand you lose only 2
>     or 3 bits and you know what the precision is. I would think most
>     applications are such that the required precision is not exactly
>     an IEEE precision, so that these 2 or 3 bits are ok. I used this
>     method once in an embedded platform, where multi precision
>     arithmetic was not an option, and this was a good compromise.
>
>     Best regards,
>
>     Michael
>
>     Intel Deutschland GmbH
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>
>
>
>
> -- 
> Tung Vu Xuan,
> Japan Advanced Institute of Science and Technology,
> Mobile: (+81) 080 4259 9135 -  (+84) 01689934381
> Hometown: Bac Ninh
> Email: toilatung90@gmail.com <mailto:toilatung90@gmail.com>


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