## Bug #14635

### Float#round(n) returns a wrong result when n is big

**Description**

**First of all, don't confuse that this is a usual floating-point error issue.**

The following looks inconsistent:

3.0e-31 #=> 3.0e-31 3.0e-31.round(31) #=> 3.0000000000000003e-31

## What it should be¶

A Float value is actually a range.

`3.0e-31`

represents a range of `0.299999999999999959315060e-30`

.. `0.300000000000000003105637e-30`

(the bounds are approximate). I call this range "A".

`3.0000000000000003e-31`

represents a range of `0.300000000000000003105637e-30`

.. `0.300000000000000046896214e-30`

. I call this range "B".

`x.round(31)`

should (1) multiple x with `10**31`

, (2) round it as an integer, and (3) divide it with `10**31`

.

In this case:

(1) `3.0e-31 * 10**31`

is a range of `2.99999999999999959315060`

.. `3.00000000000000003105637`

.

(2) The rounded result is 3, whichever value is chosen from the range above.

(3) `3.0 / 10**31`

is within the range "A", not within the range "B", so the result should be `3.0e-31`

, not `3.0000000000000003e-31`

.

## How the bug occurs¶

The reason why `3.0e-31.round(31)`

returns `3.0000000000000003e-31`

, is the implementation issue of `Float#round`

. It does the following:

(1) `f = pow(10, b)`

(2) `x = round(x * f)`

as an integer

(3) return `x / f`

However, a double variable `f`

cannot represent `pow(10, 31)`

precisely. In other words, the `10**31`

must be handled as an integer, but the implementation handles it as an inexact floating-point value. This is the issue.

## How to fix¶

The issue is simple, but it might be very difficult to fix. `strtod`

handles a string `"3.0e-31"`

correctly. So, by doing the same as `strtod`

, this issue would be fixed. However, the strtod implementation looks very difficult, at least to me. Contribution from mathematician is welcome.

(Honestly, I don't want to see such a complication in the source code. Another simpler approach would be more preferable.)

## References¶

This issue has been already reported in #5273 by marcandre. But the status of the ticket looks unclear; I cannot see how many issues remains. So, I created this ticket for just one bug that I could confirm.

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