## 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.

**Related issues**

#### Updated by mame (Yusuke Endoh) about 2 years ago

**Related to***Bug #5273: Float#round returns the wrong floats for higher precision*added

#### Updated by mame (Yusuke Endoh) about 2 years ago

**Subject**changed from*Float#round sometimes returns a wrong result*to*Float#round(n) returns a wrong result when n is big*

#### Updated by mame (Yusuke Endoh) about 2 years ago

I've found a much simpler solution: when `n`

is big, it should first translate the float to a rational, then call `Rational#round`

, and finally translate the resulting rational to a float. It is slow, only when n >= 23 for `Float#round(n)`

.

Currently, `Rational#to_f`

has the same inaccuracy issue, which can be fixed by #14637. The following patch includes the hunk for #14637.

```
diff --git a/bignum.c b/bignum.c
index b4c7560034..fd5f385cac 100644
--- a/bignum.c
+++ b/bignum.c
@@ -6178,9 +6178,7 @@ rb_big_fdiv_double(VALUE x, VALUE y)
return big_fdiv_int(x, rb_int2big(FIX2LONG(y)));
}
else if (RB_BIGNUM_TYPE_P(y)) {
- dy = rb_big2dbl(y);
- if (isinf(dx) || isinf(dy))
- return big_fdiv_int(x, y);
+ return big_fdiv_int(x, y);
}
else if (RB_FLOAT_TYPE_P(y)) {
dy = RFLOAT_VALUE(y);
diff --git a/internal.h b/internal.h
index 9b6a213151..0bf20b19b0 100644
--- a/internal.h
+++ b/internal.h
@@ -1689,6 +1689,7 @@ VALUE rb_cstr_to_rat(const char *, int);
VALUE rb_rational_abs(VALUE self);
VALUE rb_rational_cmp(VALUE self, VALUE other);
VALUE rb_numeric_quo(VALUE x, VALUE y);
+VALUE rb_flo_round_by_rational(int argc, VALUE *argv, VALUE num);
/* re.c */
VALUE rb_reg_compile(VALUE str, int options, const char *sourcefile, int sourceline);
diff --git a/numeric.c b/numeric.c
index 01856c7f20..15b27e9132 100644
--- a/numeric.c
+++ b/numeric.c
@@ -2239,6 +2239,10 @@ flo_round(int argc, VALUE *argv, VALUE num)
frexp(number, &binexp);
if (float_round_overflow(ndigits, binexp)) return num;
if (float_round_underflow(ndigits, binexp)) return DBL2NUM(0);
+ if (ndigits > DBL_MANT_DIG * log(2.0) / log(5.0)) {
+ /* In this case, pow(10, ndigits) cannot be accurate. */
+ return rb_flo_round_by_rational(argc, argv, num);
+ }
f = pow(10, ndigits);
x = ROUND_CALL(mode, round, (number, f));
return DBL2NUM(x / f);
diff --git a/rational.c b/rational.c
index d88f50f886..01bb88d1ae 100644
--- a/rational.c
+++ b/rational.c
@@ -1533,6 +1533,13 @@ nurat_round_n(int argc, VALUE *argv, VALUE self)
return f_round_common(argc, argv, self, round_func);
}
+static VALUE float_to_r(VALUE self);
+VALUE
+rb_flo_round_by_rational(int argc, VALUE *argv, VALUE num)
+{
+ return nurat_to_f(nurat_round_n(argc, argv, float_to_r(num)));
+}
+
static double
nurat_to_double(VALUE self)
{
@@ -2016,7 +2023,6 @@ integer_denominator(VALUE self)
return INT2FIX(1);
}
-static VALUE float_to_r(VALUE self);
/*
* call-seq:
* flo.numerator -> integer
```