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Rational

A rational number can be represented as a pair of integer numbers: a/b (b>0), where a is the numerator and b is the denominator. Integer a equals rational a/1 mathematically.

In Ruby, you can create rational objects with the Kernel#Rational, to_r, or rationalize methods or by suffixing r to a literal. The return values will be irreducible fractions.

Rational(1)      #=> (1/1)
Rational(2, 3)   #=> (2/3)
Rational(4, -6)  #=> (-2/3)
3.to_r           #=> (3/1)
2/3r             #=> (2/3)

You can also create rational objects from floating-point numbers or strings.

Rational(0.3)    #=> (5404319552844595/18014398509481984)
Rational('0.3')  #=> (3/10)
Rational('2/3')  #=> (2/3)

0.3.to_r         #=> (5404319552844595/18014398509481984)
'0.3'.to_r       #=> (3/10)
'2/3'.to_r       #=> (2/3)
0.3.rationalize  #=> (3/10)

A rational object is an exact number, which helps you to write programs without any rounding errors.

10.times.inject(0) {|t| t + 0.1 }              #=> 0.9999999999999999
10.times.inject(0) {|t| t + Rational('0.1') }  #=> (1/1)

However, when an expression includes an inexact component (numerical value or operation), it will produce an inexact result.

Rational(10) / 3   #=> (10/3)
Rational(10) / 3.0 #=> 3.3333333333333335

Rational(-8) ** Rational(1, 3)
                   #=> (1.0000000000000002+1.7320508075688772i)

Public Instance Methods

rat * numeric → numeric click to toggle source

Performs multiplication.

Rational(2, 3)  * Rational(2, 3)   #=> (4/9)
Rational(900)   * Rational(1)      #=> (900/1)
Rational(-2, 9) * Rational(-9, 2)  #=> (1/1)
Rational(9, 8)  * 4                #=> (9/2)
Rational(20, 9) * 9.8              #=> 21.77777777777778

 
               static VALUE
nurat_mul(VALUE self, VALUE other)
{
    if (RB_INTEGER_TYPE_P(other)) {
        {
            get_dat1(self);

            return f_muldiv(self,
                            dat->num, dat->den,
                            other, ONE, '*');
        }
    }
    else if (RB_FLOAT_TYPE_P(other)) {
        return DBL2NUM(nurat_to_double(self) * RFLOAT_VALUE(other));
    }
    else if (RB_TYPE_P(other, T_RATIONAL)) {
        {
            get_dat2(self, other);

            return f_muldiv(self,
                            adat->num, adat->den,
                            bdat->num, bdat->den, '*');
        }
    }
    else {
        return rb_num_coerce_bin(self, other, '*');
    }
}

rat ** numeric → numeric click to toggle source

Performs exponentiation.

Rational(2)    ** Rational(3)     #=> (8/1)
Rational(10)   ** -2              #=> (1/100)
Rational(10)   ** -2.0            #=> 0.01
Rational(-4)   ** Rational(1, 2)  #=> (0.0+2.0i)
Rational(1, 2) ** 0               #=> (1/1)
Rational(1, 2) ** 0.0             #=> 1.0

 
               static VALUE
nurat_expt(VALUE self, VALUE other)
{
    if (k_numeric_p(other) && k_exact_zero_p(other))
        return f_rational_new_bang1(CLASS_OF(self), ONE);

    if (k_rational_p(other)) {
        get_dat1(other);

        if (f_one_p(dat->den))
            other = dat->num; /* c14n */
    }

    /* Deal with special cases of 0**n and 1**n */
    if (k_numeric_p(other) && k_exact_p(other)) {
        get_dat1(self);
        if (f_one_p(dat->den)) {
            if (f_one_p(dat->num)) {
                return f_rational_new_bang1(CLASS_OF(self), ONE);
            }
            else if (f_minus_one_p(dat->num) && RB_INTEGER_TYPE_P(other)) {
                return f_rational_new_bang1(CLASS_OF(self), INT2FIX(f_odd_p(other) ? -1 : 1));
            }
            else if (INT_ZERO_P(dat->num)) {
                if (rb_num_negative_p(other)) {
                    rb_num_zerodiv();
                }
                else {
                    return f_rational_new_bang1(CLASS_OF(self), ZERO);
                }
            }
        }
    }

    /* General case */
    if (FIXNUM_P(other)) {
        {
            VALUE num, den;

            get_dat1(self);

            if (INT_POSITIVE_P(other)) {
                num = rb_int_pow(dat->num, other);
                den = rb_int_pow(dat->den, other);
            }
            else if (INT_NEGATIVE_P(other)) {
                num = rb_int_pow(dat->den, rb_int_uminus(other));
                den = rb_int_pow(dat->num, rb_int_uminus(other));
            }
            else {
                num = ONE;
                den = ONE;
            }
            if (RB_FLOAT_TYPE_P(num)) { /* infinity due to overflow */
                if (RB_FLOAT_TYPE_P(den)) return DBL2NUM(NAN);
                return num;
            }
            if (RB_FLOAT_TYPE_P(den)) { /* infinity due to overflow */
                num = ZERO;
                den = ONE;
            }
            return f_rational_new2(CLASS_OF(self), num, den);
        }
    }
    else if (RB_TYPE_P(other, T_BIGNUM)) {
        rb_warn("in a**b, b may be too big");
        return rb_float_pow(nurat_to_f(self), other);
    }
    else if (RB_FLOAT_TYPE_P(other) || RB_TYPE_P(other, T_RATIONAL)) {
        return rb_float_pow(nurat_to_f(self), other);
    }
    else {
        return rb_num_coerce_bin(self, other, rb_intern("**"));
    }
}

rat + numeric → numeric click to toggle source

Performs addition.

Rational(2, 3)  + Rational(2, 3)   #=> (4/3)
Rational(900)   + Rational(1)      #=> (901/1)
Rational(-2, 9) + Rational(-9, 2)  #=> (-85/18)
Rational(9, 8)  + 4                #=> (41/8)
Rational(20, 9) + 9.8              #=> 12.022222222222222

 
               VALUE
rb_rational_plus(VALUE self, VALUE other)
{
    if (RB_INTEGER_TYPE_P(other)) {
        {
            get_dat1(self);

            return f_rational_new_no_reduce2(CLASS_OF(self),
                                             rb_int_plus(dat->num, rb_int_mul(other, dat->den)),
                                             dat->den);
        }
    }
    else if (RB_FLOAT_TYPE_P(other)) {
        return DBL2NUM(nurat_to_double(self) + RFLOAT_VALUE(other));
    }
    else if (RB_TYPE_P(other, T_RATIONAL)) {
        {
            get_dat2(self, other);

            return f_addsub(self,
                            adat->num, adat->den,
                            bdat->num, bdat->den, '+');
        }
    }
    else {
        return rb_num_coerce_bin(self, other, '+');
    }
}

rat - numeric → numeric click to toggle source

Performs subtraction.

Rational(2, 3)  - Rational(2, 3)   #=> (0/1)
Rational(900)   - Rational(1)      #=> (899/1)
Rational(-2, 9) - Rational(-9, 2)  #=> (77/18)
Rational(9, 8)  - 4                #=> (-23/8)
Rational(20, 9) - 9.8              #=> -7.577777777777778

 
               static VALUE
nurat_sub(VALUE self, VALUE other)
{
    if (RB_INTEGER_TYPE_P(other)) {
        {
            get_dat1(self);

            return f_rational_new_no_reduce2(CLASS_OF(self),
                                             rb_int_minus(dat->num, rb_int_mul(other, dat->den)),
                                             dat->den);
        }
    }
    else if (RB_FLOAT_TYPE_P(other)) {
        return DBL2NUM(nurat_to_double(self) - RFLOAT_VALUE(other));
    }
    else if (RB_TYPE_P(other, T_RATIONAL)) {
        {
            get_dat2(self, other);

            return f_addsub(self,
                            adat->num, adat->den,
                            bdat->num, bdat->den, '-');
        }
    }
    else {
        return rb_num_coerce_bin(self, other, '-');
    }
}

-rat → rational click to toggle source

Negates rat.

 
               VALUE
rb_rational_uminus(VALUE self)
{
    const int unused = (assert(RB_TYPE_P(self, T_RATIONAL)), 0);
    get_dat1(self);
    (void)unused;
    return f_rational_new2(CLASS_OF(self), rb_int_uminus(dat->num), dat->den);
}

rat / numeric → numeric click to toggle source

Performs division.

Rational(2, 3)  / Rational(2, 3)   #=> (1/1)
Rational(900)   / Rational(1)      #=> (900/1)
Rational(-2, 9) / Rational(-9, 2)  #=> (4/81)
Rational(9, 8)  / 4                #=> (9/32)
Rational(20, 9) / 9.8              #=> 0.22675736961451246

 
               static VALUE
nurat_div(VALUE self, VALUE other)
{
    if (RB_INTEGER_TYPE_P(other)) {
        if (f_zero_p(other))
            rb_num_zerodiv();
        {
            get_dat1(self);

            return f_muldiv(self,
                            dat->num, dat->den,
                            other, ONE, '/');
        }
    }
    else if (RB_FLOAT_TYPE_P(other))
        return DBL2NUM(nurat_to_double(self) / RFLOAT_VALUE(other));
    else if (RB_TYPE_P(other, T_RATIONAL)) {
        if (f_zero_p(other))
            rb_num_zerodiv();
        {
            get_dat2(self, other);

            if (f_one_p(self))
                return f_rational_new_no_reduce2(CLASS_OF(self),
                                                 bdat->den, bdat->num);

            return f_muldiv(self,
                            adat->num, adat->den,
                            bdat->num, bdat->den, '/');
        }
    }
    else {
        return rb_num_coerce_bin(self, other, '/');
    }
}

rational <=> numeric → -1, 0, +1, or nil click to toggle source

Returns -1, 0, or +1 depending on whether rational is less than, equal to, or greater than numeric.

nil is returned if the two values are incomparable.

Rational(2, 3) <=> Rational(2, 3)  #=> 0
Rational(5)    <=> 5               #=> 0
Rational(2, 3) <=> Rational(1, 3)  #=> 1
Rational(1, 3) <=> 1               #=> -1
Rational(1, 3) <=> 0.3             #=> 1

Rational(1, 3) <=> "0.3"           #=> nil

 
               VALUE
rb_rational_cmp(VALUE self, VALUE other)
{
    if (RB_INTEGER_TYPE_P(other)) {
        {
            get_dat1(self);

            if (dat->den == LONG2FIX(1))
                return rb_int_cmp(dat->num, other); /* c14n */
            other = f_rational_new_bang1(CLASS_OF(self), other);
            goto other_is_rational;
        }
    }
    else if (RB_FLOAT_TYPE_P(other)) {
        return rb_dbl_cmp(nurat_to_double(self), RFLOAT_VALUE(other));
    }
    else if (RB_TYPE_P(other, T_RATIONAL)) {
        other_is_rational:
        {
            VALUE num1, num2;

            get_dat2(self, other);

            if (FIXNUM_P(adat->num) && FIXNUM_P(adat->den) &&
                FIXNUM_P(bdat->num) && FIXNUM_P(bdat->den)) {
                num1 = f_imul(FIX2LONG(adat->num), FIX2LONG(bdat->den));
                num2 = f_imul(FIX2LONG(bdat->num), FIX2LONG(adat->den));
            }
            else {
                num1 = rb_int_mul(adat->num, bdat->den);
                num2 = rb_int_mul(bdat->num, adat->den);
            }
            return rb_int_cmp(rb_int_minus(num1, num2), ZERO);
        }
    }
    else {
        return rb_num_coerce_cmp(self, other, rb_intern("<=>"));
    }
}

rat == object → true or false click to toggle source

Returns true if rat equals object numerically.

Rational(2, 3)  == Rational(2, 3)   #=> true
Rational(5)     == 5                #=> true
Rational(0)     == 0.0              #=> true
Rational('1/3') == 0.33             #=> false
Rational('1/2') == '1/2'            #=> false

 
               static VALUE
nurat_eqeq_p(VALUE self, VALUE other)
{
    if (RB_INTEGER_TYPE_P(other)) {
        {
            get_dat1(self);

            if (INT_ZERO_P(dat->num) && INT_ZERO_P(other))
                return Qtrue;

            if (!FIXNUM_P(dat->den))
                return Qfalse;
            if (FIX2LONG(dat->den) != 1)
                return Qfalse;
            return rb_int_equal(dat->num, other);
        }
    }
    else if (RB_FLOAT_TYPE_P(other)) {
        const double d = nurat_to_double(self);
        return f_boolcast(FIXNUM_ZERO_P(rb_dbl_cmp(d, RFLOAT_VALUE(other))));
    }
    else if (RB_TYPE_P(other, T_RATIONAL)) {
        {
            get_dat2(self, other);

            if (INT_ZERO_P(adat->num) && INT_ZERO_P(bdat->num))
                return Qtrue;

            return f_boolcast(rb_int_equal(adat->num, bdat->num) &&
                              rb_int_equal(adat->den, bdat->den));
        }
    }
    else {
        return rb_equal(other, self);
    }
}

abs → rational click to toggle source

Returns the absolute value of rat.

(1/2r).abs    #=> (1/2)
(-1/2r).abs   #=> (1/2)

Rational#magnitude is an alias for Rational#abs.

 
               VALUE
rb_rational_abs(VALUE self)
{
    get_dat1(self);
    if (INT_NEGATIVE_P(dat->num)) {
        VALUE num = rb_int_abs(dat->num);
        return nurat_s_canonicalize_internal_no_reduce(CLASS_OF(self), num, dat->den);
    }
    return self;
}

ceil([ndigits]) → integer or rational click to toggle source

Returns the smallest number greater than or equal to rat with a precision of ndigits decimal digits (default: 0).

When the precision is negative, the returned value is an integer with at least ndigits.abs trailing zeros.

Returns a rational when ndigits is positive, otherwise returns an integer.

Rational(3).ceil      #=> 3
Rational(2, 3).ceil   #=> 1
Rational(-3, 2).ceil  #=> -1

  #    decimal      -  1  2  3 . 4  5  6
  #                   ^  ^  ^  ^   ^  ^
  #   precision      -3 -2 -1  0  +1 +2

Rational('-123.456').ceil(+1).to_f  #=> -123.4
Rational('-123.456').ceil(-1)       #=> -120

 
               static VALUE
nurat_ceil_n(int argc, VALUE *argv, VALUE self)
{
    return f_round_common(argc, argv, self, nurat_ceil);
}

denominator → integer click to toggle source

Returns the denominator (always positive).

Rational(7).denominator             #=> 1
Rational(7, 1).denominator          #=> 1
Rational(9, -4).denominator         #=> 4
Rational(-2, -10).denominator       #=> 5

 
               static VALUE
nurat_denominator(VALUE self)
{
    get_dat1(self);
    return dat->den;
}

fdiv(numeric) → float click to toggle source

Performs division and returns the value as a Float.

Rational(2, 3).fdiv(1)       #=> 0.6666666666666666
Rational(2, 3).fdiv(0.5)     #=> 1.3333333333333333
Rational(2).fdiv(3)          #=> 0.6666666666666666

 
               static VALUE
nurat_fdiv(VALUE self, VALUE other)
{
    VALUE div;
    if (f_zero_p(other))
        return DBL2NUM(nurat_to_double(self) / 0.0);
    if (FIXNUM_P(other) && other == LONG2FIX(1))
        return nurat_to_f(self);
    div = nurat_div(self, other);
    if (RB_TYPE_P(div, T_RATIONAL))
        return nurat_to_f(div);
    if (RB_FLOAT_TYPE_P(div))
        return div;
    return rb_funcall(div, rb_intern("to_f"), 0);
}

floor([ndigits]) → integer or rational click to toggle source

Returns the largest number less than or equal to rat with a precision of ndigits decimal digits (default: 0).

When the precision is negative, the returned value is an integer with at least ndigits.abs trailing zeros.

Returns a rational when ndigits is positive, otherwise returns an integer.

Rational(3).floor      #=> 3
Rational(2, 3).floor   #=> 0
Rational(-3, 2).floor  #=> -2

  #    decimal      -  1  2  3 . 4  5  6
  #                   ^  ^  ^  ^   ^  ^
  #   precision      -3 -2 -1  0  +1 +2

Rational('-123.456').floor(+1).to_f  #=> -123.5
Rational('-123.456').floor(-1)       #=> -130

 
               static VALUE
nurat_floor_n(int argc, VALUE *argv, VALUE self)
{
    return f_round_common(argc, argv, self, nurat_floor);
}

inspect → string click to toggle source

Returns the value as a string for inspection.

Rational(2).inspect      #=> "(2/1)"
Rational(-8, 6).inspect  #=> "(-4/3)"
Rational('1/2').inspect  #=> "(1/2)"

 
               static VALUE
nurat_inspect(VALUE self)
{
    VALUE s;

    s = rb_usascii_str_new2("(");
    rb_str_concat(s, f_format(self, f_inspect));
    rb_str_cat2(s, ")");

    return s;
}

magnitude → rational click to toggle source

Returns the absolute value of rat.

(1/2r).abs    #=> (1/2)
(-1/2r).abs   #=> (1/2)

Rational#magnitude is an alias for Rational#abs.

 
               VALUE
rb_rational_abs(VALUE self)
{
    get_dat1(self);
    if (INT_NEGATIVE_P(dat->num)) {
        VALUE num = rb_int_abs(dat->num);
        return nurat_s_canonicalize_internal_no_reduce(CLASS_OF(self), num, dat->den);
    }
    return self;
}

negative? → true or false click to toggle source

Returns true if rat is less than 0.

 
               static VALUE
nurat_negative_p(VALUE self)
{
    get_dat1(self);
    return f_boolcast(INT_NEGATIVE_P(dat->num));
}

numerator → integer click to toggle source

Returns the numerator.

Rational(7).numerator        #=> 7
Rational(7, 1).numerator     #=> 7
Rational(9, -4).numerator    #=> -9
Rational(-2, -10).numerator  #=> 1

 
               static VALUE
nurat_numerator(VALUE self)
{
    get_dat1(self);
    return dat->num;
}

positive? → true or false click to toggle source

Returns true if rat is greater than 0.

 
               static VALUE
nurat_positive_p(VALUE self)
{
    get_dat1(self);
    return f_boolcast(INT_POSITIVE_P(dat->num));
}

quo(numeric) → numeric click to toggle source

Performs division.

Rational(2, 3)  / Rational(2, 3)   #=> (1/1)
Rational(900)   / Rational(1)      #=> (900/1)
Rational(-2, 9) / Rational(-9, 2)  #=> (4/81)
Rational(9, 8)  / 4                #=> (9/32)
Rational(20, 9) / 9.8              #=> 0.22675736961451246

 
               static VALUE
nurat_div(VALUE self, VALUE other)
{
    if (RB_INTEGER_TYPE_P(other)) {
        if (f_zero_p(other))
            rb_num_zerodiv();
        {
            get_dat1(self);

            return f_muldiv(self,
                            dat->num, dat->den,
                            other, ONE, '/');
        }
    }
    else if (RB_FLOAT_TYPE_P(other))
        return DBL2NUM(nurat_to_double(self) / RFLOAT_VALUE(other));
    else if (RB_TYPE_P(other, T_RATIONAL)) {
        if (f_zero_p(other))
            rb_num_zerodiv();
        {
            get_dat2(self, other);

            if (f_one_p(self))
                return f_rational_new_no_reduce2(CLASS_OF(self),
                                                 bdat->den, bdat->num);

            return f_muldiv(self,
                            adat->num, adat->den,
                            bdat->num, bdat->den, '/');
        }
    }
    else {
        return rb_num_coerce_bin(self, other, '/');
    }
}

rationalize → self click to toggle source

rationalize(eps) → rational

Returns a simpler approximation of the value if the optional argument eps is given (rat-|eps| <= result <= rat+|eps|), self otherwise.

r = Rational(5033165, 16777216)
r.rationalize                    #=> (5033165/16777216)
r.rationalize(Rational('0.01'))  #=> (3/10)
r.rationalize(Rational('0.1'))   #=> (1/3)

 
               static VALUE
nurat_rationalize(int argc, VALUE *argv, VALUE self)
{
    VALUE e, a, b, p, q;

    if (argc == 0)
        return self;

    if (nurat_negative_p(self))
        return rb_rational_uminus(nurat_rationalize(argc, argv, rb_rational_uminus(self)));

    rb_scan_args(argc, argv, "01", &e);
    e = f_abs(e);
    a = f_sub(self, e);
    b = f_add(self, e);

    if (f_eqeq_p(a, b))
        return self;

    nurat_rationalize_internal(a, b, &p, &q);
    return f_rational_new2(CLASS_OF(self), p, q);
}

round([ndigits] [, half: mode]) → integer or rational click to toggle source

Returns rat rounded to the nearest value with a precision of ndigits decimal digits (default: 0).

When the precision is negative, the returned value is an integer with at least ndigits.abs trailing zeros.

Returns a rational when ndigits is positive, otherwise returns an integer.

Rational(3).round      #=> 3
Rational(2, 3).round   #=> 1
Rational(-3, 2).round  #=> -2

  #    decimal      -  1  2  3 . 4  5  6
  #                   ^  ^  ^  ^   ^  ^
  #   precision      -3 -2 -1  0  +1 +2

Rational('-123.456').round(+1).to_f  #=> -123.5
Rational('-123.456').round(-1)       #=> -120

The optional half keyword argument is available similar to Float#round.

Rational(25, 100).round(1, half: :up)    #=> (3/10)
Rational(25, 100).round(1, half: :down)  #=> (1/5)
Rational(25, 100).round(1, half: :even)  #=> (1/5)
Rational(35, 100).round(1, half: :up)    #=> (2/5)
Rational(35, 100).round(1, half: :down)  #=> (3/10)
Rational(35, 100).round(1, half: :even)  #=> (2/5)
Rational(-25, 100).round(1, half: :up)   #=> (-3/10)
Rational(-25, 100).round(1, half: :down) #=> (-1/5)
Rational(-25, 100).round(1, half: :even) #=> (-1/5)

 
               static VALUE
nurat_round_n(int argc, VALUE *argv, VALUE self)
{
    VALUE opt;
    enum ruby_num_rounding_mode mode = (
        argc = rb_scan_args(argc, argv, "*:", NULL, &opt),
        rb_num_get_rounding_option(opt));
    VALUE (*round_func)(VALUE) = ROUND_FUNC(mode, nurat_round);
    return f_round_common(argc, argv, self, round_func);
}

to_f → float click to toggle source

Returns the value as a Float.

Rational(2).to_f      #=> 2.0
Rational(9, 4).to_f   #=> 2.25
Rational(-3, 4).to_f  #=> -0.75
Rational(20, 3).to_f  #=> 6.666666666666667

 
               static VALUE
nurat_to_f(VALUE self)
{
    return DBL2NUM(nurat_to_double(self));
}

to_i → integer click to toggle source

Returns the truncated value as an integer.

Equivalent to Rational#truncate.

Rational(2, 3).to_i    #=> 0
Rational(3).to_i       #=> 3
Rational(300.6).to_i   #=> 300
Rational(98, 71).to_i  #=> 1
Rational(-31, 2).to_i  #=> -15

 
               static VALUE
nurat_truncate(VALUE self)
{
    get_dat1(self);
    if (INT_NEGATIVE_P(dat->num))
        return rb_int_uminus(rb_int_idiv(rb_int_uminus(dat->num), dat->den));
    return rb_int_idiv(dat->num, dat->den);
}

to_r → self click to toggle source

Returns self.

Rational(2).to_r      #=> (2/1)
Rational(-8, 6).to_r  #=> (-4/3)

 
               static VALUE
nurat_to_r(VALUE self)
{
    return self;
}

to_s → string click to toggle source

Returns the value as a string.

Rational(2).to_s      #=> "2/1"
Rational(-8, 6).to_s  #=> "-4/3"
Rational('1/2').to_s  #=> "1/2"

 
               static VALUE
nurat_to_s(VALUE self)
{
    return f_format(self, f_to_s);
}

truncate([ndigits]) → integer or rational click to toggle source

Returns rat truncated (toward zero) to a precision of ndigits decimal digits (default: 0).

When the precision is negative, the returned value is an integer with at least ndigits.abs trailing zeros.

Returns a rational when ndigits is positive, otherwise returns an integer.

Rational(3).truncate      #=> 3
Rational(2, 3).truncate   #=> 0
Rational(-3, 2).truncate  #=> -1

  #    decimal      -  1  2  3 . 4  5  6
  #                   ^  ^  ^  ^   ^  ^
  #   precision      -3 -2 -1  0  +1 +2

Rational('-123.456').truncate(+1).to_f  #=> -123.4
Rational('-123.456').truncate(-1)       #=> -120

 
               static VALUE
nurat_truncate_n(int argc, VALUE *argv, VALUE self)
{
    return f_round_common(argc, argv, self, nurat_truncate);
}

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