When mathn is required, the Math module changes as follows:

Standard Math module behaviour:

Math.sqrt(4/9)     # => 0.0
Math.sqrt(4.0/9.0) # => 0.666666666666667
Math.sqrt(- 4/9)   # => Errno::EDOM: Numerical argument out of domain - sqrt

After require 'mathn', this is changed to:

require 'mathn'
Math.sqrt(4/9)      # => 2/3
Math.sqrt(4.0/9.0)  # => 0.666666666666667
Math.sqrt(- 4/9)    # => Complex(0, 2/3)

The Math module contains module functions for basic trigonometric and transcendental functions. See class Float for a list of constants that define Ruby's floating point accuracy.

Domains and codomains are given only for real (not complex) numbers.

Namespace
Methods
A
C
E
F
G
H
L
R
S
T
Constants
PI = DBL2NUM(M_PI)
 

Definition of the mathematical constant PI as a Float number.
E = DBL2NUM(M_E)
 

Definition of the mathematical constant E (e) as a Float number.
Class Public methods
Math.acos(x) → Float

Computes the arc cosine of x. Returns 0..PI.

Domain: [-1, 1]

Codomain: [0, PI]

Math.acos(0) == Math::PI/2  #=> true

Source: show 

static VALUE
math_acos(VALUE obj, VALUE x)
{
    double d;

    d = Get_Double(x);
    /* check for domain error */
    if (d < -1.0 || 1.0 < d) domain_error("acos");
    return DBL2NUM(acos(d));
}
Math.acosh(x) → Float

Computes the inverse hyperbolic cosine of x.

Domain: [1, INFINITY)

Codomain: [0, INFINITY)

Math.acosh(1) #=> 0.0

Source: show 

static VALUE
math_acosh(VALUE obj, VALUE x)
{
    double d;

    d = Get_Double(x);
    /* check for domain error */
    if (d < 1.0) domain_error("acosh");
    return DBL2NUM(acosh(d));
}
Math.asin(x) → Float

Computes the arc sine of x. Returns -PI/2..PI/2.

Domain: [-1, -1]

Codomain: [-PI/2, PI/2]

Math.asin(1) == Math::PI/2  #=> true

Source: show 

static VALUE
math_asin(VALUE obj, VALUE x)
{
    double d;

    d = Get_Double(x);
    /* check for domain error */
    if (d < -1.0 || 1.0 < d) domain_error("asin");
    return DBL2NUM(asin(d));
}
Math.asinh(x) → Float

Computes the inverse hyperbolic sine of x.

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

Math.asinh(1) #=> 0.881373587019543

Source: show 

static VALUE
math_asinh(VALUE obj, VALUE x)
{
    return DBL2NUM(asinh(Get_Double(x)));
}
Math.atan(x) → Float

Computes the arc tangent of x. Returns -PI/2..PI/2.

Domain: (-INFINITY, INFINITY)

Codomain: (-PI/2, PI/2)

Math.atan(0) #=> 0.0

Source: show 

static VALUE
math_atan(VALUE obj, VALUE x)
{
    return DBL2NUM(atan(Get_Double(x)));
}
Math.atan2(y, x) → Float

Computes the arc tangent given y and x. Returns a Float in the range -PI..PI. Return value is a angle in radians between the positive x-axis of cartesian plane and the point given by the coordinates (x, y) on it.

Domain: (-INFINITY, INFINITY)

Codomain: [-PI, PI]

Math.atan2(-0.0, -1.0) #=> -3.141592653589793
Math.atan2(-1.0, -1.0) #=> -2.356194490192345
Math.atan2(-1.0, 0.0)  #=> -1.5707963267948966
Math.atan2(-1.0, 1.0)  #=> -0.7853981633974483
Math.atan2(-0.0, 1.0)  #=> -0.0
Math.atan2(0.0, 1.0)   #=> 0.0
Math.atan2(1.0, 1.0)   #=> 0.7853981633974483
Math.atan2(1.0, 0.0)   #=> 1.5707963267948966
Math.atan2(1.0, -1.0)  #=> 2.356194490192345
Math.atan2(0.0, -1.0)  #=> 3.141592653589793
Math.atan2(INFINITY, INFINITY)   #=> 0.7853981633974483
Math.atan2(INFINITY, -INFINITY)  #=> 2.356194490192345
Math.atan2(-INFINITY, INFINITY)  #=> -0.7853981633974483
Math.atan2(-INFINITY, -INFINITY) #=> -2.356194490192345

Source: show 

static VALUE
math_atan2(VALUE obj, VALUE y, VALUE x)
{
    double dx, dy;
    dx = Get_Double(x);
    dy = Get_Double(y);
    if (dx == 0.0 && dy == 0.0) {
        if (!signbit(dx))
            return DBL2NUM(dy);
        if (!signbit(dy))
            return DBL2NUM(M_PI);
        return DBL2NUM(-M_PI);
    }
#ifndef ATAN2_INF_C99
    if (isinf(dx) && isinf(dy)) {
        /* optimization for FLONUM */
        if (dx < 0.0) {
            const double dz = (3.0 * M_PI / 4.0);
            return (dy < 0.0) ? DBL2NUM(-dz) : DBL2NUM(dz);
        }
        else {
            const double dz = (M_PI / 4.0);
            return (dy < 0.0) ? DBL2NUM(-dz) : DBL2NUM(dz);
        }
    }
#endif
    return DBL2NUM(atan2(dy, dx));
}
Math.atanh(x) → Float

Computes the inverse hyperbolic tangent of x.

Domain: (-1, 1)

Codomain: (-INFINITY, INFINITY)

Math.atanh(1) #=> Infinity

Source: show 

static VALUE
math_atanh(VALUE obj, VALUE x)
{
    double d;

    d = Get_Double(x);
    /* check for domain error */
    if (d <  -1.0 || +1.0 <  d) domain_error("atanh");
    /* check for pole error */
    if (d == -1.0) return DBL2NUM(-INFINITY);
    if (d == +1.0) return DBL2NUM(+INFINITY);
    return DBL2NUM(atanh(d));
}
Math.cbrt(x) → Float

Returns the cube root of x.

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

-9.upto(9) {|x|
  p [x, Math.cbrt(x), Math.cbrt(x)**3]
}
#=> [-9, -2.0800838230519, -9.0]
#   [-8, -2.0, -8.0]
#   [-7, -1.91293118277239, -7.0]
#   [-6, -1.81712059283214, -6.0]
#   [-5, -1.7099759466767, -5.0]
#   [-4, -1.5874010519682, -4.0]
#   [-3, -1.44224957030741, -3.0]
#   [-2, -1.25992104989487, -2.0]
#   [-1, -1.0, -1.0]
#   [0, 0.0, 0.0]
#   [1, 1.0, 1.0]
#   [2, 1.25992104989487, 2.0]
#   [3, 1.44224957030741, 3.0]
#   [4, 1.5874010519682, 4.0]
#   [5, 1.7099759466767, 5.0]
#   [6, 1.81712059283214, 6.0]
#   [7, 1.91293118277239, 7.0]
#   [8, 2.0, 8.0]
#   [9, 2.0800838230519, 9.0]

Source: show 

static VALUE
math_cbrt(VALUE obj, VALUE x)
{
    return DBL2NUM(cbrt(Get_Double(x)));
}
Math.cos(x) → Float

Computes the cosine of x (expressed in radians). Returns a Float in the range -1.0..1.0.

Domain: (-INFINITY, INFINITY)

Codomain: [-1, 1]

Math.cos(Math::PI) #=> -1.0

Source: show 

static VALUE
math_cos(VALUE obj, VALUE x)
{
    return DBL2NUM(cos(Get_Double(x)));
}
Math.cosh(x) → Float

Computes the hyperbolic cosine of x (expressed in radians).

Domain: (-INFINITY, INFINITY)

Codomain: [1, INFINITY)

Math.cosh(0) #=> 1.0

Source: show 

static VALUE
math_cosh(VALUE obj, VALUE x)
{
    return DBL2NUM(cosh(Get_Double(x)));
}
Math.erf(x) → Float

Calculates the error function of x.

Domain: (-INFINITY, INFINITY)

Codomain: (-1, 1)

Math.erf(0) #=> 0.0

Source: show 

static VALUE
math_erf(VALUE obj, VALUE x)
{
    return DBL2NUM(erf(Get_Double(x)));
}
Math.erfc(x) → Float

Calculates the complementary error function of x.

Domain: (-INFINITY, INFINITY)

Codomain: (0, 2)

Math.erfc(0) #=> 1.0

Source: show 

static VALUE
math_erfc(VALUE obj, VALUE x)
{
    return DBL2NUM(erfc(Get_Double(x)));
}
Math.exp(x) → Float

Returns e**x.

Domain: (-INFINITY, INFINITY)

Codomain: (0, INFINITY)

Math.exp(0)       #=> 1.0
Math.exp(1)       #=> 2.718281828459045
Math.exp(1.5)     #=> 4.4816890703380645

Source: show 

static VALUE
math_exp(VALUE obj, VALUE x)
{
    return DBL2NUM(exp(Get_Double(x)));
}
Math.frexp(x) → [fraction, exponent]

Returns a two-element array containing the normalized fraction (a Float) and exponent (a Fixnum) of x.

fraction, exponent = Math.frexp(1234)   #=> [0.6025390625, 11]
fraction * 2**exponent                  #=> 1234.0

Source: show 

static VALUE
math_frexp(VALUE obj, VALUE x)
{
    double d;
    int exp;

    d = frexp(Get_Double(x), &exp);
    return rb_assoc_new(DBL2NUM(d), INT2NUM(exp));
}
Math.gamma(x) → Float

Calculates the gamma function of x.

Note that gamma(n) is same as fact(n-1) for integer n > 0. However gamma(n) returns float and can be an approximation.

def fact(n) (1..n).inject(1) {|r,i| r*i } end
1.upto(26) {|i| p [i, Math.gamma(i), fact(i-1)] }
#=> [1, 1.0, 1]
#   [2, 1.0, 1]
#   [3, 2.0, 2]
#   [4, 6.0, 6]
#   [5, 24.0, 24]
#   [6, 120.0, 120]
#   [7, 720.0, 720]
#   [8, 5040.0, 5040]
#   [9, 40320.0, 40320]
#   [10, 362880.0, 362880]
#   [11, 3628800.0, 3628800]
#   [12, 39916800.0, 39916800]
#   [13, 479001600.0, 479001600]
#   [14, 6227020800.0, 6227020800]
#   [15, 87178291200.0, 87178291200]
#   [16, 1307674368000.0, 1307674368000]
#   [17, 20922789888000.0, 20922789888000]
#   [18, 355687428096000.0, 355687428096000]
#   [19, 6.402373705728e+15, 6402373705728000]
#   [20, 1.21645100408832e+17, 121645100408832000]
#   [21, 2.43290200817664e+18, 2432902008176640000]
#   [22, 5.109094217170944e+19, 51090942171709440000]
#   [23, 1.1240007277776077e+21, 1124000727777607680000]
#   [24, 2.5852016738885062e+22, 25852016738884976640000]
#   [25, 6.204484017332391e+23, 620448401733239439360000]
#   [26, 1.5511210043330954e+25, 15511210043330985984000000]

Source: show 

static VALUE
math_gamma(VALUE obj, VALUE x)
{
    static const double fact_table[] = {
        /* fact(0) */ 1.0,
        /* fact(1) */ 1.0,
        /* fact(2) */ 2.0,
        /* fact(3) */ 6.0,
        /* fact(4) */ 24.0,
        /* fact(5) */ 120.0,
        /* fact(6) */ 720.0,
        /* fact(7) */ 5040.0,
        /* fact(8) */ 40320.0,
        /* fact(9) */ 362880.0,
        /* fact(10) */ 3628800.0,
        /* fact(11) */ 39916800.0,
        /* fact(12) */ 479001600.0,
        /* fact(13) */ 6227020800.0,
        /* fact(14) */ 87178291200.0,
        /* fact(15) */ 1307674368000.0,
        /* fact(16) */ 20922789888000.0,
        /* fact(17) */ 355687428096000.0,
        /* fact(18) */ 6402373705728000.0,
        /* fact(19) */ 121645100408832000.0,
        /* fact(20) */ 2432902008176640000.0,
        /* fact(21) */ 51090942171709440000.0,
        /* fact(22) */ 1124000727777607680000.0,
        /* fact(23)=25852016738884976640000 needs 56bit mantissa which is
         * impossible to represent exactly in IEEE 754 double which have
         * 53bit mantissa. */
    };
    enum {NFACT_TABLE = numberof(fact_table)};
    double d;
    d = Get_Double(x);
    /* check for domain error */
    if (isinf(d) && signbit(d)) domain_error("gamma");
    if (d == floor(d)) {
        if (d < 0.0) domain_error("gamma");
        if (1.0 <= d && d <= (double)NFACT_TABLE) {
            return DBL2NUM(fact_table[(int)d - 1]);
        }
    }
    return DBL2NUM(tgamma(d));
}
Math.hypot(x, y) → Float

Returns sqrt(x**2 + y**2), the hypotenuse of a right-angled triangle with sides x and y.

Math.hypot(3, 4)   #=> 5.0

Source: show 

static VALUE
math_hypot(VALUE obj, VALUE x, VALUE y)
{
    return DBL2NUM(hypot(Get_Double(x), Get_Double(y)));
}
Math.ldexp(fraction, exponent) → float

Returns the value of fraction*(2**exponent).

fraction, exponent = Math.frexp(1234)
Math.ldexp(fraction, exponent)   #=> 1234.0

Source: show 

static VALUE
math_ldexp(VALUE obj, VALUE x, VALUE n)
{
    return DBL2NUM(ldexp(Get_Double(x), NUM2INT(n)));
}
Math.lgamma(x) → [float, -1 or 1]

Calculates the logarithmic gamma of x and the sign of gamma of x.

::lgamma is same as

[Math.log(Math.gamma(x).abs), Math.gamma(x) < 0 ? -1 : 1]

but avoid overflow by ::gamma for large x.

Math.lgamma(0) #=> [Infinity, 1]

Source: show 

static VALUE
math_lgamma(VALUE obj, VALUE x)
{
    double d;
    int sign=1;
    VALUE v;
    d = Get_Double(x);
    /* check for domain error */
    if (isinf(d)) {
        if (signbit(d)) domain_error("lgamma");
        return rb_assoc_new(DBL2NUM(INFINITY), INT2FIX(1));
    }
    v = DBL2NUM(lgamma_r(d, &sign));
    return rb_assoc_new(v, INT2FIX(sign));
}
Math.log(x) → Float
Math.log(x, base) → Float

Returns the logarithm of x. If additional second argument is given, it will be the base of logarithm. Otherwise it is e (for the natural logarithm).

Domain: (0, INFINITY)

Codomain: (-INFINITY, INFINITY)

Math.log(0)          #=> -Infinity
Math.log(1)          #=> 0.0
Math.log(Math::E)    #=> 1.0
Math.log(Math::E**3) #=> 3.0
Math.log(12, 3)      #=> 2.2618595071429146

Source: show 

static VALUE
math_log(int argc, const VALUE *argv, VALUE obj)
{
    VALUE x, base;
    double d;

    rb_scan_args(argc, argv, "11", &x, &base);
    d = math_log1(x);
    if (argc == 2) {
        d /= math_log1(base);
    }
    return DBL2NUM(d);
}
Math.log10(x) → Float

Returns the base 10 logarithm of x.

Domain: (0, INFINITY)

Codomain: (-INFINITY, INFINITY)

Math.log10(1)       #=> 0.0
Math.log10(10)      #=> 1.0
Math.log10(10**100) #=> 100.0

Source: show 

static VALUE
math_log10(VALUE obj, VALUE x)
{
    double d;
    size_t numbits;

    if (RB_BIGNUM_TYPE_P(x) && BIGNUM_POSITIVE_P(x) &&
            DBL_MAX_EXP <= (numbits = rb_absint_numwords(x, 1, NULL))) {
        numbits -= DBL_MANT_DIG;
        x = rb_big_rshift(x, SIZET2NUM(numbits));
    }
    else {
        numbits = 0;
    }

    d = Get_Double(x);
    /* check for domain error */
    if (d < 0.0) domain_error("log10");
    /* check for pole error */
    if (d == 0.0) return DBL2NUM(-INFINITY);

    return DBL2NUM(log10(d) + numbits * log10(2)); /* log10(d * 2 ** numbits) */
}
Math.log2(x) → Float

Returns the base 2 logarithm of x.

Domain: (0, INFINITY)

Codomain: (-INFINITY, INFINITY)

Math.log2(1)      #=> 0.0
Math.log2(2)      #=> 1.0
Math.log2(32768)  #=> 15.0
Math.log2(65536)  #=> 16.0

Source: show 

static VALUE
math_log2(VALUE obj, VALUE x)
{
    double d;
    size_t numbits;

    if (RB_BIGNUM_TYPE_P(x) && BIGNUM_POSITIVE_P(x) &&
            DBL_MAX_EXP <= (numbits = rb_absint_numwords(x, 1, NULL))) {
        numbits -= DBL_MANT_DIG;
        x = rb_big_rshift(x, SIZET2NUM(numbits));
    }
    else {
        numbits = 0;
    }

    d = Get_Double(x);
    /* check for domain error */
    if (d < 0.0) domain_error("log2");
    /* check for pole error */
    if (d == 0.0) return DBL2NUM(-INFINITY);

    return DBL2NUM(log2(d) + numbits); /* log2(d * 2 ** numbits) */
}
rsqrt(a)

Compute square root of a non negative number. This method is internally used by Math.sqrt.

Source: show | on GitHub 

# File lib/mathn.rb, line 142
def rsqrt(a)
  if a.kind_of?(Float)
    sqrt!(a)
  elsif a.kind_of?(Rational)
    rsqrt(a.numerator)/rsqrt(a.denominator)
  else
    src = a
    max = 2 ** 32
    byte_a = [src & 0xffffffff]
    # ruby's bug
    while (src >= max) and (src >>= 32)
      byte_a.unshift src & 0xffffffff
    end

    answer = 0
    main = 0
    side = 0
    for elm in byte_a
      main = (main << 32) + elm
      side <<= 16
      if answer != 0
        if main * 4  < side * side
          applo = main.div(side)
        else
          applo = ((sqrt!(side * side + 4 * main) - side)/2.0).to_i + 1
        end
      else
        applo = sqrt!(main).to_i + 1
      end

      while (x = (side + applo) * applo) > main
        applo -= 1
      end
      main -= x
      answer = (answer << 16) + applo
      side += applo * 2
    end
    if main == 0
      answer
    else
      sqrt!(a)
    end
  end
end
Math.sin(x) → Float

Computes the sine of x (expressed in radians). Returns a Float in the range -1.0..1.0.

Domain: (-INFINITY, INFINITY)

Codomain: [-1, 1]

Math.sin(Math::PI/2) #=> 1.0

Source: show 

static VALUE
math_sin(VALUE obj, VALUE x)
{
    return DBL2NUM(sin(Get_Double(x)));
}
Math.sinh(x) → Float

Computes the hyperbolic sine of x (expressed in radians).

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

Math.sinh(0) #=> 0.0

Source: show 

static VALUE
math_sinh(VALUE obj, VALUE x)
{
    return DBL2NUM(sinh(Get_Double(x)));
}
sqrt(a)

Computes the square root of a. It makes use of Complex and Rational to have no rounding errors if possible.

Math.sqrt(4/9)      # => 2/3
Math.sqrt(- 4/9)    # => Complex(0, 2/3)
Math.sqrt(4.0/9.0)  # => 0.666666666666667

Source: show | on GitHub 

# File lib/mathn.rb, line 119
def sqrt(a)
  if a.kind_of?(Complex)
    abs = sqrt(a.real*a.real + a.imag*a.imag)
    x = sqrt((a.real + abs)/Rational(2))
    y = sqrt((-a.real + abs)/Rational(2))
    if a.imag >= 0
      Complex(x, y)
    else
      Complex(x, -y)
    end
  elsif a.respond_to?(:nan?) and a.nan?
    a
  elsif a >= 0
    rsqrt(a)
  else
    Complex(0,rsqrt(-a))
  end
end
Math.tan(x) → Float

Computes the tangent of x (expressed in radians).

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

Math.tan(0) #=> 0.0

Source: show 

static VALUE
math_tan(VALUE obj, VALUE x)
{
    return DBL2NUM(tan(Get_Double(x)));
}
Math.tanh(x) → Float

Computes the hyperbolic tangent of x (expressed in radians).

Domain: (-INFINITY, INFINITY)

Codomain: (-1, 1)

Math.tanh(0) #=> 0.0

Source: show 

static VALUE
math_tanh(VALUE obj, VALUE x)
{
    return DBL2NUM(tanh(Get_Double(x)));
}