# module Math

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.

E
PI

### Public Class Methods

acos(x) → float click to toggle source

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

```static VALUE
math_acos(VALUE obj, VALUE x)
{
double d0, d;

Need_Float(x);
d0 = RFLOAT_VALUE(x);
/* check for domain error */
if (d0 < -1.0 || 1.0 < d0) domain_error("acos");
d = acos(d0);
return DBL2NUM(d);
}```
acosh(x) → float click to toggle source

Computes the inverse hyperbolic cosine of x.

```static VALUE
math_acosh(VALUE obj, VALUE x)
{
double d0, d;

Need_Float(x);
d0 = RFLOAT_VALUE(x);
/* check for domain error */
if (d0 < 1.0) domain_error("acosh");
d = acosh(d0);
return DBL2NUM(d);
}```
asin(x) → float click to toggle source

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

```static VALUE
math_asin(VALUE obj, VALUE x)
{
double d0, d;

Need_Float(x);
d0 = RFLOAT_VALUE(x);
/* check for domain error */
if (d0 < -1.0 || 1.0 < d0) domain_error("asin");
d = asin(d0);
return DBL2NUM(d);
}```
asinh(x) → float click to toggle source

Computes the inverse hyperbolic sine of x.

```static VALUE
math_asinh(VALUE obj, VALUE x)
{
Need_Float(x);
return DBL2NUM(asinh(RFLOAT_VALUE(x)));
}```
atan(x) → float click to toggle source

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

```static VALUE
math_atan(VALUE obj, VALUE x)
{
Need_Float(x);
return DBL2NUM(atan(RFLOAT_VALUE(x)));
}```
atan2(y, x) → float click to toggle source

Computes the arc tangent given y and x. Returns -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
```
```static VALUE
math_atan2(VALUE obj, VALUE y, VALUE x)
{
#ifndef M_PI
# define M_PI 3.14159265358979323846
#endif
double dx, dy;
Need_Float2(y, x);
dx = RFLOAT_VALUE(x);
dy = RFLOAT_VALUE(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);
}
if (isinf(dx) && isinf(dy)) domain_error("atan2");
return DBL2NUM(atan2(dy, dx));
}```
atanh(x) → float click to toggle source

Computes the inverse hyperbolic tangent of x.

```static VALUE
math_atanh(VALUE obj, VALUE x)
{
double d0, d;

Need_Float(x);
d0 = RFLOAT_VALUE(x);
/* check for domain error */
if (d0 <  -1.0 || +1.0 <  d0) domain_error("atanh");
/* check for pole error */
if (d0 == -1.0) return DBL2NUM(-INFINITY);
if (d0 == +1.0) return DBL2NUM(+INFINITY);
d = atanh(d0);
return DBL2NUM(d);
}```
cbrt(numeric) → float click to toggle source

Returns the cube root of numeric.

```-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]
```
```static VALUE
math_cbrt(VALUE obj, VALUE x)
{
Need_Float(x);
return DBL2NUM(cbrt(RFLOAT_VALUE(x)));
}```
cos(x) → float click to toggle source

Computes the cosine of x (expressed in radians). Returns -1..1.

```static VALUE
math_cos(VALUE obj, VALUE x)
{
Need_Float(x);
return DBL2NUM(cos(RFLOAT_VALUE(x)));
}```
cosh(x) → float click to toggle source

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

```static VALUE
math_cosh(VALUE obj, VALUE x)
{
Need_Float(x);

return DBL2NUM(cosh(RFLOAT_VALUE(x)));
}```
erf(x) → float click to toggle source

Calculates the error function of x.

```static VALUE
math_erf(VALUE obj, VALUE x)
{
Need_Float(x);
return DBL2NUM(erf(RFLOAT_VALUE(x)));
}```
erfc(x) → float click to toggle source

Calculates the complementary error function of x.

```static VALUE
math_erfc(VALUE obj, VALUE x)
{
Need_Float(x);
return DBL2NUM(erfc(RFLOAT_VALUE(x)));
}```
exp(x) → float click to toggle source

Returns e**x.

```Math.exp(0)       #=> 1.0
Math.exp(1)       #=> 2.718281828459045
Math.exp(1.5)     #=> 4.4816890703380645
```
```static VALUE
math_exp(VALUE obj, VALUE x)
{
Need_Float(x);
return DBL2NUM(exp(RFLOAT_VALUE(x)));
}```
frexp(numeric) → [ fraction, exponent ] click to toggle source

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

```fraction, exponent = Math.frexp(1234)   #=> [0.6025390625, 11]
fraction * 2**exponent                  #=> 1234.0
```
```static VALUE
math_frexp(VALUE obj, VALUE x)
{
double d;
int exp;

Need_Float(x);

d = frexp(RFLOAT_VALUE(x), &exp);
return rb_assoc_new(DBL2NUM(d), INT2NUM(exp));
}```
gamma(x) → float click to toggle source

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]
```
```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. */
};
double d0, d;
double intpart, fracpart;
Need_Float(x);
d0 = RFLOAT_VALUE(x);
/* check for domain error */
if (isinf(d0) && signbit(d0)) domain_error("gamma");
fracpart = modf(d0, &intpart);
if (fracpart == 0.0) {
if (intpart < 0) domain_error("gamma");
if (0 < intpart &&
intpart - 1 < (double)numberof(fact_table)) {
return DBL2NUM(fact_table[(int)intpart - 1]);
}
}
d = tgamma(d0);
return DBL2NUM(d);
}```
hypot(x, y) → float click to toggle source

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

```Math.hypot(3, 4)   #=> 5.0
```
```static VALUE
math_hypot(VALUE obj, VALUE x, VALUE y)
{
Need_Float2(x, y);
return DBL2NUM(hypot(RFLOAT_VALUE(x), RFLOAT_VALUE(y)));
}```
ldexp(flt, int) → float click to toggle source

Returns the value of flt*(2**int).

```fraction, exponent = Math.frexp(1234)
Math.ldexp(fraction, exponent)   #=> 1234.0
```
```static VALUE
math_ldexp(VALUE obj, VALUE x, VALUE n)
{
Need_Float(x);
return DBL2NUM(ldexp(RFLOAT_VALUE(x), NUM2INT(n)));
}```
lgamma(x) → [float, -1 or 1] click to toggle source

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.

```static VALUE
math_lgamma(VALUE obj, VALUE x)
{
double d0, d;
int sign=1;
VALUE v;
Need_Float(x);
d0 = RFLOAT_VALUE(x);
/* check for domain error */
if (isinf(d0)) {
if (signbit(d0)) domain_error("lgamma");
return rb_assoc_new(DBL2NUM(INFINITY), INT2FIX(1));
}
d = lgamma_r(d0, &sign);
v = DBL2NUM(d);
return rb_assoc_new(v, INT2FIX(sign));
}```
log(numeric) → float click to toggle source
log(num,base) → float

Returns the natural logarithm of numeric. If additional second argument is given, it will be the base of logarithm.

```Math.log(1)          #=> 0.0
Math.log(Math::E)    #=> 1.0
Math.log(Math::E**3) #=> 3.0
Math.log(12,3)       #=> 2.2618595071429146
```
```static VALUE
math_log(int argc, VALUE *argv)
{
VALUE x, base;
double d0, d;

rb_scan_args(argc, argv, "11", &x, &base);
Need_Float(x);
d0 = RFLOAT_VALUE(x);
/* check for domain error */
if (d0 < 0.0) domain_error("log");
/* check for pole error */
if (d0 == 0.0) return DBL2NUM(-INFINITY);
d = log(d0);
if (argc == 2) {
Need_Float(base);
d /= log(RFLOAT_VALUE(base));
}
return DBL2NUM(d);
}```
log10(numeric) → float click to toggle source

Returns the base 10 logarithm of numeric.

```Math.log10(1)       #=> 0.0
Math.log10(10)      #=> 1.0
Math.log10(10**100) #=> 100.0
```
```static VALUE
math_log10(VALUE obj, VALUE x)
{
double d0, d;

Need_Float(x);
d0 = RFLOAT_VALUE(x);
/* check for domain error */
if (d0 < 0.0) domain_error("log10");
/* check for pole error */
if (d0 == 0.0) return DBL2NUM(-INFINITY);
d = log10(d0);
return DBL2NUM(d);
}```
log2(numeric) → float click to toggle source

Returns the base 2 logarithm of numeric.

```Math.log2(1)      #=> 0.0
Math.log2(2)      #=> 1.0
Math.log2(32768)  #=> 15.0
Math.log2(65536)  #=> 16.0
```
```static VALUE
math_log2(VALUE obj, VALUE x)
{
double d0, d;

Need_Float(x);
d0 = RFLOAT_VALUE(x);
/* check for domain error */
if (d0 < 0.0) domain_error("log2");
/* check for pole error */
if (d0 == 0.0) return DBL2NUM(-INFINITY);
d = log2(d0);
return DBL2NUM(d);
}```
rsqrt(a) click to toggle source

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

```# File lib/mathn.rb, line 255
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

main = 0
side = 0
for elm in byte_a
main = (main << 32) + elm
side <<= 16
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
side += applo * 2
end
if main == 0
else
sqrt!(a)
end
end
end```
sin(x) → float click to toggle source

Computes the sine of x (expressed in radians). Returns -1..1.

```static VALUE
math_sin(VALUE obj, VALUE x)
{
Need_Float(x);

return DBL2NUM(sin(RFLOAT_VALUE(x)));
}```
sinh(x) → float click to toggle source

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

```static VALUE
math_sinh(VALUE obj, VALUE x)
{
Need_Float(x);
return DBL2NUM(sinh(RFLOAT_VALUE(x)));
}```
sqrt(a) click to toggle source

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
```
```# File lib/mathn.rb, line 226
def sqrt(a)
if a.kind_of?(Complex)
abs = sqrt(a.real*a.real + a.imag*a.imag)
#      if not abs.kind_of?(Rational)
#        return a**Rational(1,2)
#      end
x = sqrt((a.real + abs)/Rational(2))
y = sqrt((-a.real + abs)/Rational(2))
#      if !(x.kind_of?(Rational) and y.kind_of?(Rational))
#        return a**Rational(1,2)
#      end
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```
tan(x) → float click to toggle source

Returns the tangent of x (expressed in radians).

```static VALUE
math_tan(VALUE obj, VALUE x)
{
Need_Float(x);

return DBL2NUM(tan(RFLOAT_VALUE(x)));
}```
tanh() → float click to toggle source

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

```static VALUE
math_tanh(VALUE obj, VALUE x)
{
Need_Float(x);
return DBL2NUM(tanh(RFLOAT_VALUE(x)));
}```

### Private Instance Methods

rsqrt(a) click to toggle source

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

```# File lib/mathn.rb, line 255
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

main = 0
side = 0
for elm in byte_a
main = (main << 32) + elm
side <<= 16
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
side += applo * 2
end
if main == 0
else
sqrt!(a)
end
end
end```
sqrt(a) click to toggle source

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
```
```# File lib/mathn.rb, line 226
def sqrt(a)
if a.kind_of?(Complex)
abs = sqrt(a.real*a.real + a.imag*a.imag)
#      if not abs.kind_of?(Rational)
#        return a**Rational(1,2)
#      end
x = sqrt((a.real + abs)/Rational(2))
y = sqrt((-a.real + abs)/Rational(2))
#      if !(x.kind_of?(Rational) and y.kind_of?(Rational))
#        return a**Rational(1,2)
#      end
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```