# Module: Math

Defined in:
math.c

## Overview

Document-class: Math

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.

## Defined Under Namespace

Classes: DomainError

## Constant Summary collapse

PI =
`DBL2NUM(atan(1.0)*4.0)`
E =
`DBL2NUM(exp(1.0))`

## Class Method Summary collapse

• Computes the arc cosine of x.

• Computes the inverse hyperbolic cosine of x.

• Computes the arc sine of x.

• Computes the inverse hyperbolic sine of x.

• Computes the arc tangent of x.

• Computes the arc tangent given y and x.

• Computes the inverse hyperbolic tangent of x.

• Returns the cube root of numeric.

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

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

• Calculates the error function of x.

• Calculates the complementary error function of x.

• Returns e**x.

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

• Calculates the gamma function of x.

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

• Returns the value of flt*(2**int).

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

• Returns the natural logarithm of numeric.

• Returns the base 10 logarithm of numeric.

• Returns the base 2 logarithm of numeric.

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

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

• Returns the non-negative square root of numeric.

• Returns the tangent of x (expressed in radians).

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

## Class Method Details

### .acos(x) ⇒ Float

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

Returns:

 ``` ``` ```# File 'math.c' 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

Computes the inverse hyperbolic cosine of x.

Returns:

 ``` ``` ```# File 'math.c' 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

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

Returns:

 ``` ``` ```# File 'math.c' 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

Computes the inverse hyperbolic sine of x.

Returns:

 ``` ``` ```# File 'math.c' static VALUE math_asinh(VALUE obj, VALUE x) { Need_Float(x); return DBL2NUM(asinh(RFLOAT_VALUE(x))); }```

### .atan(x) ⇒ Float

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

Returns:

 ``` ``` ```# File 'math.c' static VALUE math_atan(VALUE obj, VALUE x) { Need_Float(x); return DBL2NUM(atan(RFLOAT_VALUE(x))); }```

### .atan2(y, x) ⇒ Float

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
``````

Returns:

 ``` ``` ```# File 'math.c' 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); }```

### .atanh(x) ⇒ Float

Computes the inverse hyperbolic tangent of x.

Returns:

 ``` ``` ```# File 'math.c' 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

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]
``````

Returns:

 ``` ``` ```# File 'math.c' static VALUE math_cbrt(VALUE obj, VALUE x) { Need_Float(x); return DBL2NUM(cbrt(RFLOAT_VALUE(x))); }```

### .cos(x) ⇒ Float

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

Returns:

 ``` ``` ```# File 'math.c' static VALUE math_cos(VALUE obj, VALUE x) { Need_Float(x); return DBL2NUM(cos(RFLOAT_VALUE(x))); }```

### .cosh(x) ⇒ Float

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

Returns:

 ``` ``` ```# File 'math.c' static VALUE math_cosh(VALUE obj, VALUE x) { Need_Float(x); return DBL2NUM(cosh(RFLOAT_VALUE(x))); }```

### .erf(x) ⇒ Float

Calculates the error function of x.

Returns:

 ``` ``` ```# File 'math.c' static VALUE math_erf(VALUE obj, VALUE x) { Need_Float(x); return DBL2NUM(erf(RFLOAT_VALUE(x))); }```

### .erfc(x) ⇒ Float

Calculates the complementary error function of x.

Returns:

 ``` ``` ```# File 'math.c' static VALUE math_erfc(VALUE obj, VALUE x) { Need_Float(x); return DBL2NUM(erfc(RFLOAT_VALUE(x))); }```

### .exp(x) ⇒ Float

Returns e**x.

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

Returns:

 ``` ``` ```# File 'math.c' static VALUE math_exp(VALUE obj, VALUE x) { Need_Float(x); return DBL2NUM(exp(RFLOAT_VALUE(x))); }```

### .frexp(numeric) ⇒ Array

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
``````

Returns:

 ``` ``` ```# File 'math.c' 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

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]
``````

Returns:

 ``` ``` ```# File 'math.c' 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. */ }```

### .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
``````

Returns:

 ``` ``` ```# File 'math.c' 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

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

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

Returns:

 ``` ``` ```# File 'math.c' static VALUE math_ldexp(VALUE obj, VALUE x, VALUE n) { Need_Float(x); return DBL2NUM(ldexp(RFLOAT_VALUE(x), NUM2INT(n))); }```

### .lgamma(x) ⇒ Array, ...

Calculates the logarithmic gamma of x and

``````the sign of gamma of x.

Math.lgamma(x) is same as
[Math.log(Math.gamma(x).abs), Math.gamma(x) < 0 ? -1 : 1]
but avoid overflow by Math.gamma(x) for large x.
``````

Returns ].

Returns:

 ``` ``` ```# File 'math.c' 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)); }```

### .log(numeric) ⇒ Float .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
``````

 ``` ``` ```# File 'math.c' 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)); }```

### .log10(numeric) ⇒ Float

Returns the base 10 logarithm of numeric.

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

Returns:

 ``` ``` ```# File 'math.c' 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

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
``````

Returns:

 ``` ``` ```# File 'math.c' 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); }```

### .sin(x) ⇒ Float

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

Returns:

 ``` ``` ```# File 'math.c' static VALUE math_sin(VALUE obj, VALUE x) { Need_Float(x); return DBL2NUM(sin(RFLOAT_VALUE(x))); }```

### .sinh(x) ⇒ Float

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

Returns:

 ``` ``` ```# File 'math.c' static VALUE math_sinh(VALUE obj, VALUE x) { Need_Float(x); return DBL2NUM(sinh(RFLOAT_VALUE(x))); }```

### .sqrt(numeric) ⇒ Float

Returns the non-negative square root of numeric.

``````0.upto(10) {|x|
p [x, Math.sqrt(x), Math.sqrt(x)**2]
}
#=>
[0, 0.0, 0.0]
[1, 1.0, 1.0]
[2, 1.4142135623731, 2.0]
[3, 1.73205080756888, 3.0]
[4, 2.0, 4.0]
[5, 2.23606797749979, 5.0]
[6, 2.44948974278318, 6.0]
[7, 2.64575131106459, 7.0]
[8, 2.82842712474619, 8.0]
[9, 3.0, 9.0]
[10, 3.16227766016838, 10.0]
``````

Returns:

 ``` ``` ```# File 'math.c' static VALUE math_sqrt(VALUE obj, VALUE x) { double d0, d; Need_Float(x); d0 = RFLOAT_VALUE(x); /* check for domain error */ if (d0 < 0.0) domain_error("sqrt"); if (d0 == 0.0) return DBL2NUM(0.0); d = sqrt(d0); return DBL2NUM(d); }```

### .tan(x) ⇒ Float

Returns the tangent of x (expressed in radians).

Returns:

 ``` ``` ```# File 'math.c' static VALUE math_tan(VALUE obj, VALUE x) { Need_Float(x); return DBL2NUM(tan(RFLOAT_VALUE(x))); }```

### .tanh ⇒ Float

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

Returns:

 ``` ``` ```# File 'math.c' static VALUE math_tanh(VALUE obj, VALUE x) { Need_Float(x); return DBL2NUM(tanh(RFLOAT_VALUE(x))); }```