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# 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.

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

### Constants

E

Definition of the mathematical constant E (e) as a Float number.

PI

Definition of the mathematical constant PI as a Float number.

### Public Class Methods

acos(x) → Float click to toggle source

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

Domain: [-1, 1]

Codomain: [0, PI]

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

Domain: [1, INFINITY)

Codomain: [0, INFINITY)

```Math.acosh(1) #=> 0.0
```
```
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.

Domain: [-1, -1]

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

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

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

```Math.asinh(1) #=> 0.881373587019543
```
```
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.

Domain: (-INFINITY, INFINITY)

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

```Math.atan(0) #=> 0.0
```
```
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 a Float in the range -PI..PI.

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
```
```
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);
}
#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));
}
```
atanh(x) → Float click to toggle source

Computes the inverse hyperbolic tangent of `x`.

Domain: (-1, 1)

Codomain: (-INFINITY, INFINITY)

```Math.atanh(1) #=> Infinity
```
```
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(x) → Float click to toggle source

Returns the cube root of `x`.

Domain: [0, INFINITY)

Codomain:[0, 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]
```
```
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 a Float in the range -1.0..1.0.

Domain: (-INFINITY, INFINITY)

Codomain: [-1, 1]

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

Domain: (-INFINITY, INFINITY)

Codomain: [1, INFINITY)

```Math.cosh(0) #=> 1.0
```
```
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`.

Domain: (-INFINITY, INFINITY)

Codomain: (-1, 1)

```Math.erf(0) #=> 0.0
```
```
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.

Domain: (-INFINITY, INFINITY)

Codomain: (0, 2)

```Math.erfc(0) #=> 1.0
```
```
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.

Domain: (-INFINITY, INFINITY)

Codomain: (0, INFINITY)

```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(x) → [fraction, exponent] click to toggle source

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
```
```
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(fraction, exponent) → float click to toggle source

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

```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.

```Math.lgamma(0) #=> [Infinity, 1]
```
```
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(x) → Float click to toggle source
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
```
```
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);
}
```
log10(x) → Float click to toggle source

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
```
```
static VALUE
math_log10(VALUE obj, VALUE x)
{
double d0, 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;
}

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);
if (numbits)
d += numbits * log10(2); /* log10(2**numbits) */
return DBL2NUM(d);
}
```
log2(x) → Float click to toggle source

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
```
```
static VALUE
math_log2(VALUE obj, VALUE x)
{
double d0, 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;
}

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);
d += numbits;
return DBL2NUM(d);
}
```
sin(x) → Float click to toggle source

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
```
```
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).

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

```Math.sinh(0) #=> 0.0
```
```
static VALUE
math_sinh(VALUE obj, VALUE x)
{
Need_Float(x);
return DBL2NUM(sinh(RFLOAT_VALUE(x)));
}
```
sqrt(x) → Float click to toggle source

Returns the non-negative square root of `x`.

Domain: [0, INFINITY)

Codomain:[0, INFINITY)

```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]
```
```
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 click to toggle source

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

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

```Math.tan(0) #=> 0.0
```
```
static VALUE
math_tan(VALUE obj, VALUE x)
{
Need_Float(x);
return DBL2NUM(tan(RFLOAT_VALUE(x)));
}
```
tanh(x) → Float click to toggle source

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

Domain: (-INFINITY, INFINITY)

Codomain: (-1, 1)

```Math.tanh(0) #=> 0.0
```
```
static VALUE
math_tanh(VALUE obj, VALUE x)
{
Need_Float(x);
return DBL2NUM(tanh(RFLOAT_VALUE(x)));
}
```