/Ruby 3

module 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` for Euler's number (e) as a `Float` number.

PI

Definition of the mathematical constant `PI` as a `Float` number.

Public Class Methods

acos(x) → Float Show source
```static VALUE
math_acos(VALUE unused_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));
}```

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

Domain: [-1, 1]

Codomain: [0, PI]

```Math.acos(0) == Math::PI/2  #=> true
```
acosh(x) → Float Show source
```static VALUE
math_acosh(VALUE unused_obj, VALUE x)
{
double d;

d = Get_Double(x);
/* check for domain error */
if (d < 1.0) domain_error("acosh");
return DBL2NUM(acosh(d));
}```

Computes the inverse hyperbolic cosine of `x`.

Domain: [1, INFINITY)

Codomain: [0, INFINITY)

```Math.acosh(1) #=> 0.0
```
asin(x) → Float Show source
```static VALUE
math_asin(VALUE unused_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));
}```

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
```
asinh(x) → Float Show source
```static VALUE
math_asinh(VALUE unused_obj, VALUE x)
{
return DBL2NUM(asinh(Get_Double(x)));
}```

Computes the inverse hyperbolic sine of `x`.

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

```Math.asinh(1) #=> 0.881373587019543
```
atan(x) → Float Show source
```static VALUE
math_atan(VALUE unused_obj, VALUE x)
{
return DBL2NUM(atan(Get_Double(x)));
}```

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

Domain: (-INFINITY, INFINITY)

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

```Math.atan(0) #=> 0.0
```
atan2(y, x) → Float Show source
```static VALUE
math_atan2(VALUE unused_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));
}```

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
```
atanh(x) → Float Show source
```static VALUE
math_atanh(VALUE unused_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(-HUGE_VAL);
if (d == +1.0) return DBL2NUM(+HUGE_VAL);
return DBL2NUM(atanh(d));
}```

Computes the inverse hyperbolic tangent of `x`.

Domain: (-1, 1)

Codomain: (-INFINITY, INFINITY)

```Math.atanh(1) #=> Infinity
```
cbrt(x) → Float Show source
```static VALUE
math_cbrt(VALUE unused_obj, VALUE x)
{
double f = Get_Double(x);
double r = cbrt(f);
#if defined __GLIBC__
if (isfinite(r)) {
r = (2.0 * r + (f / r / r)) / 3.0;
}
#endif
return DBL2NUM(r);
}```

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]
```
cos(x) → Float Show source
```static VALUE
math_cos(VALUE unused_obj, VALUE x)
{
return DBL2NUM(cos(Get_Double(x)));
}```

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
```
cosh(x) → Float Show source
```static VALUE
math_cosh(VALUE unused_obj, VALUE x)
{
return DBL2NUM(cosh(Get_Double(x)));
}```

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

Domain: (-INFINITY, INFINITY)

Codomain: [1, INFINITY)

```Math.cosh(0) #=> 1.0
```
erf(x) → Float Show source
```static VALUE
math_erf(VALUE unused_obj, VALUE x)
{
return DBL2NUM(erf(Get_Double(x)));
}```

Calculates the error function of `x`.

Domain: (-INFINITY, INFINITY)

Codomain: (-1, 1)

```Math.erf(0) #=> 0.0
```
erfc(x) → Float Show source
```static VALUE
math_erfc(VALUE unused_obj, VALUE x)
{
return DBL2NUM(erfc(Get_Double(x)));
}```

Calculates the complementary error function of x.

Domain: (-INFINITY, INFINITY)

Codomain: (0, 2)

```Math.erfc(0) #=> 1.0
```
exp(x) → Float Show source
```static VALUE
math_exp(VALUE unused_obj, VALUE x)
{
return DBL2NUM(exp(Get_Double(x)));
}```

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
```
frexp(x) → [fraction, exponent] Show source
```static VALUE
math_frexp(VALUE unused_obj, VALUE x)
{
double d;
int exp;

d = frexp(Get_Double(x), &exp);
return rb_assoc_new(DBL2NUM(d), INT2NUM(exp));
}```

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

```fraction, exponent = Math.frexp(1234)   #=> [0.6025390625, 11]
fraction * 2**exponent                  #=> 1234.0
```
gamma(x) → Float Show source
```static VALUE
math_gamma(VALUE unused_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)) {
if (signbit(d)) domain_error("gamma");
return DBL2NUM(HUGE_VAL);
}
if (d == 0.0) {
return signbit(d) ? DBL2NUM(-HUGE_VAL) : DBL2NUM(HUGE_VAL);
}
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));
}```

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]
```
hypot(x, y) → Float Show source
```static VALUE
math_hypot(VALUE unused_obj, VALUE x, VALUE y)
{
return DBL2NUM(hypot(Get_Double(x), Get_Double(y)));
}```

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

```Math.hypot(3, 4)   #=> 5.0
```
ldexp(fraction, exponent) → float Show source
```static VALUE
math_ldexp(VALUE unused_obj, VALUE x, VALUE n)
{
return DBL2NUM(ldexp(Get_Double(x), NUM2INT(n)));
}```

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

```fraction, exponent = Math.frexp(1234)
Math.ldexp(fraction, exponent)   #=> 1234.0
```
lgamma(x) → [float, -1 or 1] Show source
```static VALUE
math_lgamma(VALUE unused_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(HUGE_VAL), INT2FIX(1));
}
if (d == 0.0) {
VALUE vsign = signbit(d) ? INT2FIX(-1) : INT2FIX(+1);
return rb_assoc_new(DBL2NUM(HUGE_VAL), vsign);
}
v = DBL2NUM(lgamma_r(d, &sign));
return rb_assoc_new(v, INT2FIX(sign));
}```

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.

```Math.lgamma(0) #=> [Infinity, 1]
```
log(x) → Float Show source
log(x, base) → Float
```static VALUE
math_log(int argc, const VALUE *argv, VALUE unused_obj)
{
return rb_math_log(argc, argv);
}```

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
```
log10(x) → Float Show source
```static VALUE
math_log10(VALUE unused_obj, VALUE x)
{
size_t numbits;
double d = get_double_rshift(x, &numbits);

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

return DBL2NUM(log10(d) + numbits * log10(2)); /* log10(d * 2 ** numbits) */
}```

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
```
log2(x) → Float Show source
```static VALUE
math_log2(VALUE unused_obj, VALUE x)
{
size_t numbits;
double d = get_double_rshift(x, &numbits);

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

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

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
```
sin(x) → Float Show source
```static VALUE
math_sin(VALUE unused_obj, VALUE x)
{
return DBL2NUM(sin(Get_Double(x)));
}```

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
```
sinh(x) → Float Show source
```static VALUE
math_sinh(VALUE unused_obj, VALUE x)
{
return DBL2NUM(sinh(Get_Double(x)));
}```

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

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

```Math.sinh(0) #=> 0.0
```
sqrt(x) → Float Show source
```static VALUE
math_sqrt(VALUE unused_obj, VALUE x)
{
return rb_math_sqrt(x);
}```

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]
```

Note that the limited precision of floating point arithmetic might lead to surprising results:

```Math.sqrt(10**46).to_i  #=> 99999999999999991611392 (!)
```

See also `BigDecimal#sqrt` and `Integer.sqrt`.

tan(x) → Float Show source
```static VALUE
math_tan(VALUE unused_obj, VALUE x)
{
return DBL2NUM(tan(Get_Double(x)));
}```

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

Domain: (-INFINITY, INFINITY)

Codomain: (-INFINITY, INFINITY)

```Math.tan(0) #=> 0.0
```
tanh(x) → Float Show source
```static VALUE
math_tanh(VALUE unused_obj, VALUE x)
{
return DBL2NUM(tanh(Get_Double(x)));
}```

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

Domain: (-INFINITY, INFINITY)

Codomain: (-1, 1)

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

Ruby Core © 1993–2020 Yukihiro Matsumoto