# module Math

Module Math provides methods for basic trigonometric, logarithmic, and transcendental functions, and for extracting roots.

You can write its constants and method calls thus:

Math::PI      # => 3.141592653589793
Math::E       # => 2.718281828459045
Math.sin(0.0) # => 0.0
Math.cos(0.0) # => 1.0

If you include module Math, you can write simpler forms:

include Math
PI       # => 3.141592653589793
E        # => 2.718281828459045
sin(0.0) # => 0.0
cos(0.0) # => 1.0

For simplicity, the examples here assume:

include Math
INFINITY = Float::INFINITY

The domains and ranges for the methods are denoted by open or closed intervals, using, respectively, parentheses or square brackets:

• An open interval does not include the endpoints:

(-INFINITY, INFINITY)
• A closed interval includes the endpoints:

[-1.0, 1.0]

• A half-open interval includes one endpoint, but not the other:

[1.0, INFINITY)

Many values returned by Math methods are numerical approximations. This is because many such values are, in mathematics, of infinite precision, while in numerical computation the precision is finite.

Thus, in mathematics, cos(π/2) is exactly zero, but in our computation cos(PI/2) is a number very close to zero:

cos(PI/2) # => 6.123031769111886e-17

For very large and very small returned values, we have added formatted numbers for clarity:

tan(PI/2)  # => 1.633123935319537e+16   # 16331239353195370.0
tan(PI)    # => -1.2246467991473532e-16 # -0.0000000000000001

See class Float for the constants that affect Ruby’s floating-point arithmetic.

### What’s Here¶ ↑

#### Trigonometric Functions¶ ↑

• ::cos: Returns the cosine of the given argument.

• ::sin: Returns the sine of the given argument.

• ::tan: Returns the tangent of the given argument.

#### Inverse Trigonometric Functions¶ ↑

• ::acos: Returns the arc cosine of the given argument.

• ::asin: Returns the arc sine of the given argument.

• ::atan: Returns the arc tangent of the given argument.

• ::atan2: Returns the arg tangent of two given arguments.

#### Hyperbolic Trigonometric Functions¶ ↑

• ::cosh: Returns the hyperbolic cosine of the given argument.

• ::sinh: Returns the hyperbolic sine of the given argument.

• ::tanh: Returns the hyperbolic tangent of the given argument.

#### Inverse Hyperbolic Trigonometric Functions¶ ↑

• ::acosh: Returns the inverse hyperbolic cosine of the given argument.

• ::asinh: Returns the inverse hyperbolic sine of the given argument.

• ::atanh: Returns the inverse hyperbolic tangent of the given argument.

#### Exponentiation and Logarithmic Functions¶ ↑

• ::exp: Returns the value of a given value raised to a given power.

• ::log: Returns the logarithm of a given value in a given base.

• ::log10: Returns the base 10 logarithm of the given argument.

• ::log2: Returns the base 2 logarithm of the given argument.

#### Fraction and Exponent Functions¶ ↑

• ::frexp: Returns the fraction and exponent of the given argument.

• ::ldexp: Returns the value for a given fraction and exponent.

#### Root Functions¶ ↑

• ::cbrt: Returns the cube root of the given argument.

• ::sqrt: Returns the square root of the given argument.

#### Error Functions¶ ↑

• ::erf: Returns the value of the Gauss error function for the given argument.

• ::erfc: Returns the value of the complementary error function for the given argument.

#### Gamma Functions¶ ↑

• ::gamma: Returns the value of the gamma function for the given argument.

• ::lgamma: Returns the value of the logarithmic gamma function for the given argument.

#### Hypotenuse Function¶ ↑

• ::hypot: Returns sqrt(a**2 + b**2) for the given a and b.

### 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 click to toggle source

Returns the arc cosine of x.

• Domain: [-1, 1].

• Range: [0, PI].

Examples:

acos(-1.0) # => 3.141592653589793  # PI
acos(0.0)  # => 1.5707963267948966 # PI/2
acos(1.0)  # => 0.0
static VALUE
math_acos(VALUE unused_obj, VALUE x)
{
double d;

d = Get_Double(x);
domain_check_range(d, -1.0, 1.0, "acos");
return DBL2NUM(acos(d));
}
acosh(x) → float click to toggle source

Returns the inverse hyperbolic cosine of x.

• Domain: [1, INFINITY].

• Range: [0, INFINITY].

Examples:

acosh(1.0)      # => 0.0
acosh(INFINITY) # => Infinity
static VALUE
math_acosh(VALUE unused_obj, VALUE x)
{
double d;

d = Get_Double(x);
domain_check_min(d, 1.0, "acosh");
return DBL2NUM(acosh(d));
}
asin(x) → float click to toggle source

Returns the arc sine of x.

• Domain: [-1, -1].

• Range: [-PI/2, PI/2].

Examples:

asin(-1.0) # => -1.5707963267948966 # -PI/2
asin(0.0)  # => 0.0
asin(1.0)  # => 1.5707963267948966  # PI/2
static VALUE
math_asin(VALUE unused_obj, VALUE x)
{
double d;

d = Get_Double(x);
domain_check_range(d, -1.0, 1.0, "asin");
return DBL2NUM(asin(d));
}
asinh(x) → float click to toggle source

Returns the inverse hyperbolic sine of x.

• Domain: [-INFINITY, INFINITY].

• Range: [-INFINITY, INFINITY].

Examples:

asinh(-INFINITY) # => -Infinity
asinh(0.0)       # => 0.0
asinh(INFINITY)  # => Infinity
static VALUE
math_asinh(VALUE unused_obj, VALUE x)
{
return DBL2NUM(asinh(Get_Double(x)));
}
atan(x) → Float click to toggle source

Returns the arc tangent of x.

• Domain: [-INFINITY, INFINITY].

• Range: [-PI/2, PI/2] .

Examples:

atan(-INFINITY) # => -1.5707963267948966 # -PI2
atan(-PI)       # => -1.2626272556789115
atan(-PI/2)     # => -1.0038848218538872
atan(0.0)       # => 0.0
atan(PI/2)      # => 1.0038848218538872
atan(PI)        # => 1.2626272556789115
atan(INFINITY)  # => 1.5707963267948966  # PI/2
static VALUE
math_atan(VALUE unused_obj, VALUE x)
{
return DBL2NUM(atan(Get_Double(x)));
}
atan2(y, x) → float click to toggle source

Returns the arc tangent of y and x in radians.

• Domain of y: [-INFINITY, INFINITY].

• Domain of x: [-INFINITY, INFINITY].

• Range: [-PI, PI].

Examples:

atan2(-1.0, -1.0) # => -2.356194490192345  # -3*PI/4
atan2(-1.0, 0.0)  # => -1.5707963267948966 # -PI/2
atan2(-1.0, 1.0)  # => -0.7853981633974483 # -PI/4
atan2(0.0, -1.0)  # => 3.141592653589793   # PI
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));
}
atanh(x) → float click to toggle source

Returns the inverse hyperbolic tangent of x.

• Domain: [-1, 1].

• Range: [-INFINITY, INFINITY].

Examples:

atanh(-1.0) # => -Infinity
atanh(0.0)  # => 0.0
atanh(1.0)  # => Infinity
static VALUE
math_atanh(VALUE unused_obj, VALUE x)
{
double d;

d = Get_Double(x);
domain_check_range(d, -1.0, +1.0, "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));
}
cbrt(x) → float click to toggle source

Returns the cube root of x.

• Domain: [-INFINITY, INFINITY].

• Range: [-INFINITY, INFINITY].

Examples:

cbrt(-INFINITY) # => -Infinity
cbrt(-27.0)     # => -3.0
cbrt(-8.0)      # => -2.0
cbrt(-2.0)      # => -1.2599210498948732
cbrt(1.0)       # => 1.0
cbrt(0.0)       # => 0.0
cbrt(1.0)       # => 1.0
cbrt(2.0)       # => 1.2599210498948732
cbrt(8.0)       # => 2.0
cbrt(27.0)      # => 3.0
cbrt(INFINITY)  # => Infinity
static VALUE
math_cbrt(VALUE unused_obj, VALUE x)
{
double f = Get_Double(x);
double r = cbrt(f);
#if defined __GLIBC__
if (isfinite(r) && !(f == 0.0 && r == 0.0)) {
r = (2.0 * r + (f / r / r)) / 3.0;
}
#endif
return DBL2NUM(r);
}
cos(x) → float click to toggle source

Returns the cosine of x in radians.

• Domain: (-INFINITY, INFINITY).

• Range: [-1.0, 1.0].

Examples:

cos(-PI)   # => -1.0
cos(-PI/2) # => 6.123031769111886e-17 # 0.0000000000000001
cos(0.0)   # => 1.0
cos(PI/2)  # => 6.123031769111886e-17 # 0.0000000000000001
cos(PI)    # => -1.0
static VALUE
math_cos(VALUE unused_obj, VALUE x)
{
return DBL2NUM(cos(Get_Double(x)));
}
cosh(x) → float click to toggle source

Returns the hyperbolic cosine of x in radians.

• Domain: [-INFINITY, INFINITY].

• Range: [1, INFINITY].

Examples:

cosh(-INFINITY) # => Infinity
cosh(0.0)       # => 1.0
cosh(INFINITY)  # => Infinity
static VALUE
math_cosh(VALUE unused_obj, VALUE x)
{
return DBL2NUM(cosh(Get_Double(x)));
}
erf(x) → float click to toggle source

Returns the value of the Gauss error function for x.

• Domain: [-INFINITY, INFINITY].

• Range: [-1, 1].

Examples:

erf(-INFINITY) # => -1.0
erf(0.0)       # => 0.0
erf(INFINITY)  # => 1.0

Related: Math.erfc.

static VALUE
math_erf(VALUE unused_obj, VALUE x)
{
return DBL2NUM(erf(Get_Double(x)));
}
erfc(x) → Float click to toggle source

Returns the value of the complementary error function for x.

• Domain: [-INFINITY, INFINITY].

• Range: [0, 2].

Examples:

erfc(-INFINITY) # => 2.0
erfc(0.0)       # => 1.0
erfc(INFINITY)  # => 0.0

Related: Math.erf.

static VALUE
math_erfc(VALUE unused_obj, VALUE x)
{
return DBL2NUM(erfc(Get_Double(x)));
}
exp(x) → float click to toggle source

Returns e raised to the x power.

• Domain: [-INFINITY, INFINITY].

• Range: [0, INFINITY].

Examples:

exp(-INFINITY) # => 0.0
exp(-1.0)      # => 0.36787944117144233 # 1.0/E
exp(0.0)       # => 1.0
exp(0.5)       # => 1.6487212707001282  # sqrt(E)
exp(1.0)       # => 2.718281828459045   # E
exp(2.0)       # => 7.38905609893065    # E**2
exp(INFINITY)  # => Infinity
static VALUE
math_exp(VALUE unused_obj, VALUE x)
{
return DBL2NUM(exp(Get_Double(x)));
}
frexp(x) → [fraction, exponent] click to toggle source

Returns a 2-element array containing the normalized signed float fraction and integer exponent of x such that:

x = fraction * 2**exponent
• Domain: [-INFINITY, INFINITY].

• Range [-INFINITY, INFINITY].

Examples:

frexp(-INFINITY) # => [-Infinity, -1]
frexp(-2.0)      # => [-0.5, 2]
frexp(-1.0)      # => [-0.5, 1]
frexp(0.0)       # => [0.0, 0]
frexp(1.0)       # => [0.5, 1]
frexp(2.0)       # => [0.5, 2]
frexp(INFINITY)  # => [Infinity, -1]

Related: Math.ldexp (inverse of Math.frexp).

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));
}
gamma(x) → float click to toggle source

Returns the value of the gamma function for x.

• Domain: (-INFINITY, INFINITY] excluding negative integers.

• Range: [-INFINITY, INFINITY].

Examples:

gamma(-2.5)      # => -0.9453087204829431
gamma(-1.5)      # => 2.3632718012073513
gamma(-0.5)      # => -3.5449077018110375
gamma(0.0)      # => Infinity
gamma(1.0)      # => 1.0
gamma(2.0)      # => 1.0
gamma(3.0)      # => 2.0
gamma(4.0)      # => 6.0
gamma(5.0)      # => 24.0

Related: Math.lgamma.

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)) {
domain_check_min(d, 0.0, "gamma");
if (1.0 <= d && d <= (double)NFACT_TABLE) {
return DBL2NUM(fact_table[(int)d - 1]);
}
}
return DBL2NUM(tgamma(d));
}
hypot(a, b) → float click to toggle source

Returns sqrt(a**2 + b**2), which is the length of the longest side c (the hypotenuse) of the right triangle whose other sides have lengths a and b.

• Domain of a: [-INFINITY, INFINITY].

• Domain of +ab: [-INFINITY, INFINITY].

• Range: [0, INFINITY].

Examples:

hypot(0.0, 1.0)       # => 1.0
hypot(1.0, 1.0)       # => 1.4142135623730951 # sqrt(2.0)
hypot(3.0, 4.0)       # => 5.0
hypot(5.0, 12.0)      # => 13.0
hypot(1.0, sqrt(3.0)) # => 1.9999999999999998 # Near 2.0

Note that if either argument is INFINITY or -INFINITY, the result is Infinity.

static VALUE
math_hypot(VALUE unused_obj, VALUE x, VALUE y)
{
return DBL2NUM(hypot(Get_Double(x), Get_Double(y)));
}
ldexp(fraction, exponent) → float click to toggle source

Returns the value of fraction * 2**exponent.

• Domain of fraction: [0.0, 1.0).

• Domain of exponent: [0, 1024] (larger values are equivalent to 1024).

Examples:

ldexp(-INFINITY, -1) # => -Infinity
ldexp(-0.5, 2)       # => -2.0
ldexp(-0.5, 1)       # => -1.0
ldexp(0.0, 0)        # => 0.0
ldexp(-0.5, 1)       # => 1.0
ldexp(-0.5, 2)       # => 2.0
ldexp(INFINITY, -1)  # => Infinity

Related: Math.frexp (inverse of Math.ldexp).

static VALUE
math_ldexp(VALUE unused_obj, VALUE x, VALUE n)
{
return DBL2NUM(ldexp(Get_Double(x), NUM2INT(n)));
}
lgamma(x) → [float, -1 or 1] click to toggle source

Returns a 2-element array equivalent to:

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

• Range of first element: (-INFINITY, INFINITY].

• Second element is -1 or 1.

Examples:

lgamma(-4.0) # => [Infinity, -1]
lgamma(-3.0) # => [Infinity, -1]
lgamma(-2.0) # => [Infinity, -1]
lgamma(-1.0) # => [Infinity, -1]
lgamma(0.0)  # => [Infinity, 1]

lgamma(1.0)  # => [0.0, 1]
lgamma(2.0)  # => [0.0, 1]
lgamma(3.0)  # => [0.6931471805599436, 1]
lgamma(4.0)  # => [1.7917594692280545, 1]

lgamma(-2.5) # => [-0.05624371649767279, -1]
lgamma(-1.5) # => [0.8600470153764797, 1]
lgamma(-0.5) # => [1.265512123484647, -1]
lgamma(0.5)  # => [0.5723649429247004, 1]
lgamma(1.5)  # => [-0.12078223763524676, 1]
lgamma(2.5)      # => [0.2846828704729205, 1]

Related: Math.gamma.

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));
}
log(x, base = Math::E) → Float click to toggle source

Returns the base base logarithm of x.

• Domain: [0, INFINITY].

• Range: [-INFINITY, INFINITY)].

Examples:

log(0.0)        # => -Infinity
log(1.0)        # => 0.0
log(E)          # => 1.0
log(INFINITY)   # => Infinity

log(0.0, 2.0)   # => -Infinity
log(1.0, 2.0)   # => 0.0
log(2.0, 2.0)   # => 1.0

log(0.0, 10.0)  # => -Infinity
log(1.0, 10.0)  # => 0.0
log(10.0, 10.0) # => 1.0
static VALUE
math_log(int argc, const VALUE *argv, VALUE unused_obj)
{
return rb_math_log(argc, argv);
}
log10(x) → float click to toggle source

Returns the base 10 logarithm of x.

• Domain: [0, INFINITY].

• Range: [-INFINITY, INFINITY].

Examples:

log10(0.0)      # => -Infinity
log10(1.0)      # => 0.0
log10(10.0)     # => 1.0
log10(INFINITY) # => Infinity
static VALUE
math_log10(VALUE unused_obj, VALUE x)
{
size_t numbits;
double d = get_double_rshift(x, &numbits);

domain_check_min(d, 0.0, "log10");
/* check for pole error */
if (d == 0.0) return DBL2NUM(-HUGE_VAL);

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

Returns the base 2 logarithm of x.

• Domain: [0, INFINITY].

• Range: [-INFINITY, INFINITY].

Examples:

log2(0.0)      # => -Infinity
log2(1.0)      # => 0.0
log2(2.0)      # => 1.0
log2(INFINITY) # => Infinity
static VALUE
math_log2(VALUE unused_obj, VALUE x)
{
size_t numbits;
double d = get_double_rshift(x, &numbits);

domain_check_min(d, 0.0, "log2");
/* check for pole error */
if (d == 0.0) return DBL2NUM(-HUGE_VAL);

return DBL2NUM(log2(d) + numbits); /* log2(d * 2 ** numbits) */
}
sin(x) → float click to toggle source

Returns the sine of x in radians.

• Domain: (-INFINITY, INFINITY).

• Range: [-1.0, 1.0].

Examples:

sin(-PI)   # => -1.2246063538223773e-16 # -0.0000000000000001
sin(-PI/2) # => -1.0
sin(0.0)   # => 0.0
sin(PI/2)  # => 1.0
sin(PI)    # => 1.2246063538223773e-16  # 0.0000000000000001
static VALUE
math_sin(VALUE unused_obj, VALUE x)
{
return DBL2NUM(sin(Get_Double(x)));
}
sinh(x) → float click to toggle source

Returns the hyperbolic sine of x in radians.

• Domain: [-INFINITY, INFINITY].

• Range: [-INFINITY, INFINITY].

Examples:

sinh(-INFINITY) # => -Infinity
sinh(0.0)       # => 0.0
sinh(INFINITY)  # => Infinity
static VALUE
math_sinh(VALUE unused_obj, VALUE x)
{
return DBL2NUM(sinh(Get_Double(x)));
}
sqrt(x) → float click to toggle source

Returns the principal (non-negative) square root of x.

• Domain: [0, INFINITY].

• Range: [0, INFINITY].

Examples:

sqrt(0.0)      # => 0.0
sqrt(0.5)      # => 0.7071067811865476
sqrt(1.0)      # => 1.0
sqrt(2.0)      # => 1.4142135623730951
sqrt(4.0)      # => 2.0
sqrt(9.0)      # => 3.0
sqrt(INFINITY) # => Infinity
static VALUE
math_sqrt(VALUE unused_obj, VALUE x)
{
return rb_math_sqrt(x);
}
tan(x) → float click to toggle source

Returns the tangent of x in radians.

• Domain: (-INFINITY, INFINITY).

• Range: (-INFINITY, INFINITY).

Examples:

tan(-PI)   # => 1.2246467991473532e-16  # -0.0000000000000001
tan(-PI/2) # => -1.633123935319537e+16  # -16331239353195370.0
tan(0.0)   # => 0.0
tan(PI/2)  # => 1.633123935319537e+16   # 16331239353195370.0
tan(PI)    # => -1.2246467991473532e-16 # -0.0000000000000001
static VALUE
math_tan(VALUE unused_obj, VALUE x)
{
return DBL2NUM(tan(Get_Double(x)));
}
tanh(x) → float click to toggle source

Returns the hyperbolic tangent of x in radians.

• Domain: [-INFINITY, INFINITY].

• Range: [-1, 1].

Examples:

tanh(-INFINITY) # => -1.0
tanh(0.0)       # => 0.0
tanh(INFINITY)  # => 1.0
static VALUE
math_tanh(VALUE unused_obj, VALUE x)
{
return DBL2NUM(tanh(Get_Double(x)));
}