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<b>2 + b</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

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 167
def self.acos: (double x) -> Float

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

acosh

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 184
def self.acosh: (double x) -> Float

Returns the inverse hyperbolic cosine of x.

Domain: [1, INFINITY].
Range: [0, INFINITY].

Examples:

acosh(1.0)      # => 0.0
acosh(INFINITY) # => Infinity

asin

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 202
def self.asin: (double x) -> Float

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

asinh

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 220
def self.asinh: (double x) -> Float

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

atan

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 243
def self.atan: (double x) -> Float

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

atan2

(double y, double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 265
def self.atan2: (double y, double x) -> Float

Returns the arc tangent of y and x in [radians](en.wikipedia.org/wiki/Trigonometric_functions#Radians_versus _degrees).

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

atanh

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 283
def self.atanh: (double x) -> Float

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

cbrt

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 308
def self.cbrt: (double x) -> Float

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

cos

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 329
def self.cos: (double x) -> Float

Returns the cosine of x in [radians](en.wikipedia.org/wiki/Trigonometric_functions#Radians_versus _degrees).

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

cosh

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 349
def self.cosh: (double x) -> Float

Returns the hyperbolic cosine of x in [radians](en.wikipedia.org/wiki/Trigonometric_functions#Radians_versus _degrees).

Domain: [-INFINITY, INFINITY].
Range: [1, INFINITY].

Examples:

cosh(-INFINITY) # => Infinity
cosh(0.0)       # => 1.0
cosh(INFINITY)  # => Infinity

erf

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 369
def self.erf: (double x) -> Float

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.

erfc

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 390
def self.erfc: (double x) -> Float

Returns the value of the [complementary error function](en.wikipedia.org/wiki/Error_function#Complementary_error_fun ction) 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.

exp

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 411
def self.exp: (double x) -> Float

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

expm1

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 432
def self.expm1: (double x) -> Float

Returns “exp(x) - 1”, e raised to the x power, minus 1.

Domain: [-INFINITY, INFINITY].
Range: [-1.0, INFINITY].

Examples:

expm1(-INFINITY) # => 0.0
expm1(-1.0)      # => -0.6321205588285577 # 1.0/E - 1
expm1(0.0)       # => 0.0
expm1(0.5)       # => 0.6487212707001282  # sqrt(E) - 1
expm1(1.0)       # => 1.718281828459045   # E - 1
expm1(2.0)       # => 6.38905609893065    # E**2 - 1
expm1(INFINITY)  # => Infinity

frexp

(double x) → [ Float, Integer ]

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 462
def self.frexp: (double x) -> [Float, Integer]

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

x = fraction * 2**exponent

See [IEEE 754 double-precision binary floating-point format: binary64](en.wikipedia.org/wiki/Double-precision_floating-point_format

#IEEE_754_double-precision_binary_floating-point_format:_binary64).

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).

gamma

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 488
def self.gamma: (double x) -> Float

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.

hypot

(double x, double y) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 513
def self.hypot: (double x, double y) -> Float

Returns sqrt(a<b>2 + b</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.

ldexp

(double fraction, int exponent) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 541
def self.ldexp: (double fraction, int exponent) -> Float

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).

See [IEEE 754 double-precision binary floating-point format: binary64](en.wikipedia.org/wiki/Double-precision_floating-point_format

#IEEE_754_double-precision_binary_floating-point_format:_binary64).

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).

lgamma

(double x) → [ Float, -1 | 1 ]

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 580
def self.lgamma: (double x) -> [Float, -1 | 1]

Returns a 2-element array equivalent to:

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

See log gamma function.

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.

log

(double x, ?double base) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 607
def self.log: (double x, ?double base) -> Float

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

log10

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 626
def self.log10: (double x) -> Float

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

log1p

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 645
def self.log1p: (double x) -> Float

Returns “log(x + 1)”, the base E logarithm of (x + 1).

Domain: [-1, INFINITY].
Range: [-INFINITY, INFINITY].

Examples:

log1p(-1.0)     # => -Infinity
log1p(0.0)      # => 0.0
log1p(E - 1)    # => 1.0
log1p(INFINITY) # => Infinity

log2

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 664
def self.log2: (double x) -> Float

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

sin

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 685
def self.sin: (double x) -> Float

Returns the sine of x in [radians](en.wikipedia.org/wiki/Trigonometric_functions#Radians_versus _degrees).

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

sinh

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 705
def self.sinh: (double x) -> Float

Returns the hyperbolic sine of x in [radians](en.wikipedia.org/wiki/Trigonometric_functions#Radians_versus _degrees).

Domain: [-INFINITY, INFINITY].
Range: [-INFINITY, INFINITY].

Examples:

sinh(-INFINITY) # => -Infinity
sinh(0.0)       # => 0.0
sinh(INFINITY)  # => Infinity

sqrt

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 727
def self.sqrt: (double x) -> Float

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

tan

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 749
def self.tan: (double x) -> Float

Returns the tangent of x in [radians](en.wikipedia.org/wiki/Trigonometric_functions#Radians_versus _degrees).

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

tanh

(double x) → Float

Source

# File vendor/bundle/ruby/4.0.0/gems/rbs-4.0.3/core/math.rbs, line 769
def self.tanh: (double x) -> Float

Returns the hyperbolic tangent of x in [radians](en.wikipedia.org/wiki/Trigonometric_functions#Radians_versus _degrees).

Domain: [-INFINITY, INFINITY].
Range: [-1, 1].

Examples:

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