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[DOC] Enhanced RDoc for Math module (#5837)

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Revises intro.
    Adds "What's Here".
    Revises methods doc.
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Burdette Lamar 2022-04-25 10:07:21 -05:00 • committed by GitHub
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117
doc/math/math.rdoc Normal file
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@ -0,0 +1,117 @@
\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, <i>cos(π/2)</i> is exactly zero,
but in our computation <tt>cos(PI/2)</tt> 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 <tt>sqrt(a**2 + b**2)</tt> for the given +a+ and +b+.

560
math.c
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@ -40,31 +40,21 @@ VALUE rb_eMathDomainError;
/*
* call-seq:
* Math.atan2(y, x) -> Float
* Math.atan2(y, x) -> float
*
* 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.
* Returns the {arc tangent}[https://en.wikipedia.org/wiki/Atan2] of +y+ and +x+
* in {radians}[https://en.wikipedia.org/wiki/Trigonometric_functions#Radians_versus_degrees].
*
* Domain: (-INFINITY, INFINITY)
* - Domain of +y+: <tt>[-INFINITY, INFINITY]</tt>.
* - Domain of +x+: <tt>[-INFINITY, INFINITY]</tt>.
* - Range: <tt>[-PI, PI]</tt>.
*
* Codomain: [-PI, PI]
* Examples:
*
* 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
* 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
*
*/
@ -100,16 +90,22 @@ math_atan2(VALUE unused_obj, VALUE y, VALUE x)
/*
* call-seq:
* Math.cos(x) -> Float
* Math.cos(x) -> float
*
* Computes the cosine of +x+ (expressed in radians).
* Returns a Float in the range -1.0..1.0.
* Returns the
* {cosine}[https://en.wikipedia.org/wiki/Sine_and_cosine] of +x+
* in {radians}[https://en.wikipedia.org/wiki/Trigonometric_functions#Radians_versus_degrees].
*
* Domain: (-INFINITY, INFINITY)
* - Domain: <tt>(-INFINITY, INFINITY)</tt>.
* - Range: <tt>[-1.0, 1.0]</tt>.
*
* Codomain: [-1, 1]
* Examples:
*
* Math.cos(Math::PI) #=> -1.0
* 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
*
*/
@ -121,16 +117,22 @@ math_cos(VALUE unused_obj, VALUE x)
/*
* call-seq:
* Math.sin(x) -> Float
* Math.sin(x) -> float
*
* Computes the sine of +x+ (expressed in radians).
* Returns a Float in the range -1.0..1.0.
* Returns the
* {sine}[https://en.wikipedia.org/wiki/Sine_and_cosine] of +x+
* in {radians}[https://en.wikipedia.org/wiki/Trigonometric_functions#Radians_versus_degrees].
*
* Domain: (-INFINITY, INFINITY)
* - Domain: <tt>(-INFINITY, INFINITY)</tt>.
* - Range: <tt>[-1.0, 1.0]</tt>.
*
* Codomain: [-1, 1]
* Examples:
*
* Math.sin(Math::PI/2) #=> 1.0
* 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
*
*/
@ -143,15 +145,22 @@ math_sin(VALUE unused_obj, VALUE x)
/*
* call-seq:
* Math.tan(x) -> Float
* Math.tan(x) -> float
*
* Computes the tangent of +x+ (expressed in radians).
* Returns the
* {tangent}[https://en.wikipedia.org/wiki/Trigonometric_functions] of +x+
* in {radians}[https://en.wikipedia.org/wiki/Trigonometric_functions#Radians_versus_degrees].
*
* Domain: (-INFINITY, INFINITY)
* - Domain: <tt>(-INFINITY, INFINITY)</tt>.
* - Range: <tt>(-INFINITY, INFINITY)</tt>.
*
* Codomain: (-INFINITY, INFINITY)
* Examples:
*
* Math.tan(0) #=> 0.0
* 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
*
*/
@ -163,15 +172,18 @@ math_tan(VALUE unused_obj, VALUE x)
/*
* call-seq:
* Math.acos(x) -> Float
* Math.acos(x) -> float
*
* Computes the arc cosine of +x+. Returns 0..PI.
* Returns the {arc cosine}[https://en.wikipedia.org/wiki/Inverse_trigonometric_functions] of +x+.
*
* Domain: [-1, 1]
* - Domain: <tt>[-1, 1]</tt>.
* - Range: <tt>[0, PI]</tt>.
*
* Codomain: [0, PI]
* Examples:
*
* Math.acos(0) == Math::PI/2 #=> true
* acos(-1.0) # => 3.141592653589793 # PI
* acos(0.0) # => 1.5707963267948966 # PI/2
* acos(1.0) # => 0.0
*
*/
@ -187,15 +199,19 @@ math_acos(VALUE unused_obj, VALUE x)
/*
* call-seq:
* Math.asin(x) -> Float
* Math.asin(x) -> float
*
* Computes the arc sine of +x+. Returns -PI/2..PI/2.
* Returns the {arc sine}[https://en.wikipedia.org/wiki/Inverse_trigonometric_functions] of +x+.
*
* Domain: [-1, -1]
* - Domain: <tt>[-1, -1]</tt>.
* - Range: <tt>[-PI/2, PI/2]</tt>.
*
* Codomain: [-PI/2, PI/2]
* Examples:
*
* asin(-1.0) # => -1.5707963267948966 # -PI/2
* asin(0.0) # => 0.0
* asin(1.0) # => 1.5707963267948966 # PI/2
*
* Math.asin(1) == Math::PI/2 #=> true
*/
static VALUE
@ -212,13 +228,21 @@ math_asin(VALUE unused_obj, VALUE x)
* call-seq:
* Math.atan(x) -> Float
*
* Computes the arc tangent of +x+. Returns -PI/2..PI/2.
* Returns the {arc tangent}[https://en.wikipedia.org/wiki/Inverse_trigonometric_functions] of +x+.
*
* Domain: (-INFINITY, INFINITY)
* - Domain: <tt>[-INFINITY, INFINITY]</tt>.
* - Range: <tt>[-PI/2, PI/2] </tt>.
*
* Codomain: (-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
*
* Math.atan(0) #=> 0.0
*/
static VALUE
@ -237,15 +261,19 @@ cosh(double x)
/*
* call-seq:
* Math.cosh(x) -> Float
* Math.cosh(x) -> float
*
* Computes the hyperbolic cosine of +x+ (expressed in radians).
* Returns the {hyperbolic cosine}[https://en.wikipedia.org/wiki/Hyperbolic_functions] of +x+
* in {radians}[https://en.wikipedia.org/wiki/Trigonometric_functions#Radians_versus_degrees].
*
* Domain: (-INFINITY, INFINITY)
* - Domain: <tt>[-INFINITY, INFINITY]</tt>.
* - Range: <tt>[1, INFINITY]</tt>.
*
* Codomain: [1, INFINITY)
* Examples:
*
* Math.cosh(0) #=> 1.0
* cosh(-INFINITY) # => Infinity
* cosh(0.0) # => 1.0
* cosh(INFINITY) # => Infinity
*
*/
@ -265,15 +293,19 @@ sinh(double x)
/*
* call-seq:
* Math.sinh(x) -> Float
* Math.sinh(x) -> float
*
* Computes the hyperbolic sine of +x+ (expressed in radians).
* Returns the {hyperbolic sine}[https://en.wikipedia.org/wiki/Hyperbolic_functions] of +x+
* in {radians}[https://en.wikipedia.org/wiki/Trigonometric_functions#Radians_versus_degrees].
*
* Domain: (-INFINITY, INFINITY)
* - Domain: <tt>[-INFINITY, INFINITY]</tt>.
* - Range: <tt>[-INFINITY, INFINITY]</tt>.
*
* Codomain: (-INFINITY, INFINITY)
* Examples:
*
* Math.sinh(0) #=> 0.0
* sinh(-INFINITY) # => -Infinity
* sinh(0.0) # => 0.0
* sinh(INFINITY) # => Infinity
*
*/
@ -300,15 +332,19 @@ tanh(double x)
/*
* call-seq:
* Math.tanh(x) -> Float
* Math.tanh(x) -> float
*
* Computes the hyperbolic tangent of +x+ (expressed in radians).
* Returns the {hyperbolic tangent}[https://en.wikipedia.org/wiki/Hyperbolic_functions] of +x+
* in {radians}[https://en.wikipedia.org/wiki/Trigonometric_functions#Radians_versus_degrees].
*
* Domain: (-INFINITY, INFINITY)
* - Domain: <tt>[-INFINITY, INFINITY]</tt>.
* - Range: <tt>[-1, 1]</tt>.
*
* Codomain: (-1, 1)
* Examples:
*
* Math.tanh(0) #=> 0.0
* tanh(-INFINITY) # => -1.0
* tanh(0.0) # => 0.0
* tanh(INFINITY) # => 1.0
*
*/
@ -320,15 +356,17 @@ math_tanh(VALUE unused_obj, VALUE x)
/*
* call-seq:
* Math.acosh(x) -> Float
* Math.acosh(x) -> float
*
* Computes the inverse hyperbolic cosine of +x+.
* Returns the {inverse hyperbolic cosine}[https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions] of +x+.
*
* Domain: [1, INFINITY)
* - Domain: <tt>[1, INFINITY]</tt>.
* - Range: <tt>[0, INFINITY]</tt>.
*
* Codomain: [0, INFINITY)
* Examples:
*
* Math.acosh(1) #=> 0.0
* acosh(1.0) # => 0.0
* acosh(INFINITY) # => Infinity
*
*/
@ -344,15 +382,18 @@ math_acosh(VALUE unused_obj, VALUE x)
/*
* call-seq:
* Math.asinh(x) -> Float
* Math.asinh(x) -> float
*
* Computes the inverse hyperbolic sine of +x+.
* Returns the {inverse hyperbolic sine}[https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions] of +x+.
*
* Domain: (-INFINITY, INFINITY)
* - Domain: <tt>[-INFINITY, INFINITY]</tt>.
* - Range: <tt>[-INFINITY, INFINITY]</tt>.
*
* Codomain: (-INFINITY, INFINITY)
* Examples:
*
* Math.asinh(1) #=> 0.881373587019543
* asinh(-INFINITY) # => -Infinity
* asinh(0.0) # => 0.0
* asinh(INFINITY) # => Infinity
*
*/
@ -364,15 +405,18 @@ math_asinh(VALUE unused_obj, VALUE x)
/*
* call-seq:
* Math.atanh(x) -> Float
* Math.atanh(x) -> float
*
* Computes the inverse hyperbolic tangent of +x+.
* Returns the {inverse hyperbolic tangent}[https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions] of +x+.
*
* Domain: (-1, 1)
* - Domain: <tt>[-1, 1]</tt>.
* - Range: <tt>[-INFINITY, INFINITY]</tt>.
*
* Codomain: (-INFINITY, INFINITY)
* Examples:
*
* Math.atanh(1) #=> Infinity
* atanh(-1.0) # => -Infinity
* atanh(0.0) # => 0.0
* atanh(1.0) # => Infinity
*
*/
@ -391,17 +435,22 @@ math_atanh(VALUE unused_obj, VALUE x)
/*
* call-seq:
* Math.exp(x) -> Float
* Math.exp(x) -> float
*
* Returns e**x.
* Returns +e+ raised to the +x+ power.
*
* Domain: (-INFINITY, INFINITY)
* - Domain: <tt>[-INFINITY, INFINITY]</tt>.
* - Range: <tt>[0, INFINITY]</tt>.
*
* Codomain: (0, INFINITY)
* Examples:
*
* Math.exp(0) #=> 1.0
* Math.exp(1) #=> 2.718281828459045
* Math.exp(1.5) #=> 4.4816890703380645
* 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
*
*/
@ -432,22 +481,27 @@ FUNC_MINIMIZED(static VALUE math_log(int, const VALUE *, VALUE));
/*
* call-seq:
* Math.log(x) -> Float
* Math.log(x, base) -> Float
* Math.log(x, base = Math::E) -> 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).
* Returns the base +base+ {logarithm}[https://en.wikipedia.org/wiki/Logarithm] of +x+.
*
* Domain: (0, INFINITY)
* - Domain: <tt>[0, INFINITY]</tt>.
* - Range: <tt>[-INFINITY, INFINITY)]</tt>.
*
* Codomain: (-INFINITY, INFINITY)
* Examples:
*
* 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
* 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
*
*/
@ -515,18 +569,19 @@ extern double log2(double);
/*
* call-seq:
* Math.log2(x) -> Float
* Math.log2(x) -> float
*
* Returns the base 2 logarithm of +x+.
* Returns the base 2 {logarithm}[https://en.wikipedia.org/wiki/Logarithm] of +x+.
*
* Domain: (0, INFINITY)
* - Domain: <tt>[0, INFINITY]</tt>.
* - Range: <tt>[-INFINITY, INFINITY]</tt>.
*
* Codomain: (-INFINITY, INFINITY)
* Examples:
*
* Math.log2(1) #=> 0.0
* Math.log2(2) #=> 1.0
* Math.log2(32768) #=> 15.0
* Math.log2(65536) #=> 16.0
* log2(0.0) # => -Infinity
* log2(1.0) # => 0.0
* log2(2.0) # => 1.0
* log2(INFINITY) # => Infinity
*
*/
@ -545,17 +600,19 @@ math_log2(VALUE unused_obj, VALUE x)
/*
* call-seq:
* Math.log10(x) -> Float
* Math.log10(x) -> float
*
* Returns the base 10 logarithm of +x+.
* Returns the base 10 {logarithm}[https://en.wikipedia.org/wiki/Logarithm] of +x+.
*
* Domain: (0, INFINITY)
* - Domain: <tt>[0, INFINITY]</tt>.
* - Range: <tt>[-INFINITY, INFINITY]</tt>.
*
* Codomain: (-INFINITY, INFINITY)
* Examples:
*
* Math.log10(1) #=> 0.0
* Math.log10(10) #=> 1.0
* Math.log10(10**100) #=> 100.0
* log10(0.0) # => -Infinity
* log10(1.0) # => 0.0
* log10(10.0) # => 1.0
* log10(INFINITY) # => Infinity
*
*/
@ -576,35 +633,23 @@ static VALUE rb_math_sqrt(VALUE x);
/*
* call-seq:
* Math.sqrt(x) -> Float
* Math.sqrt(x) -> float
*
* Returns the non-negative square root of +x+.
* Returns the principal (non-negative) {square root}[https://en.wikipedia.org/wiki/Square_root] of +x+.
*
* Domain: [0, INFINITY)
* - Domain: <tt>[0, INFINITY]</tt>.
* - Range: <tt>[0, INFINITY]</tt>.
*
* Codomain:[0, INFINITY)
* Examples:
*
* 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]
* 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
*
* 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.
*/
static VALUE
@ -652,36 +697,26 @@ rb_math_sqrt(VALUE x)
/*
* call-seq:
* Math.cbrt(x) -> Float
* Math.cbrt(x) -> float
*
* Returns the cube root of +x+.
* Returns the {cube root}[https://en.wikipedia.org/wiki/Cube_root] of +x+.
*
* Domain: (-INFINITY, INFINITY)
* - Domain: <tt>[-INFINITY, INFINITY]</tt>.
* - Range: <tt>[-INFINITY, INFINITY]</tt>.
*
* Codomain: (-INFINITY, INFINITY)
* Examples:
*
* -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]
* 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
*
*/
@ -700,13 +735,30 @@ math_cbrt(VALUE unused_obj, VALUE x)
/*
* call-seq:
* Math.frexp(x) -> [fraction, exponent]
* Math.frexp(x) -> [fraction, exponent]
*
* Returns a two-element array containing the normalized fraction (a Float)
* and exponent (an Integer) of +x+.
* 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}[https://en.wikipedia.org/wiki/Double-precision_floating-point_format#IEEE_754_double-precision_binary_floating-point_format:_binary64].
*
* - Domain: <tt>[-INFINITY, INFINITY]</tt>.
* - Range <tt>[-INFINITY, INFINITY]</tt>.
*
* 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).
*
* fraction, exponent = Math.frexp(1234) #=> [0.6025390625, 11]
* fraction * 2**exponent #=> 1234.0
*/
static VALUE
@ -721,12 +773,28 @@ math_frexp(VALUE unused_obj, VALUE x)
/*
* call-seq:
* Math.ldexp(fraction, exponent) -> float
* Math.ldexp(fraction, exponent) -> float
*
* Returns the value of +fraction+*(2**+exponent+).
* Returns the value of <tt>fraction * 2**exponent</tt>.
*
* - Domain of +fraction+: <tt>[0.0, 1.0)</tt>.
* - Domain of +exponent+: <tt>[0, 1024]</tt>
* (larger values are equivalent to 1024).
*
* See {IEEE 754 double-precision binary floating-point format: binary64}[https://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).
*
* fraction, exponent = Math.frexp(1234)
* Math.ldexp(fraction, exponent) #=> 1234.0
*/
static VALUE
@ -737,12 +805,27 @@ math_ldexp(VALUE unused_obj, VALUE x, VALUE n)
/*
* call-seq:
* Math.hypot(x, y) -> Float
* Math.hypot(a, b) -> float
*
* Returns sqrt(x**2 + y**2), the hypotenuse of a right-angled triangle with
* sides +x+ and +y+.
* Returns <tt>sqrt(a**2 + b**2)</tt>,
* 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+: <tt>[-INFINITY, INFINITY]</tt>.
* - Domain of +ab: <tt>[-INFINITY, INFINITY]</tt>.
* - Range: <tt>[0, INFINITY]</tt>.
*
* 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 <tt>-INFINITY</tt>,
* the result is +Infinity+.
*
* Math.hypot(3, 4) #=> 5.0
*/
static VALUE
@ -753,15 +836,20 @@ math_hypot(VALUE unused_obj, VALUE x, VALUE y)
/*
* call-seq:
* Math.erf(x) -> Float
* Math.erf(x) -> float
*
* Calculates the error function of +x+.
* Returns the value of the {Gauss error function}[https://en.wikipedia.org/wiki/Error_function] for +x+.
*
* Domain: (-INFINITY, INFINITY)
* - Domain: <tt>[-INFINITY, INFINITY]</tt>.
* - Range: <tt>[-1, 1]</tt>.
*
* Codomain: (-1, 1)
* Examples:
*
* Math.erf(0) #=> 0.0
* erf(-INFINITY) # => -1.0
* erf(0.0) # => 0.0
* erf(INFINITY) # => 1.0
*
* Related: Math.erfc.
*
*/
@ -775,13 +863,18 @@ math_erf(VALUE unused_obj, VALUE x)
* call-seq:
* Math.erfc(x) -> Float
*
* Calculates the complementary error function of x.
* Returns the value of the {complementary error function}[https://en.wikipedia.org/wiki/Error_function#Complementary_error_function] for +x+.
*
* Domain: (-INFINITY, INFINITY)
* - Domain: <tt>[-INFINITY, INFINITY]</tt>.
* - Range: <tt>[0, 2]</tt>.
*
* Codomain: (0, 2)
* Examples:
*
* Math.erfc(0) #=> 1.0
* erfc(-INFINITY) # => 2.0
* erfc(0.0) # => 1.0
* erfc(INFINITY) # => 0.0
*
* Related: Math.erf.
*
*/
@ -793,41 +886,26 @@ math_erfc(VALUE unused_obj, VALUE x)
/*
* call-seq:
* Math.gamma(x) -> Float
* Math.gamma(x) -> float
*
* Calculates the gamma function of x.
* Returns the value of the {gamma function}[https://en.wikipedia.org/wiki/Gamma_function] for +x+.
*
* Note that gamma(n) is the same as fact(n-1) for integer n > 0.
* However gamma(n) returns float and can be an approximation.
* - Domain: <tt>(-INFINITY, INFINITY]</tt> excluding negative integers.
* - Range: <tt>[-INFINITY, INFINITY]</tt>.
*
* 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]
* 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.
*
*/
@ -884,15 +962,39 @@ math_gamma(VALUE unused_obj, VALUE x)
/*
* call-seq:
* Math.lgamma(x) -> [float, -1 or 1]
* Math.lgamma(x) -> [float, -1 or 1]
*
* Calculates the logarithmic gamma of +x+ and the sign of gamma of +x+.
* Returns a 2-element array equivalent to:
*
* Math.lgamma(x) is the same as
* [Math.log(Math.gamma(x).abs), Math.gamma(x) < 0 ? -1 : 1]
* but avoids overflow by Math.gamma(x) for large x.
* [Math.log(Math.gamma(x).abs), Math.gamma(x) < 0 ? -1 : 1]
*
* Math.lgamma(0) #=> [Infinity, 1]
* See {logarithmic gamma function}[https://en.wikipedia.org/wiki/Gamma_function#The_log-gamma_function].
*
* - Domain: <tt>(-INFINITY, INFINITY]</tt>.
* - Range of first element: <tt>(-INFINITY, INFINITY]</tt>.
* - 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.
*
*/
@ -962,12 +1064,8 @@ exp1(sqrt)
/*
* 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.
* :include: doc/math/math.rdoc
*
* Domains and codomains are given only for real (not complex) numbers.
*/