# Float objects represent inexact real numbers using the native architecture's
# double-precision floating point representation.
#
# Floating point has a different arithmetic and is an inexact number. So you
# should know its esoteric system. See following:
#
# *   http://docs.sun.com/source/806-3568/ncg_goldberg.html
# *   https://github.com/rdp/ruby_tutorials_core/wiki/Ruby-Talk-FAQ#floats_impre
#     cise
# *   http://en.wikipedia.org/wiki/Floating_point#Accuracy_problems
#
#
class Float < Numeric
  public

  # Returns the modulo after division of `float` by `other`.
  #
  #     6543.21.modulo(137)      #=> 104.21000000000004
  #     6543.21.modulo(137.24)   #=> 92.92999999999961
  #
  def %: (Integer) -> Float
       | (Float) -> Float
       | (Rational) -> Float
       | (Numeric) -> Numeric

  # Returns a new Float which is the product of `float` and `other`.
  #
  def *: (Complex) -> Complex
       | (Numeric) -> Float

  # Raises `float` to the power of `other`.
  #
  #     2.0**3   #=> 8.0
  #
  def **: (Complex) -> Complex
        | (Numeric) -> Float

  # Returns a new Float which is the sum of `float` and `other`.
  #
  def +: (Complex) -> Complex
       | (Numeric) -> Float

  def +@: () -> Float

  # Returns a new Float which is the difference of `float` and `other`.
  #
  def -: (Complex) -> Complex
       | (Numeric) -> Float

  # Returns `float`, negated.
  #
  def -@: () -> Float

  # Returns a new Float which is the result of dividing `float` by `other`.
  #
  def /: (Complex) -> Complex
       | (Numeric) -> Float

  # Returns `true` if `float` is less than `real`.
  #
  # The result of `NaN < NaN` is undefined, so an implementation-dependent value
  # is returned.
  #
  def <: (Numeric) -> bool

  # Returns `true` if `float` is less than or equal to `real`.
  #
  # The result of `NaN <= NaN` is undefined, so an implementation-dependent value
  # is returned.
  #
  def <=: (Numeric) -> bool

  # Returns -1, 0, or +1 depending on whether `float` is less than, equal to, or
  # greater than `real`. This is the basis for the tests in the Comparable module.
  #
  # The result of `NaN <=> NaN` is undefined, so an implementation-dependent value
  # is returned.
  #
  # `nil` is returned if the two values are incomparable.
  #
  def <=>: (Numeric) -> Integer?

  # Returns `true` only if `obj` has the same value as `float`. Contrast this with
  # Float#eql?, which requires `obj` to be a Float.
  #
  #     1.0 == 1   #=> true
  #
  # The result of `NaN == NaN` is undefined, so an implementation-dependent value
  # is returned.
  #
  def ==: (untyped) -> bool

  # Returns `true` only if `obj` has the same value as `float`. Contrast this with
  # Float#eql?, which requires `obj` to be a Float.
  #
  #     1.0 == 1   #=> true
  #
  # The result of `NaN == NaN` is undefined, so an implementation-dependent value
  # is returned.
  #
  def ===: (untyped) -> bool

  # Returns `true` if `float` is greater than `real`.
  #
  # The result of `NaN > NaN` is undefined, so an implementation-dependent value
  # is returned.
  #
  def >: (Numeric) -> bool

  # Returns `true` if `float` is greater than or equal to `real`.
  #
  # The result of `NaN >= NaN` is undefined, so an implementation-dependent value
  # is returned.
  #
  def >=: (Numeric) -> bool

  # Returns the absolute value of `float`.
  #
  #     (-34.56).abs   #=> 34.56
  #     -34.56.abs     #=> 34.56
  #     34.56.abs      #=> 34.56
  #
  # Float#magnitude is an alias for Float#abs.
  #
  def abs: () -> Float

  def abs2: () -> Float

  # Returns 0 if the value is positive, pi otherwise.
  #
  def angle: () -> (Integer | Float)

  # Returns 0 if the value is positive, pi otherwise.
  #
  alias arg angle

  # Returns the smallest number greater than or equal to `float` with a precision
  # of `ndigits` decimal digits (default: 0).
  #
  # When the precision is negative, the returned value is an integer with at least
  # `ndigits.abs` trailing zeros.
  #
  # Returns a floating point number when `ndigits` is positive, otherwise returns
  # an integer.
  #
  #     1.2.ceil      #=> 2
  #     2.0.ceil      #=> 2
  #     (-1.2).ceil   #=> -1
  #     (-2.0).ceil   #=> -2
  #
  #     1.234567.ceil(2)   #=> 1.24
  #     1.234567.ceil(3)   #=> 1.235
  #     1.234567.ceil(4)   #=> 1.2346
  #     1.234567.ceil(5)   #=> 1.23457
  #
  #     34567.89.ceil(-5)  #=> 100000
  #     34567.89.ceil(-4)  #=> 40000
  #     34567.89.ceil(-3)  #=> 35000
  #     34567.89.ceil(-2)  #=> 34600
  #     34567.89.ceil(-1)  #=> 34570
  #     34567.89.ceil(0)   #=> 34568
  #     34567.89.ceil(1)   #=> 34567.9
  #     34567.89.ceil(2)   #=> 34567.89
  #     34567.89.ceil(3)   #=> 34567.89
  #
  # Note that the limited precision of floating point arithmetic might lead to
  # surprising results:
  #
  #     (2.1 / 0.7).ceil  #=> 4 (!)
  #
  def ceil: () -> Integer
          | (int digits) -> (Integer | Float)

  def clone: (?freeze: bool) -> self

  # Returns an array with both `numeric` and `float` represented as Float objects.
  #
  # This is achieved by converting `numeric` to a Float.
  #
  #     1.2.coerce(3)       #=> [3.0, 1.2]
  #     2.5.coerce(1.1)     #=> [1.1, 2.5]
  #
  def coerce: (Numeric) -> [Float, Float]

  def conj: () -> Float

  def conjugate: () -> Float

  # Returns the denominator (always positive).  The result is machine dependent.
  #
  # See also Float#numerator.
  #
  def denominator: () -> Integer

  def div: (Numeric) -> Integer

  # See Numeric#divmod.
  #
  #     42.0.divmod(6)   #=> [7, 0.0]
  #     42.0.divmod(5)   #=> [8, 2.0]
  #
  def divmod: (Numeric) -> [Numeric, Numeric]

  def dup: () -> self

  # Returns `true` only if `obj` is a Float with the same value as `float`.
  # Contrast this with Float#==, which performs type conversions.
  #
  #     1.0.eql?(1)   #=> false
  #
  # The result of `NaN.eql?(NaN)` is undefined, so an implementation-dependent
  # value is returned.
  #
  def eql?: (untyped) -> bool

  # Returns `float / numeric`, same as Float#/.
  #
  def fdiv: (Complex) -> Complex
          | (Numeric) -> Float

  # Returns `true` if `float` is a valid IEEE floating point number, i.e. it is
  # not infinite and Float#nan? is `false`.
  #
  def finite?: () -> bool

  # Returns the largest number less than or equal to `float` with a precision of
  # `ndigits` decimal digits (default: 0).
  #
  # When the precision is negative, the returned value is an integer with at least
  # `ndigits.abs` trailing zeros.
  #
  # Returns a floating point number when `ndigits` is positive, otherwise returns
  # an integer.
  #
  #     1.2.floor      #=> 1
  #     2.0.floor      #=> 2
  #     (-1.2).floor   #=> -2
  #     (-2.0).floor   #=> -2
  #
  #     1.234567.floor(2)   #=> 1.23
  #     1.234567.floor(3)   #=> 1.234
  #     1.234567.floor(4)   #=> 1.2345
  #     1.234567.floor(5)   #=> 1.23456
  #
  #     34567.89.floor(-5)  #=> 0
  #     34567.89.floor(-4)  #=> 30000
  #     34567.89.floor(-3)  #=> 34000
  #     34567.89.floor(-2)  #=> 34500
  #     34567.89.floor(-1)  #=> 34560
  #     34567.89.floor(0)   #=> 34567
  #     34567.89.floor(1)   #=> 34567.8
  #     34567.89.floor(2)   #=> 34567.89
  #     34567.89.floor(3)   #=> 34567.89
  #
  # Note that the limited precision of floating point arithmetic might lead to
  # surprising results:
  #
  #     (0.3 / 0.1).floor  #=> 2 (!)
  #
  def floor: () -> Integer
           | (int digits) -> (Integer | Numeric)

  # Returns a hash code for this float.
  #
  # See also Object#hash.
  #
  def hash: () -> Integer

  def i: () -> Complex

  def imag: () -> Integer

  def imaginary: () -> Integer

  # Returns `nil`, -1, or 1 depending on whether the value is finite, `-Infinity`,
  # or `+Infinity`.
  #
  #     (0.0).infinite?        #=> nil
  #     (-1.0/0.0).infinite?   #=> -1
  #     (+1.0/0.0).infinite?   #=> 1
  #
  def infinite?: () -> Integer?

  alias inspect to_s

  def integer?: () -> bool

  # Returns the absolute value of `float`.
  #
  #     (-34.56).abs   #=> 34.56
  #     -34.56.abs     #=> 34.56
  #     34.56.abs      #=> 34.56
  #
  # Float#magnitude is an alias for Float#abs.
  #
  alias magnitude abs

  # Returns the modulo after division of `float` by `other`.
  #
  #     6543.21.modulo(137)      #=> 104.21000000000004
  #     6543.21.modulo(137.24)   #=> 92.92999999999961
  #
  def modulo: (Numeric) -> Float

  # Returns `true` if `float` is an invalid IEEE floating point number.
  #
  #     a = -1.0      #=> -1.0
  #     a.nan?        #=> false
  #     a = 0.0/0.0   #=> NaN
  #     a.nan?        #=> true
  #
  def nan?: () -> bool

  # Returns `true` if `float` is less than 0.
  #
  def negative?: () -> bool

  # Returns the next representable floating point number.
  #
  # Float::MAX.next_float and Float::INFINITY.next_float is Float::INFINITY.
  #
  # Float::NAN.next_float is Float::NAN.
  #
  # For example:
  #
  #     0.01.next_float    #=> 0.010000000000000002
  #     1.0.next_float     #=> 1.0000000000000002
  #     100.0.next_float   #=> 100.00000000000001
  #
  #     0.01.next_float - 0.01     #=> 1.734723475976807e-18
  #     1.0.next_float - 1.0       #=> 2.220446049250313e-16
  #     100.0.next_float - 100.0   #=> 1.4210854715202004e-14
  #
  #     f = 0.01; 20.times { printf "%-20a %s\n", f, f.to_s; f = f.next_float }
  #     #=> 0x1.47ae147ae147bp-7 0.01
  #     #   0x1.47ae147ae147cp-7 0.010000000000000002
  #     #   0x1.47ae147ae147dp-7 0.010000000000000004
  #     #   0x1.47ae147ae147ep-7 0.010000000000000005
  #     #   0x1.47ae147ae147fp-7 0.010000000000000007
  #     #   0x1.47ae147ae148p-7  0.010000000000000009
  #     #   0x1.47ae147ae1481p-7 0.01000000000000001
  #     #   0x1.47ae147ae1482p-7 0.010000000000000012
  #     #   0x1.47ae147ae1483p-7 0.010000000000000014
  #     #   0x1.47ae147ae1484p-7 0.010000000000000016
  #     #   0x1.47ae147ae1485p-7 0.010000000000000018
  #     #   0x1.47ae147ae1486p-7 0.01000000000000002
  #     #   0x1.47ae147ae1487p-7 0.010000000000000021
  #     #   0x1.47ae147ae1488p-7 0.010000000000000023
  #     #   0x1.47ae147ae1489p-7 0.010000000000000024
  #     #   0x1.47ae147ae148ap-7 0.010000000000000026
  #     #   0x1.47ae147ae148bp-7 0.010000000000000028
  #     #   0x1.47ae147ae148cp-7 0.01000000000000003
  #     #   0x1.47ae147ae148dp-7 0.010000000000000031
  #     #   0x1.47ae147ae148ep-7 0.010000000000000033
  #
  #     f = 0.0
  #     100.times { f += 0.1 }
  #     f                           #=> 9.99999999999998       # should be 10.0 in the ideal world.
  #     10-f                        #=> 1.9539925233402755e-14 # the floating point error.
  #     10.0.next_float-10          #=> 1.7763568394002505e-15 # 1 ulp (unit in the last place).
  #     (10-f)/(10.0.next_float-10) #=> 11.0                   # the error is 11 ulp.
  #     (10-f)/(10*Float::EPSILON)  #=> 8.8                    # approximation of the above.
  #     "%a" % 10                   #=> "0x1.4p+3"
  #     "%a" % f                    #=> "0x1.3fffffffffff5p+3" # the last hex digit is 5.  16 - 5 = 11 ulp.
  #
  def next_float: () -> Float

  def nonzero?: () -> self?

  # Returns the numerator.  The result is machine dependent.
  #
  #     n = 0.3.numerator    #=> 5404319552844595
  #     d = 0.3.denominator  #=> 18014398509481984
  #     n.fdiv(d)            #=> 0.3
  #
  # See also Float#denominator.
  #
  def numerator: () -> Integer

  # Returns 0 if the value is positive, pi otherwise.
  #
  alias phase angle

  def polar: () -> [ Float, Integer | Float ]

  # Returns `true` if `float` is greater than 0.
  #
  def positive?: () -> bool

  # Returns the previous representable floating point number.
  #
  # (-Float::MAX).prev_float and (-Float::INFINITY).prev_float is
  # -Float::INFINITY.
  #
  # Float::NAN.prev_float is Float::NAN.
  #
  # For example:
  #
  #     0.01.prev_float    #=> 0.009999999999999998
  #     1.0.prev_float     #=> 0.9999999999999999
  #     100.0.prev_float   #=> 99.99999999999999
  #
  #     0.01 - 0.01.prev_float     #=> 1.734723475976807e-18
  #     1.0 - 1.0.prev_float       #=> 1.1102230246251565e-16
  #     100.0 - 100.0.prev_float   #=> 1.4210854715202004e-14
  #
  #     f = 0.01; 20.times { printf "%-20a %s\n", f, f.to_s; f = f.prev_float }
  #     #=> 0x1.47ae147ae147bp-7 0.01
  #     #   0x1.47ae147ae147ap-7 0.009999999999999998
  #     #   0x1.47ae147ae1479p-7 0.009999999999999997
  #     #   0x1.47ae147ae1478p-7 0.009999999999999995
  #     #   0x1.47ae147ae1477p-7 0.009999999999999993
  #     #   0x1.47ae147ae1476p-7 0.009999999999999992
  #     #   0x1.47ae147ae1475p-7 0.00999999999999999
  #     #   0x1.47ae147ae1474p-7 0.009999999999999988
  #     #   0x1.47ae147ae1473p-7 0.009999999999999986
  #     #   0x1.47ae147ae1472p-7 0.009999999999999985
  #     #   0x1.47ae147ae1471p-7 0.009999999999999983
  #     #   0x1.47ae147ae147p-7  0.009999999999999981
  #     #   0x1.47ae147ae146fp-7 0.00999999999999998
  #     #   0x1.47ae147ae146ep-7 0.009999999999999978
  #     #   0x1.47ae147ae146dp-7 0.009999999999999976
  #     #   0x1.47ae147ae146cp-7 0.009999999999999974
  #     #   0x1.47ae147ae146bp-7 0.009999999999999972
  #     #   0x1.47ae147ae146ap-7 0.00999999999999997
  #     #   0x1.47ae147ae1469p-7 0.009999999999999969
  #     #   0x1.47ae147ae1468p-7 0.009999999999999967
  #
  def prev_float: () -> Float

  # Returns `float / numeric`, same as Float#/.
  #
  def quo: (Complex) -> Complex
         | (Numeric) -> Float

  # Returns a simpler approximation of the value (flt-|eps| <= result <=
  # flt+|eps|).  If the optional argument `eps` is not given, it will be chosen
  # automatically.
  #
  #     0.3.rationalize          #=> (3/10)
  #     1.333.rationalize        #=> (1333/1000)
  #     1.333.rationalize(0.01)  #=> (4/3)
  #
  # See also Float#to_r.
  #
  def rationalize: (?Numeric eps) -> Rational

  def real: () -> Float

  def real?: () -> true

  def rect: () -> [ Float, Numeric ]

  alias rectangular rect

  def remainder: (Numeric) -> Float

  # Returns `float` rounded to the nearest value with a precision of `ndigits`
  # decimal digits (default: 0).
  #
  # When the precision is negative, the returned value is an integer with at least
  # `ndigits.abs` trailing zeros.
  #
  # Returns a floating point number when `ndigits` is positive, otherwise returns
  # an integer.
  #
  #     1.4.round      #=> 1
  #     1.5.round      #=> 2
  #     1.6.round      #=> 2
  #     (-1.5).round   #=> -2
  #
  #     1.234567.round(2)   #=> 1.23
  #     1.234567.round(3)   #=> 1.235
  #     1.234567.round(4)   #=> 1.2346
  #     1.234567.round(5)   #=> 1.23457
  #
  #     34567.89.round(-5)  #=> 0
  #     34567.89.round(-4)  #=> 30000
  #     34567.89.round(-3)  #=> 35000
  #     34567.89.round(-2)  #=> 34600
  #     34567.89.round(-1)  #=> 34570
  #     34567.89.round(0)   #=> 34568
  #     34567.89.round(1)   #=> 34567.9
  #     34567.89.round(2)   #=> 34567.89
  #     34567.89.round(3)   #=> 34567.89
  #
  # If the optional `half` keyword argument is given, numbers that are half-way
  # between two possible rounded values will be rounded according to the specified
  # tie-breaking `mode`:
  #
  # *   `:up` or `nil`: round half away from zero (default)
  # *   `:down`: round half toward zero
  # *   `:even`: round half toward the nearest even number
  #
  #         2.5.round(half: :up)      #=> 3
  #         2.5.round(half: :down)    #=> 2
  #         2.5.round(half: :even)    #=> 2
  #         3.5.round(half: :up)      #=> 4
  #         3.5.round(half: :down)    #=> 3
  #         3.5.round(half: :even)    #=> 4
  #         (-2.5).round(half: :up)   #=> -3
  #         (-2.5).round(half: :down) #=> -2
  #         (-2.5).round(half: :even) #=> -2
  #
  def round: (?half: :up | :down | :even) -> Integer
           | (int digits, ?half: :up | :down | :even) -> (Integer | Float)

  def step: (?Numeric limit, ?Numeric step) { (Float) -> void } -> self
          | (?Numeric limit, ?Numeric step) -> Enumerator[Float, self]
          | (?by: Numeric, ?to: Numeric) { (Float) -> void } -> self
          | (?by: Numeric, ?to: Numeric) -> Enumerator[Float, self]

  def to_c: () -> Complex

  # Since `float` is already a Float, returns `self`.
  #
  def to_f: () -> Float

  # Returns the `float` truncated to an Integer.
  #
  #     1.2.to_i      #=> 1
  #     (-1.2).to_i   #=> -1
  #
  # Note that the limited precision of floating point arithmetic might lead to
  # surprising results:
  #
  #     (0.3 / 0.1).to_i  #=> 2 (!)
  #
  # #to_int is an alias for #to_i.
  #
  def to_i: () -> Integer

  # Returns the `float` truncated to an Integer.
  #
  #     1.2.to_i      #=> 1
  #     (-1.2).to_i   #=> -1
  #
  # Note that the limited precision of floating point arithmetic might lead to
  # surprising results:
  #
  #     (0.3 / 0.1).to_i  #=> 2 (!)
  #
  # #to_int is an alias for #to_i.
  #
  alias to_int to_i

  # Returns the value as a rational.
  #
  #     2.0.to_r    #=> (2/1)
  #     2.5.to_r    #=> (5/2)
  #     -0.75.to_r  #=> (-3/4)
  #     0.0.to_r    #=> (0/1)
  #     0.3.to_r    #=> (5404319552844595/18014398509481984)
  #
  # NOTE: 0.3.to_r isn't the same as "0.3".to_r.  The latter is equivalent to
  # "3/10".to_r, but the former isn't so.
  #
  #     0.3.to_r   == 3/10r  #=> false
  #     "0.3".to_r == 3/10r  #=> true
  #
  # See also Float#rationalize.
  #
  def to_r: () -> Rational

  # Returns a string containing a representation of `self`. As well as a fixed or
  # exponential form of the `float`, the call may return `NaN`, `Infinity`, and
  # `-Infinity`.
  #
  def to_s: () -> String

  # Returns `float` truncated (toward zero) to a precision of `ndigits` decimal
  # digits (default: 0).
  #
  # When the precision is negative, the returned value is an integer with at least
  # `ndigits.abs` trailing zeros.
  #
  # Returns a floating point number when `ndigits` is positive, otherwise returns
  # an integer.
  #
  #     2.8.truncate           #=> 2
  #     (-2.8).truncate        #=> -2
  #     1.234567.truncate(2)   #=> 1.23
  #     34567.89.truncate(-2)  #=> 34500
  #
  # Note that the limited precision of floating point arithmetic might lead to
  # surprising results:
  #
  #     (0.3 / 0.1).truncate  #=> 2 (!)
  #
  def truncate: () -> Integer
              | (Integer ndigits) -> (Integer | Float)

  # Returns `true` if `float` is 0.0.
  #
  def zero?: () -> bool
end

# The minimum number of significant decimal digits in a double-precision
# floating point.
#
# Usually defaults to 15.
#
Float::DIG: Integer

# The difference between 1 and the smallest double-precision floating point
# number greater than 1.
#
# Usually defaults to 2.2204460492503131e-16.
#
Float::EPSILON: Float

# An expression representing positive infinity.
#
Float::INFINITY: Float

# The number of base digits for the `double` data type.
#
# Usually defaults to 53.
#
Float::MANT_DIG: Integer

# The largest possible integer in a double-precision floating point number.
#
# Usually defaults to 1.7976931348623157e+308.
#
Float::MAX: Float

# The largest positive exponent in a double-precision floating point where 10
# raised to this power minus 1.
#
# Usually defaults to 308.
#
Float::MAX_10_EXP: Integer

# The largest possible exponent value in a double-precision floating point.
#
# Usually defaults to 1024.
#
Float::MAX_EXP: Integer

# The smallest positive normalized number in a double-precision floating point.
#
# Usually defaults to 2.2250738585072014e-308.
#
# If the platform supports denormalized numbers, there are numbers between zero
# and Float::MIN. 0.0.next_float returns the smallest positive floating point
# number including denormalized numbers.
#
Float::MIN: Float

# The smallest negative exponent in a double-precision floating point where 10
# raised to this power minus 1.
#
# Usually defaults to -307.
#
Float::MIN_10_EXP: Integer

# The smallest possible exponent value in a double-precision floating point.
#
# Usually defaults to -1021.
#
Float::MIN_EXP: Integer

# An expression representing a value which is "not a number".
#
Float::NAN: Float

# The base of the floating point, or number of unique digits used to represent
# the number.
#
# Usually defaults to 2 on most systems, which would represent a base-10
# decimal.
#
Float::RADIX: Integer

# Deprecated, do not use.
#
# Represents the rounding mode for floating point addition at the start time.
#
# Usually defaults to 1, rounding to the nearest number.
#
# Other modes include:
#
# -1
# :   Indeterminable
# 0
# :   Rounding towards zero
# 1
# :   Rounding to the nearest number
# 2
# :   Rounding towards positive infinity
# 3
# :   Rounding towards negative infinity
#
#
Float::ROUNDS: Integer