1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
//! # traits::realfloat
//!
//! Types that implement this trait can be considered as real floating-point numbers.
use crate::number::{
instances::{int::Int, integer::Integer},
traits::{floating::Floating, realfrac::RealFrac, zero::Zero},
utils::{clamp, decimal_to_binary, from_integral, integral_power, non_negative_integral_power},
};
/// Concepts of RealFloat.
pub trait RealFloat: RealFrac + Floating {
/// The base of the numerical system.
///
/// The base, also known as "radix" in standards,
/// is typically `2`, which represents binary representation.
/// However, for decimal numbers, the base may be `10`.
const FLOAT_RADIX: Int = Int::of(2);
/// Number of digits in the radix used, including any implicit digit, but not counting the sign bit.
const FLOAT_DIGITS: Int;
/// In the standard representation of floating-point numbers,
/// the range of the exponent is defined as `[-m + 2, m + 1)`
/// if the maximum value of the floating-point exponent is `m`.
const FLOAT_RANGE: (Int, Int);
/// Decode a real floating-point number into its significand and exponent.
///
/// # Examples
///
/// For example, for the `Double`:
///
/// ```rust
/// use rmatrix_ks::number::{
/// instances::{double::Double, int::Int, integer::Integer},
/// traits::realfloat::RealFloat,
/// };
///
/// fn main() {
/// let d1 = Double::of(3.14);
/// assert_eq!(
/// d1.decode_float(),
/// (Integer::of_str("7070651414971679").unwrap(), Int::of(-51))
/// );
///
/// let d2 = Double::of(-13.14);
/// assert_eq!(
/// d2.decode_float(),
/// (Integer::of_str("-7397162387956040").unwrap(), Int::of(-49))
/// );
/// }
/// ```
fn decode_float(self) -> (Integer, Int) {
let sign = self >= Self::zero();
let rfp = self.absolute_value();
let (exponent_digits, float_digits) = decimal_to_binary(rfp).expect(&format!(
"Error[RealFloat::decode_float]: Failed to convert ({}) to binary format.",
self
));
let exponent = Int::of(exponent_digits.len() as i32) - Self::FLOAT_DIGITS;
let significand_digits = vec![exponent_digits, float_digits].concat();
let two = Integer::of(true, &[2])
.expect("Error[RealFloat::decode_float]: Failed to obtain integer two.");
let mut significand = Integer::zero();
for (index, &base) in significand_digits.iter().enumerate().rev() {
if base == 1u8 {
let exp = significand_digits.len() - index - 1;
significand = significand
+ non_negative_integral_power(two.clone(), Int::of(exp as i32)).expect(
&format!(
"Error[RealFloat::decode_float]: Failed to compute pow(2, {})",
exp
),
);
}
}
significand.sign = sign;
(significand, exponent)
}
/// Encode the given significand and exponent into a floating-point number.
fn encode_float(significand: Integer, exponent: Int) -> Self {
integral_power(from_integral(Self::FLOAT_RADIX), exponent)
.map(|p: Self| p * Self::from_integer(significand))
.expect(concat!(
"Error[RealFloat::encode_float]: ",
"Should be able to produce the correct result."
))
}
/// Return the actual exponent in the floating-point representation.
///
/// In Haskell, it defines like:
///
/// ```haskell
/// exponent 0 = 0
/// exponent x = snd (decodeFloat x) + floatDigits x
/// ```
fn exponent(self) -> Int {
if self.is_zero() {
Int::zero()
} else {
self.decode_float().1 + Self::FLOAT_DIGITS
}
}
/// Return the actual significand in the floating-point representation.
fn significand(self) -> Self {
Self::encode_float(self.decode_float().0, -Self::FLOAT_DIGITS)
}
/// Multiplies a real floating-point number by an integer power of the radix.
fn scale_float(self, factor: Int) -> Self {
if self.is_zero() || self.is_not_a_number() || self.is_infinite_number() {
self
} else {
let (significand, exponent) = self.decode_float();
let (lower_boundary, upper_boundary) = Self::FLOAT_RANGE;
let factor_p = upper_boundary - lower_boundary + Int::of(4) * Self::FLOAT_DIGITS;
Self::encode_float(significand, exponent + clamp(factor_p, factor))
}
}
/// Validate whether a given real floating-point number is NaN.
fn is_not_a_number(&self) -> bool;
/// Validate whether a given real floating-point number is Inf.
fn is_infinite_number(&self) -> bool;
/// Validate whether a given real floating-point number is in a denormalized form.
fn is_denormalized(&self) -> bool;
/// Validate whether a given real floating-point number is "negative zero".
fn is_negative_zero(&self) -> bool;
/// The two-parameter arctangent function, atan2(y, x).
fn arc_tangent_2(y: Self, x: Self) -> Self {
if x > Self::zero() {
(y / x).arc_tangent()
} else if x.is_zero() && y > Self::zero() {
Self::PI * Self::half()
} else if x < Self::zero() && y > Self::zero() {
Self::PI + (y / x).arc_tangent()
} else if (x <= Self::zero() && y < Self::zero())
|| (x < Self::zero() && y.is_negative_zero())
|| (x.is_negative_zero() && y.is_negative_zero())
{
-Self::arc_tangent_2(-y, x)
} else if y.is_zero() && (x < Self::zero() || x.is_negative_zero()) {
Self::PI
} else if x.is_zero() && y.is_zero() {
y
} else {
x + y
}
}
}