abax 0.1.45

A lightweight Rust library providing high-precision mathematical constants and special functions, including Bernoulli numbers, Riemann Zeta values, robust incomplete gamma functions, and probability distribution functions like normal and lognormal CDF.
Documentation 
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

/// Weibull cumulative distribution function (CDF) for scalar values.
///
/// Evaluates the CDF of the Weibull distribution with scale parameter `a` 
/// and shape parameter `b` at the value `x`.
///
/// # Arguments
/// * `x` - The value at which to evaluate the CDF.
/// * `a` - Scale parameter (a > 0).
/// * `b` - Shape parameter (b > 0).
/// * `upper` - If true, computes the upper tail probability (survival function).
///             If false, computes the standard lower tail probability.
///
/// # Returns
/// The cumulative probability at `x`. Returns `f64::NAN` if parameters are invalid.
///
/// # Mathematical Formula
/// For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi><mo>&#x2265;</mo><mn>0</mn></math>, 
/// <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>&gt;</mo><mn>0</mn></math>, 
/// <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi><mo>&gt;</mo><mn>0</mn></math>:
///
/// Standard lower tail (`upper = false`):
/// <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
///   <mi>F</mi>
///   <mo stretchy="false">(</mo>
///   <mi>x</mi>
///   <mo>;</mo>
///   <mi>a</mi>
///   <mo>,</mo>
///   <mi>b</mi>
///   <mo stretchy="false">)</mo>
///   <mo>=</mo>
///   <mn>1</mn>
///   <mo>&#x2212;</mo>
///   <msup>
///     <mi>e</mi>
///     <mrow>
///       <mo>&#x2212;</mo>
///       <msup>
///         <mrow>
///           <mo>(</mo>
///           <mfrac>
///             <mi>x</mi>
///             <mi>a</mi>
///           </mfrac>
///           <mo>)</mo>
///         </mrow>
///         <mi>b</mi>
///       </msup>
///     </mrow>
///   </msup>
/// </math>
///
/// Upper tail (`upper = true`):
/// <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
///   <mi>S</mi>
///   <mo stretchy="false">(</mo>
///   <mi>x</mi>
///   <mo>;</mo>
///   <mi>a</mi>
///   <mo>,</mo>
///   <mi>b</mi>
///   <mo stretchy="false">)</mo>
///   <mo>=</mo>
///   <msup>
///     <mi>e</mi>
///     <mrow>
///       <mo>&#x2212;</mo>
///       <msup>
///         <mrow>
///           <mo>(</mo>
///           <mfrac>
///             <mi>x</mi>
///             <mi>a</mi>
///           </mfrac>
///           <mo>)</mo>
///         </mrow>
///         <mi>b</mi>
///       </msup>
///     </mrow>
///   </msup>
/// </math>
pub fn wblcdf(x: f64, a: f64, b: f64, upper: bool) -> f64 {
    // 1. Return NaN for out of range parameters or incoming NaNs
    if x.is_nan() || a.is_nan() || b.is_nan() || a <= 0.0 || b <= 0.0 {
        return f64::NAN;
    }

    // 2. Handle negative x values safely.
    // The probability mass below 0 is exactly 0.0. 
    // Therefore, lower tail is 0.0 and upper tail is 1.0.
    if x < 0.0 {
        return if upper { 1.0 } else { 0.0 };
    }

    let z = (x / a).powf(b);

    if upper {
        // Upper tail probability: exp(-z)
        (-z).exp()
    } else {
        // Standard lower tail probability: 1 - exp(-z)
        // expm1(-z) computes exp(-z) - 1 accurately for small values of z.
        // Therefore, -(exp(-z) - 1) equals 1 - exp(-z).
        -(-z).exp_m1()
    }
}

#[cfg(test)]
mod tests {
    use super::*;

    #[test]
    fn test_standard_weibull_cdf() {
        // For a=1, b=1 (Exponential distribution), F(1) = 1 - exp(-1)
        let res_lower = wblcdf(1.0, 1.0, 1.0, false);
        let expected_lower = 1.0 - (-1.0f64).exp();
        assert!((res_lower - expected_lower).abs() < 1e-15);

        // Survival curve version: S(1) = exp(-1)
        let res_upper = wblcdf(1.0, 1.0, 1.0, true);
        let expected_upper = (-1.0f64).exp();
        assert!((res_upper - expected_upper).abs() < 1e-15);
        
        // Sum of complements must be exactly 1.0
        assert_eq!(res_lower + res_upper, 1.0);
    }

    #[test]
    fn test_negative_x() {
        // Left boundary edge case checks
        assert_eq!(wblcdf(-2.5, 1.0, 2.0, false), 0.0);
        assert_eq!(wblcdf(-2.5, 1.0, 2.0, true), 1.0);
        assert_eq!(wblcdf(-0.0, 1.0, 1.0, false), 0.0);
    }

    #[test]
    fn test_zero_x() {
        // At exactly x = 0, lower tail should be 0, upper tail should be 1
        assert_eq!(wblcdf(0.0, 5.0, 1.5, false), 0.0);
        assert_eq!(wblcdf(0.0, 5.0, 1.5, true), 1.0);
    }

    #[test]
    fn test_invalid_parameters() {
        assert!(wblcdf(1.0, 0.0, 1.0, false).is_nan());
        assert!(wblcdf(1.0, 1.0, -1.0, false).is_nan());
        assert!(wblcdf(f64::NAN, 1.0, 1.0, false).is_nan());
    }

    #[test]
    fn test_extreme_right_tail() {
        // Far right tail limit evaluation
        assert_eq!(wblcdf(100.0, 1.0, 2.0, false), 1.0);
        assert_eq!(wblcdf(100.0, 1.0, 2.0, true), 0.0);
    }
}