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
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206

use crate::{BigInt, Ratio, pi_iter};

/// It returns `sin(x)`. It gets more accurate as `iter` gets bigger.
pub fn sin_iter(x: &Ratio, iter: usize) -> Ratio {

    // -inf ~ 0 -> sin(x) = -sin(-x)
    // 0 ~ pi/4 -> good to go
    // pi/4 ~ pi/2 -> sin(x) = cos(pi/2 - x)
    // pi/2 ~ pi -> sin(x) = sin(pi - x)
    // pi ~ 2pi -> sin(x) = -sin(x - pi)
    // 2pi ~ inf -> sin(x) = sin(x - 2pi)
    let pi = pi_iter(iter);
    let mut pi_div = x.div_rat(&pi);
    let mut negate = false;

    if pi_div.is_neg() {
        negate = !negate;
        pi_div.neg_mut();
    }

    if pi_div.gt_i32(2) {
        pi_div.sub_bi_mut(&pi_div.truncate_bi().div_i32(2).mul_i32(2));
    }

    if pi_div.gt_one() {
        pi_div.sub_i32_mut(1);
        negate = !negate;
    }

    if pi_div.mul_i32(2).gt_one() {
        pi_div = Ratio::one().sub_rat(&pi_div);
    }

    if pi_div.mul_i32(4).lt_one() {
        let mut result = sin_iter_worker(&pi_div.mul_rat(&pi), iter);
        if negate { result.neg_mut(); }
        result
    }

    else {
        let mut result = cos_iter_worker(&Ratio::from_denom_and_numer_i32(2, 1).sub_rat(&pi_div).mul_rat(&pi), iter);
        if negate { result.neg_mut(); }
        result
    }

}

// x - x^3/3! + x^5/5! - x^7/7! + ...
fn sin_iter_worker(x: &Ratio, iter: usize) -> Ratio {
    let mut result = x.clone();
    let mut numer = x.pow_i32(3);
    let x_sq = x.mul_rat(x);
    let mut denom = BigInt::from_i32(6);

    for i in 0..iter {
        result.sub_rat_mut(
            &numer.div_bi(&denom)
        );

        denom.mul_i32_mut((i * 4 + 4) as i32);
        denom.mul_i32_mut((i * 4 + 5) as i32);
        numer.mul_rat_mut(&x_sq);

        result.add_rat_mut(
            &numer.div_bi(&denom)
        );

        denom.mul_i32_mut((i * 4 + 6) as i32);
        denom.mul_i32_mut((i * 4 + 7) as i32);
        numer.mul_rat_mut(&x_sq);
    }

    result
}

/// It returns `cos(x)`. It gets more accurate as `iter` gets bigger.
pub fn cos_iter(x: &Ratio, iter: usize) -> Ratio {
    // -inf ~ 0 -> cos(x) = cos(-x)
    // 0 ~ pi/4 -> good to go
    // pi/4 ~ pi/2 -> cos(x) = sin(pi/2 - x)
    // pi/2 ~ pi -> cos(x) = -cos(pi - x)
    // pi ~ 2pi -> cos(x) = cos(2pi - x)
    // 2pi ~ inf -> cos(x) = cos(x - 2pi)
    let pi = pi_iter(iter);
    let mut pi_div = x.div_rat(&pi);
    let mut negate = false;

    if pi_div.is_neg() {
        pi_div.neg_mut();
    }

    if pi_div.gt_i32(2) {
        pi_div.sub_bi_mut(&pi_div.truncate_bi().div_i32(2).mul_i32(2));
    }

    if pi_div.gt_one() {
        pi_div = Ratio::from_i32(2).sub_rat(&pi_div);
    }

    if pi_div.mul_i32(2).gt_one() {
        pi_div = Ratio::one().sub_rat(&pi_div);
        negate = !negate;
    }

    if pi_div.mul_i32(4).lt_one() {
        let mut result = cos_iter_worker(&pi_div.mul_rat(&pi), iter);
        if negate { result.neg_mut(); }
        result
    }

    else {
        let mut result = sin_iter_worker(&Ratio::from_denom_and_numer_i32(2, 1).sub_rat(&pi_div).mul_rat(&pi), iter);
        if negate { result.neg_mut(); }
        result
    }
}

// 1 - x^2/2! + x^4/4! - x^6/6! + ...
fn cos_iter_worker(x: &Ratio, iter: usize) -> Ratio {
    let mut result = Ratio::one();
    let x_sq = x.mul_rat(x);
    let mut numer = x_sq.clone();
    let mut denom = BigInt::from_i32(2);

    for i in 0..iter {
        result.sub_rat_mut(
            &numer.div_bi(&denom)
        );

        denom.mul_i32_mut((i * 4 + 3) as i32);
        denom.mul_i32_mut((i * 4 + 4) as i32);
        numer.mul_rat_mut(&x_sq);

        result.add_rat_mut(
            &numer.div_bi(&denom)
        );

        denom.mul_i32_mut((i * 4 + 5) as i32);
        denom.mul_i32_mut((i * 4 + 6) as i32);
        numer.mul_rat_mut(&x_sq);
    }

    result
}

/// It returns `tan(x)`. It gets more accurate as `iter` gets bigger.
pub fn tan_iter(x: &Ratio, iter: usize) -> Ratio {
    sin_iter(x, iter).div_rat(&cos_iter(x, iter))
}

#[cfg(test)]
mod tests {
    use crate::{Ratio, sqrt_iter, sin_iter, cos_iter, pi_iter};
    use crate::utils::are_close;
    use crate::consts::RUN_ALL_TESTS;

    #[test]
    fn sin_test() {
        if !RUN_ALL_TESTS { return; }
        let iter = 9;
        let samples = vec![
            // (a, b, c) -> sin(a * pi / b) = c
            (-2, 12, Ratio::from_denom_and_numer_i32(2, 1).neg()),       // sin(-pi/6) = -0.5
            (-1, 12, sqrt_iter(&Ratio::from_i32(6), iter).sub_rat(&sqrt_iter(&Ratio::from_i32(2), iter)).div_i32(4).neg()),  // sin(-pi/12) = -(sqrt(6) - sqrt(2))/4
            ( 0, 12, Ratio::zero()),                                     // sin(0) = 0
            ( 1, 12, sqrt_iter(&Ratio::from_i32(6), iter).sub_rat(&sqrt_iter(&Ratio::from_i32(2), iter)).div_i32(4)),  // sin(pi/12) = (sqrt(6) - sqrt(2))/4
            ( 2, 12, Ratio::from_denom_and_numer_i32(2, 1)),             // sin(pi/6) = 0.5
            ( 3, 12, sqrt_iter(&Ratio::from_i32(2), iter).reci()),       // sin(pi/4) = 1/sqrt(2)
            ( 4, 12, sqrt_iter(&Ratio::from_i32(3), iter).div_i32(2)),   // sin(pi/3) = sqrt(3)/2
            ( 5, 12, sqrt_iter(&Ratio::from_i32(6), iter).add_rat(&sqrt_iter(&Ratio::from_i32(2), iter)).div_i32(4)),  // sin(5*pi/12) = (sqrt(6) + sqrt(2))/4
            ( 6, 12, Ratio::one()),                                      // sin(pi/2) = 1
            ( 7, 12, sqrt_iter(&Ratio::from_i32(6), iter).add_rat(&sqrt_iter(&Ratio::from_i32(2), iter)).div_i32(4)),  // sin(7*pi/12) = (sqrt(6) + sqrt(2))/4
            ( 8, 12, sqrt_iter(&Ratio::from_i32(3), iter).div_i32(2)),   // sin(2*pi/3) = sqrt(3)/2
            ( 9, 12, sqrt_iter(&Ratio::from_i32(2), iter).reci()),       // sin(3*pi/4) = 1/sqrt(2)
            (10, 12, Ratio::from_denom_and_numer_i32(2, 1)),             // sin(5*pi/6) = 0.5
            (11, 12, sqrt_iter(&Ratio::from_i32(6), iter).sub_rat(&sqrt_iter(&Ratio::from_i32(2), iter)).div_i32(4)),  // sin(11*pi/12) = (sqrt(6) - sqrt(2))/4
            (12, 12, Ratio::zero()),                                     // sin(pi) = 0
            (13, 12, sqrt_iter(&Ratio::from_i32(6), iter).sub_rat(&sqrt_iter(&Ratio::from_i32(2), iter)).div_i32(4).neg()),  // sin(13*pi/12) = -(sqrt(6) - sqrt(2))/4
            (14, 12, Ratio::from_denom_and_numer_i32(2, -1)),            // sin(7*pi/6) = -0.5
        ];
        let pi = pi_iter(iter);
        let cos_coeff = Ratio::from_denom_and_numer_i32(-2, 1);
        let accuracy = 3e-6;

        for (numer, denom, value) in samples.into_iter() {
            let sin_val = sin_iter(&Ratio::from_denom_and_numer_i32(denom, numer).mul_rat(&pi), iter);
            let sin_val2 = sin_iter(&Ratio::from_denom_and_numer_i32(denom, numer).add_i32(6).mul_rat(&pi), iter);
            let sin_val3 = sin_iter(&Ratio::from_denom_and_numer_i32(denom, numer).add_i32(-6).mul_rat(&pi), iter);
            let cos_val = cos_iter(&Ratio::from_denom_and_numer_i32(denom, numer).add_rat(&cos_coeff).mul_rat(&pi), iter);
            let cos_val2 = cos_iter(&Ratio::from_denom_and_numer_i32(denom, numer).add_rat(&cos_coeff).add_i32(6).mul_rat(&pi), iter);
            let cos_val3 = cos_iter(&Ratio::from_denom_and_numer_i32(denom, numer).add_rat(&cos_coeff).add_i32(-6).mul_rat(&pi), iter);
            let ans_f64 = (3.14159265358979f64 * numer as f64 / denom as f64).sin();
            let ans_f64 = Ratio::from_ieee754_f64(ans_f64).unwrap();

            assert!(are_close(&ans_f64, &value, accuracy));
            assert!(are_close(&sin_val, &value, accuracy));
            assert!(are_close(&sin_val2, &value, accuracy));
            assert!(are_close(&sin_val3, &value, accuracy));
            assert!(are_close(&cos_val, &value, accuracy));
            assert!(are_close(&cos_val2, &value, accuracy));
            assert!(are_close(&cos_val3, &value, accuracy));
        }

    }

}