symplex 0.2.0

Exact symbolic mathematics for Rust: calculus, summation, solving, linear algebra, transforms, compile-time dimensional analysis, and Rust/C code generation
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
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
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345

//! Comprehensive Taylor/Maclaurin series tests with numerical accuracy verification.

mod common;

use symplex::prelude::*;

// ═══════════════════════════════════════════════════════════════════════════
// Helper
// ═══════════════════════════════════════════════════════════════════════════

/// Evaluate a series expression at a rational point x = p/q and return the f64 value.
fn eval_series_at(series: &Ex, var: &Ex, p: i64, q: i64) -> f64 {
    let ctx = series.context();
    let pt = ctx.rational(p, q);
    series
        .subs(var, &pt)
        .eval()
        .eval_f64()
        .expect("series numerical evaluation should succeed")
}

// ═══════════════════════════════════════════════════════════════════════════
// 1. exp(x) Maclaurin series (around x=0), order 6
// ═══════════════════════════════════════════════════════════════════════════

#[test]
fn taylor_exp_at_0() {
    let ctx = Context::new();
    let x = ctx.symbol("x");
    let series = x.exp().maclaurin(&x, 6);
    assert!(!series.has_unevaluated(), "exp(x) maclaurin failed");
    let expanded = series.expand().eval();

    // Evaluate at x = 1/2
    let val = eval_series_at(&expanded, &x, 1, 2);
    let exact = 0.5_f64.exp();
    assert!(
        (val - exact).abs() < 1e-3,
        "exp series at x=0.5: got {val}, expected {exact}, err={}",
        (val - exact).abs()
    );

    // Also verify at x = 1
    let val_1 = eval_series_at(&expanded, &x, 1, 1);
    let exact_1 = 1.0_f64.exp();
    assert!(
        (val_1 - exact_1).abs() < 0.01,
        "exp series at x=1: got {val_1}, expected {exact_1}, err={}",
        (val_1 - exact_1).abs()
    );
}

// ═══════════════════════════════════════════════════════════════════════════
// 2. sin(x) Maclaurin series (around x=0), order 8
// ═══════════════════════════════════════════════════════════════════════════

#[test]
fn taylor_sin_at_0() {
    let ctx = Context::new();
    let x = ctx.symbol("x");
    let series = x.sin().maclaurin(&x, 8);
    assert!(!series.has_unevaluated(), "sin(x) maclaurin failed");
    let expanded = series.expand().eval();

    // Evaluate at x = 1/2
    let val = eval_series_at(&expanded, &x, 1, 2);
    let exact = 0.5_f64.sin();
    assert!(
        (val - exact).abs() < 1e-6,
        "sin series at x=0.5: got {val}, expected {exact}, err={}",
        (val - exact).abs()
    );

    // Evaluate at x = 1
    let val_1 = eval_series_at(&expanded, &x, 1, 1);
    let exact_1 = 1.0_f64.sin();
    assert!(
        (val_1 - exact_1).abs() < 1e-4,
        "sin series at x=1: got {val_1}, expected {exact_1}, err={}",
        (val_1 - exact_1).abs()
    );
}

// ═══════════════════════════════════════════════════════════════════════════
// 3. cos(x) Maclaurin series (around x=0), order 8
// ═══════════════════════════════════════════════════════════════════════════

#[test]
fn taylor_cos_at_0() {
    let ctx = Context::new();
    let x = ctx.symbol("x");
    let series = x.cos().maclaurin(&x, 8);
    assert!(!series.has_unevaluated(), "cos(x) maclaurin failed");
    let expanded = series.expand().eval();

    // Evaluate at x = 1/2
    let val = eval_series_at(&expanded, &x, 1, 2);
    let exact = 0.5_f64.cos();
    assert!(
        (val - exact).abs() < 1e-6,
        "cos series at x=0.5: got {val}, expected {exact}, err={}",
        (val - exact).abs()
    );

    // Evaluate at x = 1
    let val_1 = eval_series_at(&expanded, &x, 1, 1);
    let exact_1 = 1.0_f64.cos();
    assert!(
        (val_1 - exact_1).abs() < 1e-4,
        "cos series at x=1: got {val_1}, expected {exact_1}, err={}",
        (val_1 - exact_1).abs()
    );
}

// ═══════════════════════════════════════════════════════════════════════════
// 4. ln(x) Taylor series around x=1, order 8
// ═══════════════════════════════════════════════════════════════════════════

#[test]
fn taylor_ln_at_1() {
    let ctx = Context::new();
    let x = ctx.symbol("x");
    // ln(x) expanded around x = 1
    let series = x.ln().series(&x, &ctx.int(1), 8);
    assert!(!series.has_unevaluated(), "ln(x) series at 1 failed");
    let expanded = series.expand().eval();

    // Evaluate at x = 11/10 (i.e. x = 1.1, close to the expansion point)
    let val = eval_series_at(&expanded, &x, 11, 10);
    let exact = 1.1_f64.ln();
    assert!(
        (val - exact).abs() < 1e-6,
        "ln series at x=1.1: got {val}, expected {exact}, err={}",
        (val - exact).abs()
    );

    // Also verify at x = 1 itself: ln(1) = 0
    let val_center = eval_series_at(&expanded, &x, 1, 1);
    assert!(
        val_center.abs() < 1e-12,
        "ln series at x=1 (expansion point) should be 0: got {val_center}"
    );
}

// ═══════════════════════════════════════════════════════════════════════════
// 5. exp(x) Taylor series around x=1, order 6
// ═══════════════════════════════════════════════════════════════════════════

#[test]
fn taylor_exp_at_1() {
    let ctx = Context::new();
    let x = ctx.symbol("x");
    // exp(x) expanded around x = 1
    let series = x.exp().series(&x, &ctx.int(1), 6);
    assert!(!series.has_unevaluated(), "exp(x) series at 1 failed");
    let expanded = series.expand().eval();

    // Evaluate at x = 11/10 (x = 1.1)
    let val = eval_series_at(&expanded, &x, 11, 10);
    let exact = 1.1_f64.exp();
    assert!(
        (val - exact).abs() < 1e-4,
        "exp series(at 1) at x=1.1: got {val}, expected {exact}, err={}",
        (val - exact).abs()
    );

    // Verify at expansion point x=1: exp(1) = e
    let val_center = eval_series_at(&expanded, &x, 1, 1);
    let exact_center = 1.0_f64.exp();
    assert!(
        (val_center - exact_center).abs() < 1e-10,
        "exp series at x=1 (expansion point) should be e: got {val_center}, expected {exact_center}"
    );
}

// ═══════════════════════════════════════════════════════════════════════════
// 6. sinh(x) Maclaurin series, order 8
// ═══════════════════════════════════════════════════════════════════════════

#[test]
fn taylor_sinh_at_0() {
    let ctx = Context::new();
    let x = ctx.symbol("x");
    let series = x.sinh().maclaurin(&x, 8);
    assert!(!series.has_unevaluated(), "sinh(x) maclaurin failed");
    let expanded = series.expand().eval();

    // sinh(x) = x + x^3/6 + x^5/120 + x^7/5040 + ...
    // Evaluate at x = 1/2
    let val = eval_series_at(&expanded, &x, 1, 2);
    let exact = 0.5_f64.sinh();
    assert!(
        (val - exact).abs() < 1e-6,
        "sinh series at x=0.5: got {val}, expected {exact}, err={}",
        (val - exact).abs()
    );

    // Evaluate at x = 1
    let val_1 = eval_series_at(&expanded, &x, 1, 1);
    let exact_1 = 1.0_f64.sinh();
    assert!(
        (val_1 - exact_1).abs() < 1e-4,
        "sinh series at x=1: got {val_1}, expected {exact_1}, err={}",
        (val_1 - exact_1).abs()
    );
}

// ═══════════════════════════════════════════════════════════════════════════
// 7. cosh(x) Maclaurin series, order 8
// ═══════════════════════════════════════════════════════════════════════════

#[test]
fn taylor_cosh_at_0() {
    let ctx = Context::new();
    let x = ctx.symbol("x");
    let series = x.cosh().maclaurin(&x, 8);
    assert!(!series.has_unevaluated(), "cosh(x) maclaurin failed");
    let expanded = series.expand().eval();

    // cosh(x) = 1 + x^2/2 + x^4/24 + x^6/720 + ...
    // Evaluate at x = 1/2
    let val = eval_series_at(&expanded, &x, 1, 2);
    let exact = 0.5_f64.cosh();
    assert!(
        (val - exact).abs() < 1e-6,
        "cosh series at x=0.5: got {val}, expected {exact}, err={}",
        (val - exact).abs()
    );

    // Evaluate at x = 1
    let val_1 = eval_series_at(&expanded, &x, 1, 1);
    let exact_1 = 1.0_f64.cosh();
    assert!(
        (val_1 - exact_1).abs() < 1e-4,
        "cosh series at x=1: got {val_1}, expected {exact_1}, err={}",
        (val_1 - exact_1).abs()
    );
}

// ═══════════════════════════════════════════════════════════════════════════
// 8. Composition: exp(sin(x)) — no fast path, order 6
// ═══════════════════════════════════════════════════════════════════════════

#[test]
fn taylor_composition_exp_sin() {
    let ctx = Context::new();
    let x = ctx.symbol("x");
    // exp(sin(x)) — a composition that doesn't match a single known pattern,
    // requiring the general Taylor expansion machinery.
    let expr = x.sin().exp();
    let series = expr.maclaurin(&x, 6);
    assert!(!series.has_unevaluated(), "exp(sin(x)) maclaurin failed");
    let expanded = series.expand().eval();

    // exp(sin(x)) at x=0: exp(sin(0)) = exp(0) = 1
    let val_0 = eval_series_at(&expanded, &x, 0, 1);
    assert!(
        (val_0 - 1.0).abs() < 1e-10,
        "exp(sin(x)) series at x=0 should be 1: got {val_0}"
    );

    // exp(sin(x)) at x = 1/2
    let val = eval_series_at(&expanded, &x, 1, 2);
    let exact = 0.5_f64.sin().exp();
    assert!(
        (val - exact).abs() < 1e-3,
        "exp(sin(x)) series at x=0.5: got {val}, expected {exact}, err={}",
        (val - exact).abs()
    );

    // exp(sin(x)) at x = 1/5 (smaller value, should be very accurate)
    let val_small = eval_series_at(&expanded, &x, 1, 5);
    let exact_small = 0.2_f64.sin().exp();
    assert!(
        (val_small - exact_small).abs() < 1e-5,
        "exp(sin(x)) series at x=0.2: got {val_small}, expected {exact_small}, err={}",
        (val_small - exact_small).abs()
    );
}

// ═══════════════════════════════════════════════════════════════════════════
// 9. Verify sin(x) series error decreases with order
// ═══════════════════════════════════════════════════════════════════════════

#[test]
fn series_numerical_accuracy_sin() {
    let ctx = Context::new();
    // At x = 1/2, increasing the order of the Maclaurin series for sin(x)
    // should produce strictly decreasing approximation error.
    let x = ctx.symbol("x");
    let exact = 0.5_f64.sin();
    let orders = [3u32, 5, 7, 9, 11];
    let mut prev_err = f64::MAX;

    for &order in &orders {
        let series = x.sin().maclaurin(&x, order);
        let expanded = series.expand().eval();
        if let Ok(val) = expanded.subs(&x, &ctx.rational(1, 2)).eval().eval_f64() {
            let err = (val - exact).abs();
            assert!(
                err < prev_err,
                "sin series error should decrease: order {order} err={err} >= prev_err={prev_err}"
            );
            prev_err = err;
        }
    }
    // After all orders, the error should be very small
    assert!(
        prev_err < 1e-6,
        "sin series at order 11 should be very accurate, err={prev_err}"
    );
}

// ═══════════════════════════════════════════════════════════════════════════
// 10. Verify exp(x) series error decreases with order
// ═══════════════════════════════════════════════════════════════════════════

#[test]
fn series_numerical_accuracy_exp() {
    let ctx = Context::new();
    // At x = 1/2, increasing the order of the Maclaurin series for exp(x)
    // should produce strictly decreasing approximation error.
    let x = ctx.symbol("x");
    let exact = 0.5_f64.exp();
    let orders = [2u32, 4, 6, 8, 10];
    let mut prev_err = f64::MAX;

    for &order in &orders {
        let series = x.exp().maclaurin(&x, order);
        let expanded = series.expand().eval();
        if let Ok(val) = expanded.subs(&x, &ctx.rational(1, 2)).eval().eval_f64() {
            let err = (val - exact).abs();
            assert!(
                err < prev_err,
                "exp series error should decrease: order {order} err={err} >= prev_err={prev_err}"
            );
            prev_err = err;
        }
    }
    // After all orders, the error should be very small
    assert!(
        prev_err < 1e-8,
        "exp series at order 10 should be very accurate, err={prev_err}"
    );
}