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
use std::collections::HashMap;
use arael::sym::*;
use arael::sym;
fn main() {
// FunctionBag for use at parse_with_functions
let mut bag = FunctionBag::new();
// ============================================================
// Basics
// ============================================================
println!("=== Basics ===\n");
sym! {
let (x, y, a, b) = symbols!(x, y, a, b);
println!("Symbols: x = {x}, y = {y}");
println!("Constants: constant(3.0) = {}, c(3.0) = {}", constant(3.0), c(3.0));
println!("Arithmetic: x + y = {}", x + y);
println!(" x * y - 1 = {}", x * y - 1.0);
println!(" x^2 = {}", pow(x, 2.0));
// Auto-simplification (like SymPy)
println!("\n--- Auto-simplification ---");
println!("(x + y) / (x + y) = {}", (x + y) / (x + y));
println!("x + 0 = {}", x + 0.0);
println!("x * 1 = {}", x * 1.0);
println!("x * 0 = {}", x * 0.0);
println!("-(-x) = {}", -(-x));
println!("x - x = {}", x - x);
println!("3*x + 2*x = {}", 3.0 * x + 2.0 * x);
println!("x * x = {}", x * x);
// ============================================================
// Derivatives — all rules (no .simplify() needed!)
// ============================================================
println!("\n=== Derivatives ===\n");
// Power rule
let x4 = pow(x, 4.0);
let dx4 = x4.diff(x);
println!("Power rule: d/dx(x^4) = {dx4}");
let d2x4 = dx4.diff(x);
println!(" 2nd deriv: d²/dx²(x^4) = {d2x4}");
// Product rule
let xsinx = x * sin(x);
println!("Product rule: d/dx(x·sin(x)) = {}", xsinx.diff(x));
// Quotient rule
let sinx_over_x = sin(x) / x;
println!("Quotient rule: d/dx(sin(x)/x) = {}", sinx_over_x.diff(x));
// Chain rule
let exp_sin = exp(sin(x));
println!("Chain rule: d/dx(exp(sin(x))) = {}", exp_sin.diff(x));
let ln_sqrt = ln(sqrt(pow(x, 2.0) + c(1.0)));
println!(" nested: d/dx(ln(√(x²+1))) = {}", ln_sqrt.diff(x));
// General power: x^x
let xx = pow(x, x);
println!("General power: d/dx(x^x) = {}", xx.diff(x));
// Trig derivatives
println!("Trig: d/dx(sin(x)) = {}", sin(x).diff(x));
println!(" d/dx(cos(x)) = {}", cos(x).diff(x));
println!(" d/dx(tan(x)) = {}", tan(x).diff(x));
// Inverse trig
println!("Inv trig: d/dx(asin(x)) = {}", asin(x).diff(x));
println!(" d/dx(acos(x)) = {}", acos(x).diff(x));
println!(" d/dx(atan(x)) = {}", atan(x).diff(x));
// ============================================================
// Expansion
// ============================================================
println!("\n=== Expansion ===\n");
let dist = x * (a + b);
println!("Distributive: ({dist}).expand() = {}", dist.expand());
let sq = pow(x + y, 2.0);
println!("(x + y)^2: {}", sq.expand());
let cb = pow(x - y, 3.0);
println!("(x - y)^3: {}", cb.expand());
// ============================================================
// Collection
// ============================================================
println!("\n=== Collection ===\n");
let scattered = a * x + b * x;
println!("({scattered}).collect(x) = {}", scattered.collect(&x));
// ============================================================
// Evaluation & Substitution
// ============================================================
println!("\n=== Evaluation & Substitution ===\n");
let expr = pow(x, 2.0) + 3.0 * x + 1.0;
let vars = HashMap::from([("x", 2.0)]);
println!("f(x) = {expr}");
println!("f(2) = {}", expr.eval(&vars).unwrap());
let subst = expr.subs(x, &(y + 1.0));
println!("f(y+1) = {subst}");
// Kinematics example
println!("\n--- Kinematics ---");
let (t, v0, a) = symbols!(t, v0, a);
let s = v0 * t + 0.5 * a * pow(t, 2.0);
println!("Position: s(t) = {s}");
let v = s.diff(t);
println!("Velocity: v(t) = ds/dt = {v}");
let a_deriv = v.diff(t);
println!("Accel: a(t) = dv/dt = {a_deriv}");
// ============================================================
// Free variables
// ============================================================
println!("\n=== Free Variables ===\n");
let fv_expr = x * y + sin(symbol("z"));
println!("free_vars({fv_expr}) = {:?}", fv_expr.free_vars());
// ============================================================
// Linear Algebra
// ============================================================
println!("\n=== Linear Algebra ===\n");
let v = SymVec::new([x, y]);
println!("v = {v}");
println!("v·v = {}", v.dot(&v));
let m = SymMat::new(2, 2, [1.0, 2.0, 3.0, 4.0]);
let mv = m.clone() * v;
println!("M = {m}");
println!("M·v = {mv}");
println!("M^T = {}", m.transpose());
// Jacobian
let f1 = x * y;
let f2 = pow(x, 2.0) + sin(y);
let j = jacobian(&[f1, f2], &["x", "y"]);
println!("\nJ([x·y, x²+sin(y)]) = {}", j.simplify());
// ============================================================
// Output formatting / code generation
// ============================================================
println!("\n=== Output Formatting / Code Generation ===\n");
// Rosenbrock: a benchmark with shared subterms to motivate CSE
// below. Each of f's two partial derivatives reuses x*x and
// (y - x*x).
let f = pow(1.0 - x, 2.0) + 100.0 * pow(y - x * x, 2.0);
println!("Display: {f}");
println!("LaTeX: {}", f.to_latex());
println!("Rust f64: {}", f.to_rust("f64"));
println!("Rust f32: {}", f.to_rust("f32"));
println!("\nLaTeX samples:");
println!(" sqrt(x) → {}", sqrt(x).to_latex());
println!(" |x| → {}", abs(x).to_latex());
println!(" exp(x) → {}", exp(x).to_latex());
println!(" M (LaTeX) → {}", m.to_latex());
// ============================================================
// Common Subexpression Elimination
// ============================================================
println!("\n=== Common Subexpression Elimination ===\n");
// cse() factors subtrees that appear more than once across a
// batch into named intermediates. Here, f and its two
// partial derivatives share `y - x*x` and `1 - x`.
let batch = [f, f.diff(x), f.diff(y)];
let (intermediates, simplified) = cse::cse(&batch);
for (name, val) in &intermediates {
println!("let {name} = {};", val.to_rust("f64"));
}
let names = ["f", "df_dx", "df_dy"];
for (i, s) in simplified.iter().enumerate() {
println!("let {} = {};", names[i], s.to_rust("f64"));
}
// ============================================================
// Error suppression (SymPy comparison)
// ============================================================
println!("\n=== Error suppression ===\n");
let (gamma, a, b, sigma) = symbols!(gamma, a, b, sigma);
let r = (y - (a * x + b)) / sigma;
let rs = gamma * atan(r / gamma);
#[allow(non_snake_case)]
let S = pow(rs, 2.0);
println!("S = {S}");
println!("dS/dx = {}", S.diff(x));
// ============================================================
// User-defined functions
// ============================================================
//
// `simple_func1` / `simple_func2` wrap a symbolic body as a
// named callable. Diff and codegen see the body when they need
// to (chain rule, inlined emission) and the name otherwise --
// the function participates in expression building like a
// built-in. `simple_func*_derivs` and `extern_func*` provide
// explicit derivatives and native eval when the body is
// opaque or brittle to auto-diff.
println!("\n=== User-defined functions ===\n");
let sq = simple_func1("sq", |t| t * t);
let hypot = simple_func2("hypot",
|a, b| sqrt(a * a + b * b));
let f = sq(x + 1.0) + hypot(x, y);
println!("f = {f}");
println!("df/dx = {}", f.diff(x));
// ============================================================
// Parsing expressions from strings
// ============================================================
println!("\n=== Parsing ===\n");
let e1 = parse("x^2 + 3*x + 1").unwrap();
println!("parse(\"x^2 + 3*x + 1\") = {e1}");
let e2 = parse("sqrt(atan2(y, x) + pi)").unwrap();
println!("parse(\"sqrt(atan2(y, x) + pi)\") = {e2}");
let e3 = parse("exp(sin(x)) * cos(x)").unwrap();
println!("parse(\"exp(sin(x)) * cos(x)\") = {e3}");
println!(" d/dx = {}", e3.diff(x));
let e4: E = "-a * x^2 + b".parse().unwrap();
println!("FromStr: \"-a * x^2 + b\" = {e4}");
// `parse_with_functions` consults a FunctionBag before
// falling back to built-ins, so expressions typed at runtime
// can reference user-defined `sq` / `hypot`.
bag.add1(simple_func1("sq", |t| t * t)).unwrap();
bag.add2(simple_func2("hypot",
|a, b| sqrt(a * a + b * b))).unwrap();
let e5 = parse_with_functions("sq(3) + hypot(3, 4)", &bag).unwrap();
println!("parse_with_functions(\"sq(3) + hypot(3, 4)\") = {e5}");
println!(" eval = {}", e5.eval(&HashMap::new()).unwrap());
// Parsed user-defined functions diff + codegen like built-ins.
let e6 = parse_with_functions("sq(x) + hypot(x, y)", &bag).unwrap();
println!("d/dx(sq(x) + hypot(x, y)) = {}", e6.diff(x));
}
}