symplex 0.3.0

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

//! 0.2 numeric back-ends: common subexpression elimination (`cse`, `cse_many`).

use symplex::prelude::*;

/// Substitute the bindings back and check value preservation numerically.
fn assert_cse_preserves(expr: &Ex, bindings: &[(Ex, Ex)], rewritten: &Ex, vars: &[&str]) {
    let mut restored = rewritten.clone();
    for (name, value) in bindings.iter().rev() {
        restored = restored.subs(name, value);
    }
    let f = expr.compile(vars).unwrap();
    let g = restored.compile(vars).unwrap();
    for i in 0..5 {
        let args: Vec<f64> = (0..vars.len())
            .map(|k| 0.3 + i as f64 * 0.7 + k as f64)
            .collect();
        let (a, b) = (f(&args), g(&args));
        assert!(
            (a - b).abs() <= 1e-12 * (1.0 + a.abs()),
            "CSE changed value at {args:?}: {a} vs {b}"
        );
    }
}

#[test]
fn bindings_are_deterministic_and_ordered_by_first_occurrence() {
    let ctx = Context::new();
    let x = ctx.symbol("x");
    let s = x.sin();
    let es = s.exp();
    // exp(sin(x)) + exp(sin(x))^2 + sin(x)*cos(x)
    let e = &(&es + &es.powi(2)) + &(&s * &x.cos());
    let (b1, r1) = e.cse();
    let (b2, r2) = e.cse();
    assert_eq!(b1.len(), b2.len());
    for ((n1, v1), (n2, v2)) in b1.iter().zip(&b2) {
        assert_eq!(format!("{n1}"), format!("{n2}"));
        assert_eq!(format!("{v1}"), format!("{v2}"));
    }
    assert_eq!(format!("{r1}"), format!("{r2}"));
    // sin(x) occurs before exp(sin(x)) in post-order → __cse_0 = sin(x).
    assert_eq!(format!("{}", b1[0].0), "__cse_0");
    assert_eq!(format!("{}", b1[0].1), "sin(x)");
    assert_eq!(format!("{}", b1[1].1), "exp(__cse_0)");
    assert_cse_preserves(&e, &b1, &r1, &["x"]);
}

#[test]
fn cheap_nodes_need_three_uses() {
    let ctx = Context::new();
    let x = ctx.symbol("x");
    let y = ctx.symbol("y");
    // x^2 used twice: not worth a temporary.
    let e = &x.powi(2).sin() + &x.powi(2).cos();
    let (b, _) = e.cse();
    assert!(
        b.iter().all(|(_, v)| format!("{v}") != "x^2"),
        "x^2 used twice should stay inline: {b:?}"
    );
    // x^2 used three times: extracted.
    let e3 = &(&x.powi(2).sin() + &x.powi(2).cos()) + &x.powi(2).exp();
    let (b3, r3) = e3.cse();
    assert!(
        b3.iter().any(|(_, v)| format!("{v}") == "x^2"),
        "x^2 used thrice should be extracted: {b3:?}"
    );
    assert_cse_preserves(&e3, &b3, &r3, &["x"]);
    // Expensive nodes are extracted at two uses.
    let e2 = &(&x * &y).sin() + &(&x * &y).cos();
    let (b2, _) = e2.cse();
    assert!(b2.iter().any(|(_, v)| format!("{v}") == "x*y"), "{b2:?}");
    // Negated atoms are never worth it at two uses.
    let en = &(-&x).exp() + &(-&x).sin();
    let (bn, _) = en.cse();
    assert!(bn.is_empty(), "negations are cheap: {bn:?}");
}

#[test]
fn boolean_nodes_are_never_extracted() {
    let ctx = Context::new();
    let x = ctx.symbol("x");
    let y = ctx.symbol("y");
    let c = x.gt(&y);
    let pw = Ex::piecewise(&[
        (&x, &c.and(&x.gt(&ctx.zero()))),
        (&y, &c.not()),
        (&ctx.zero(), &c),
    ]);
    let (b, r) = pw.cse();
    assert!(b.is_empty(), "conditions must stay native booleans: {b:?}");
    let code = r.to_rust_fn("pw", &["x", "y"]).unwrap();
    assert!(!code.contains("let t"), "{code}");
    let ccode = pw.to_c_fn("pw", &["x", "y"]).unwrap();
    assert!(!ccode.contains("const double t"), "{ccode}");
}

#[test]
fn cse_many_shares_across_expressions() {
    let ctx = Context::new();
    let x = ctx.symbol("x");
    let y = ctx.symbol("y");
    let f = &(&x * &y).sin() * &x.exp();
    let fx = f.diff(&x);
    let fy = f.diff(&y);
    let (bindings, exprs) = Ex::cse_many(&[&fx, &fy]);
    assert_eq!(exprs.len(), 2);
    assert!(
        !bindings.is_empty(),
        "gradient components share cos(x*y), exp(x), x*y"
    );
    let names: Vec<String> = bindings.iter().map(|(n, _)| format!("{n}")).collect();
    for (i, n) in names.iter().enumerate() {
        assert_eq!(n, &format!("__cse_{i}"), "sequential naming");
    }
    // Every binding value only references earlier bindings.
    for (i, (_, v)) in bindings.iter().enumerate() {
        let s = format!("{v}");
        for j in i..bindings.len() {
            assert!(
                !s.contains(&format!("__cse_{j}")),
                "binding {i} refers to later binding {j}: {s}"
            );
        }
    }
    // Value preservation for each output.
    let fxc = fx.compile(&["x", "y"]).unwrap();
    let fyc = fy.compile(&["x", "y"]).unwrap();
    let mut rx = exprs[0].clone();
    let mut ry = exprs[1].clone();
    for (n, v) in bindings.iter().rev() {
        rx = rx.subs(n, v);
        ry = ry.subs(n, v);
    }
    let rxc = rx.compile(&["x", "y"]).unwrap();
    let ryc = ry.compile(&["x", "y"]).unwrap();
    for p in [(0.3, 1.1), (2.0, -0.7), (-1.5, 0.4)] {
        let a = [p.0, p.1];
        assert!((fxc(&a) - rxc(&a)).abs() < 1e-12 * (1.0 + fxc(&a).abs()));
        assert!((fyc(&a) - ryc(&a)).abs() < 1e-12 * (1.0 + fyc(&a).abs()));
    }
    // Empty input.
    let (b0, e0) = Ex::cse_many(&[]);
    assert!(b0.is_empty() && e0.is_empty());
    // Single input agrees with cse().
    let (b1, e1) = Ex::cse_many(&[&fx]);
    let (b1s, e1s) = fx.cse();
    assert_eq!(b1.len(), b1s.len());
    assert_eq!(format!("{}", e1[0]), format!("{e1s}"));
}

#[test]
fn compile_many_uses_shared_cse() {
    // Compare instruction counts: the shared program should be smaller than
    // the sum of the independent programs.
    let ctx = Context::new();
    let x = ctx.symbol("x");
    let y = ctx.symbol("y");
    let f = &(&x * &y).sin() * &(&x * &y).exp() + &(&x * &y).cos();
    let fx = f.diff(&x);
    let fy = f.diff(&y);
    let shared = Ex::compile_many(&[&fx, &fy], &["x", "y"]).unwrap();
    let separate = fx.compile(&["x", "y"]).unwrap().instruction_count()
        + fy.compile(&["x", "y"]).unwrap().instruction_count();
    assert!(
        shared.instruction_count() < separate,
        "shared {} vs separate {}",
        shared.instruction_count(),
        separate
    );
}