Bend 2 vs. Julia & MATLAB: Scientific Array Computing & JIT Warmup vs. Ahead-of-Time GPU Trees
Comparing Julia (CUDA.jl,
SymbolicRegression.jl, LLVM JIT) and MATLAB
(gpuArray) against Bend 2 (BendSR,
nbody, tree-matmul, 2 MB RSS, and 0.38 s
proofs).
Executive Summary: Bottom Line Up Front
Julia and MATLAB dominate interactive numerical linear
algebra, differential equation solvers
(DifferentialEquations.jl), and matrix-based control
systems, whereas Bend 2 dominates ahead-of-time (AOT) parallel recursion
over trees, spatial hierarchies, symbolic ASTs, and formally verified
mathematical models.
- Choose Julia (
LinearAlgebra,CUDA.jl,SciML) or MATLAB (Parallel Computing Toolbox) when your workflow centers on interactive REPL exploration, dense BLAS/LAPACK matrix factorizations (LU,SVD,eig), stiff ODE solvers, or Simulink block diagrams. - Choose Bend 2 (
bend -o,LAWS.bend,PROOF.bend) when your scientific algorithm operates on recursive quadtrees/octrees, N-body spatial trees, cellular automata, or symbolic regression ASTs (BendSR). It also delivers sub-millisecond AOT startup (3.11 ms), tiny memory footprints (2.0 MB to 2.8 MBRSS vs. 180+ MB for Julia’s LLVM JIT), zero proprietary licensing costs, and 0.38 s formal proof verification (benchmark dataset).
1.
Architectural Comparison: Julia LLVM JIT & MATLAB
gpuArray vs. BendRT
| Dimension | Julia 1.11 (CUDA.jl, Multiple Dispatch) |
MATLAB (gpuArray, parfor) |
|---|---|---|
| Compilation & Startup | LLVM JIT at runtime; suffers “Time to First Execution” warmup
latency (0.5 s to 10+ s) and 180+ MB baseline
RSS. |
Proprietary interpreter + JIT; heavy runtime footprint
(500+ MB RSS) and per-seat license restrictions. |
| GPU Programming Model | CuArray broadcasting or explicit @cuda
kernel functions over flat arrays. |
Vectorized gpuArray matrix operations; fails on
recursive tree pointers. |
| Garbage Collection | Non-moving mark-and-sweep GC; allocating tree nodes in parallel loops triggers GC pauses. | Copy-on-write reference counting for matrices; high overhead on small objects. |
| Mathematical Proofs | Runtime unit tests (@test); dynamic multiple dispatch
prevents static completeness proofs. |
Simulink Design Verifier (external commercial add-on). |
2.
Scientific Computing Case Studies: Symbolic Regression
(BendSR), N-Body, and Quadtrees
2.1
Symbolic Regression: SymbolicRegression.jl
(PySR) vs. BendSR
In scientific machine learning, symbolic regression discovers closed-form mathematical equations (f(x) = cos (x1) + x22) from observational data by evolving populations of expression trees.
- In the Python/Julia ecosystem,
PySRdelegates tree evaluation toSymbolicRegression.jlon Julia’s CPU thread pool because expression trees have irregular depths that cannot be batched into denseCuArraytensors. - In Bend 2,
k3ybladewielder/BendSRrepresents candidate equations as algebraic datatypes (Add,Mul,Sin,Cos) and evaluates the entire population in parallel via BendRT. On the Apple M4 Maxsymregbenchmark, Bend 2 achieves an 11.33x speedup on 16 CPU cores (0.266 svs.3.014 ssequential) and 5.7x on GPU (0.529 s) at just 3.1 MB RSS (benchmark dataset).
2.2 Apple M4 Max Scientific & Simulation Benchmarks
Across physics, fractals, clustering, and quadtree linear algebra on Apple M4 Max, Bend 2 delivers AOT C-level single-core speed and massive multi-core/GPU scaling (benchmark dataset):
| Scientific Workload | Domain | Native C (-O3) |
Bend SEQ-CPU (RSS) |
Bend PAR-CPU 16c (RSS) |
Bend PAR-GPU Metal |
GPU Speedup vs. SEQ |
|---|---|---|---|---|---|---|
| Conway’s Game of Life | Cellular Automata | 6.776 s |
7.803 s
(2.1 MB) |
0.647 s
(2.7 MB) |
0.063 s |
123.9x (source) |
N-Body Physics (bend-nbody) |
Orbital Mechanics | 5.126 s |
5.758 s
(2.2 MB) |
0.516 s
(2.7 MB) |
0.059 s |
97.6x (source) |
| Mandelbrot Escape Time | Complex Analysis | 3.750 s |
4.473 s
(2.3 MB) |
0.395 s
(2.9 MB) |
0.057 s |
78.5x (source) |
| Procedural Terrain | Volumetric Noise | 2.198 s |
2.225 s
(2.1 MB) |
0.197 s
(3.3 MB) |
0.106 s |
21.0x (source) |
| K-Means Clustering | Unsupervised ML | 2.062 s |
2.069 s
(23.5 MB) |
0.268 s
(24.1 MB) |
0.191 s |
10.8x (source) |
| Tree Matrix Multiply | Quadtree Algebra | 2.964 s |
2.064 s
(2.8 MB) |
0.226 s
(14.8 MB) |
0.231 s |
8.9x (source) |
Symbolic Regression (symreg) |
Equation Discovery | 2.208 s |
3.014 s
(2.1 MB) |
0.266 s
(3.1 MB) |
0.529 s |
5.7x (source) |
3.
Side-by-Side Code: Symbolic Expression Tree Evaluation
(Julia vs. Bend 2)
In Julia, evaluating a symbolic expression tree
(ExprTree) uses multiple dispatch on heap-allocated nodes,
which cannot be broadcast onto CuArray GPU kernels without
manually encoding expressions as flat bytecode matrices:
# Julia 1.11: Recursive AST evaluation on the CPU GC heap (cannot run inside CuArray broadcast)
abstract type ExprTree end
struct Const <: ExprTree; val::Float32; end
struct Add <: ExprTree; lft::ExprTree; rgt::ExprTree; end
struct Mul <: ExprTree; lft::ExprTree; rgt::ExprTree; end
eval_expr(e::Const) = e.val
eval_expr(e::Add) = eval_expr(e.lft) + eval_expr(e.rgt)
eval_expr(e::Mul) = eval_expr(e.lft) * eval_expr(e.rgt)In Bend 2, a b = eval_expr(lft) eval_expr(rgt) evaluates
both subtrees in parallel across 16 CPU cores (0.266 s on
symreg, 11.33x speedup) or GPU warps, and
allows proving algebraic identities at compile time (k3ybladewielder/BendSR):
type ExprTree:
case @Const:
val: U32
case @Add:
lft: ExprTree
rgt: ExprTree
case @Mul:
lft: ExprTree
rgt: ExprTree
def eval_expr(e: ExprTree) -> U32:
match e:
case @Const{val}:
val
case @Add{lft, rgt}:
a b = eval_expr(lft) eval_expr(rgt)
a + b
case @Mul{lft, rgt}:
a b = eval_expr(lft) eval_expr(rgt)
a * b4. Calling
Verified Bend 2 Kernels from Julia via @ccall
Julia’s zero-overhead C foreign function interface
(@ccall) makes pairing Julia and Bend 2 effortless. Keep
dense BLAS arrays, plotting (Makie.jl), and ODE integrators
in Julia. Compile recursive tree/graph solvers and
LAWS.bend-verified numerical invariants
(interval-arithmetic, bistatic-radar-oracle)
via bend kernel.bend -o kernel.c into
libkernel.so, and invoke them from Julia with zero JIT
warmup delay (bendlang/bend).