Benchmarks

What ER2 costs against SymPy and against raw PARI. These record what is true today, not a promise (see Stability, N6).

ER2 benchmarks

Generated by uv run python benchmarks/run.py --write; do not edit by hand.

Python 3.14.6, SymPy 1.14.0, cypari2 2.2.4, x86_64. Times are medians per call; ×1.00 means ER2 is as fast as the reference.

Notes: isprime proves primality (PARI’s APRCL); sympy.isprime and ispseudoprime run the BPSW probable-prime test, so they are compared with each other. Number theory calls get a different argument each time, because SymPy caches results.

Startup

Benchmark	ER2	Reference	Reference time	ER2 / reference
er2 startup, number theory only	145 ms	Python	24.7 ms	×5.87
er2 startup, with symbols (loads SymPy)	539 ms	Python	24.7 ms	×21.80

Dispatch overhead (§2.1)

Benchmark	ER2	Reference	Reference time	ER2 / reference
choose a backend: phi(n), an integer	292 ns	the whole call	5.36 µs	×0.05
choose a backend: det, 10x10 over Z	1.52 µs	the whole call	115 µs	×0.01
choose a backend: det, 10x10 with a symbol	19.5 µs	the whole call	503 ms	×0.00
choose a backend: factor, a polynomial over Q	52.8 µs	the whole call	467 µs	×0.11

Integer arithmetic (D2)

Benchmark	ER2	Reference	Reference time	ER2 / reference
numeric loop, 20000 iterations (ER2 literals)	22.4 ms	Python int	1.62 ms	×13.78
a + b	288 ns	int a + b	35.1 ns	×8.22

Number theory

Benchmark	ER2	Reference	Reference time	ER2 / reference
phi(n), n ≈ 1.2e8	5.35 µs	cypari2	2.99 µs	×1.79
phi(n), n ≈ 1.2e8	5.35 µs	SymPy	89.4 µs	×0.06
sigma(n), n ≈ 1.2e8	5.56 µs	cypari2	3.07 µs	×1.81
sigma(n), n ≈ 1.2e8	5.56 µs	SymPy	90.1 µs	×0.06
isprime(2^127 - 1), proven	381 µs	cypari2	374 µs	×1.02
ispseudoprime (BPSW), Mersenne primes	20.6 µs	cypari2	17.6 µs	×1.17
ispseudoprime (BPSW), Mersenne primes	20.6 µs	SymPy	63.6 µs	×0.32
factor(n), n ≈ 1e18 semiprime	224 µs	cypari2	205 µs	×1.09
factor(n), n ≈ 1e18 semiprime	224 µs	SymPy	12.3 ms	×0.02

Polynomial factorization

Benchmark	ER2	Reference	Reference time	ER2 / reference
factor, degree 4	341 µs	SymPy factor	737 µs	×0.46
factor, degree 9	716 µs	SymPy factor	2.24 ms	×0.32
factor, degree 22	1.61 ms	SymPy factor	10 ms	×0.16
factor, degree 60	10.6 ms	SymPy factor	263 ms	×0.04

Polynomials over F_p (M5)

Benchmark	ER2	Reference	Reference time	ER2 / reference
factor mod 7, degree 12	965 µs	cypari2	20.5 µs	×47.07
factor mod 7, degree 12	965 µs	SymPy	2.69 ms	×0.36
factor mod 7, degree 48	3.11 ms	cypari2	604 µs	×5.14
factor mod 7, degree 48	3.11 ms	SymPy	19.3 ms	×0.16

Linear algebra (M5)

Benchmark	ER2	Reference	Reference time	ER2 / reference
det, 10x10 over Z	110 µs	cypari2	14 µs	×7.82
det, 10x10 over Z	110 µs	SymPy	5.08 ms	×0.02
kernel, 10x10 over Z	354 µs	cypari2	22.1 µs	×16.03
kernel, 10x10 over Z	354 µs	SymPy	899 µs	×0.39
hermite_form, 10x10 over Z	261 µs	cypari2	32.3 µs	×8.09
hermite_form, 10x10 over Z	261 µs	SymPy	356 µs	×0.73
smith_form, 10x10 over Z	257 µs	cypari2	55.5 µs	×4.62
smith_form, 10x10 over Z	257 µs	SymPy	764 µs	×0.34
det, 20x20 over Z	518 µs	cypari2	152 µs	×3.42
det, 20x20 over Z	518 µs	SymPy	40.1 ms	×0.01
kernel, 20x20 over Z	987 µs	cypari2	268 µs	×3.68
kernel, 20x20 over Z	987 µs	SymPy	3.32 ms	×0.30
hermite_form, 20x20 over Z	971 µs	cypari2	237 µs	×4.09
hermite_form, 20x20 over Z	971 µs	SymPy	2.84 ms	×0.34
smith_form, 20x20 over Z	1.2 ms	cypari2	484 µs	×2.48
smith_form, 20x20 over Z	1.2 ms	SymPy	3.52 ms	×0.34