Benchmarks

Here we present benchmarks of Symbolica versus other computer algebra systems and libraries.

Algebra performance

First, we compare Symbolica with other full computer algebra systems: Mathematica and SymPy. We test polynomial arithmetic through the computation of a greatest common divisor operation on multivariate polynomials. It is a useful test that stresses many areas, including memory management.

On this standard polynomial GCD benchmark, Symbolica comfortably outpaces both: 2 seconds, compared with 89 seconds and over an hour.

Code and timing provenance

The inputs have a known common factor. Construction is outside the timer; only the GCD call is timed, with verification afterward (allowing constant normalization of the common factor). Symbolica uses its polynomial representation, Mathematica receives expanded expressions, and SymPy performs any conversion required by its expression-level gcd call.

Symbolica

from time import perf_counter
from symbolica import S

xs = S("x1", "x2", "x3", "x4", "x5", "x6", "x7")
coefficients = [3, 5, 7, 9, 11, 13, 15]
def base(signs):
    return 1 + sum(c*s*x for c, s, x in zip(coefficients, signs, xs))

degree = 7
a = (base([1]*7)**degree - 1).to_polynomial()
b = (base([-1,-1,-1,1,-1,-1,1])**degree + 1).to_polynomial()
g = (base([1,1,1,1,1,1,-1])**degree + 3).to_polynomial()
ag, bg = a*g, b*g
start = perf_counter()
result = ag.gcd(bg)
seconds = perf_counter() - start
assert result.monic() == g.monic()
print(seconds)

Mathematica

xs = Symbol /@ Table["x" <> ToString[i], {i, 7}];
coefficients = Range[3, 15, 2];
base[signs_] := 1 + (coefficients signs).xs;
degree = 7;
a = base[ConstantArray[1, 7]]^degree - 1;
b = base[{-1,-1,-1,1,-1,-1,1}]^degree + 1;
g = base[{1,1,1,1,1,1,-1}]^degree + 3;
ag = Expand[a g]; bg = Expand[b g];
{seconds, result} = AbsoluteTiming[PolynomialGCD[ag, bg]];
If[Expand[result - g] =!= 0, Abort[]];
Print[seconds];

SymPy

from time import perf_counter
import sympy as sp

xs = sp.symbols("x1:8")
coefficients = [3, 5, 7, 9, 11, 13, 15]
def base(signs):
    return 1 + sum(c*s*x for c, s, x in zip(coefficients, signs, xs))

degree = 7
a = base([1]*7)**degree - 1
b = base([-1,-1,-1,1,-1,-1,1])**degree + 1
g = base([1,1,1,1,1,1,-1])**degree + 3
ag, bg = a*g, b*g
start = perf_counter()
result = sp.gcd(ag, bg)
seconds = perf_counter() - start
assert sp.expand(result - g) == 0
print(seconds)

Symbolica vs FLINT

Next, compare Symbolica with FLINT, a highly specialized C library for polynomial arithmetic and number theory.

Symbolica is faster in 179 of 291 paired cases in this suite. The table shows wins and the median of the individual FLINT / Symbolica timing ratios for each operation. A ratio above 1 favors Symbolica; below 1 favors FLINT. These ratios describe individual cases, not total-suite speedups.

Operation	Cases	Symbolica wins	Median FLINT / Symbolica
Multiplication	100	62	1.05×
Exact division	33	2	0.34×
GCD	61	44	1.87×
Factorization	53	40	1.78×
Resultant	38	27	1.22×
Discriminant	6	4	1.05×

The suite covers 291 paired cases: polynomial multiplication, exact division, GCD, factorization, resultants, and discriminants in native Rust and C, with both generated inputs and fixed polynomial fixtures.

MultiplicationExact divisionGCDFactorizationResultantDiscriminant

Both axes are logarithmic. The dashed line means equal time.

Hover, focus, or tap a point to inspect a case.

291 cases shown

Explore the exact timings

Median elapsed time per operation in milliseconds. Ratio is FLINT time divided by Symbolica time.

Case	Symbolica (ms)	FLINT (ms)	FLINT / Symbolica	Samples
rational/sparse8_d128/b12/qfalse/multiply	0.007945	0.007193	0.905×	3
rational/sparse8_d128/b12/qfalse/exact_division	0.074881	0.009308	0.124×	3
rational/sparse8_d128/b12/qtrue/multiply	0.014538	0.007094	0.488×	3
rational/sparse8_d128/b12/qtrue/exact_division	0.090441	0.009027	0.0998×	3
rational/sparse8_d128/b63/qfalse/multiply	0.007755	0.009178	1.18×	3
rational/sparse8_d128/b63/qfalse/exact_division	0.070874	0.016381	0.231×	3
rational/sparse8_d128/b63/qtrue/multiply	0.016711	0.010720	0.641×	3
rational/sparse8_d128/b63/qtrue/exact_division	0.089109	0.019817	0.222×	3
rational/sparse8_d128/b127/qfalse/multiply	0.015960	0.010981	0.688×	3
rational/sparse8_d128/b127/qfalse/exact_division	0.091112	0.020870	0.229×	3
rational/sparse8_d128/b127/qtrue/multiply	0.050074	0.014417	0.288×	3
rational/sparse8_d128/b127/qtrue/exact_division	0.139534	0.024677	0.177×	3
rational/dense3_d12/b12/qfalse/multiply	0.854596	0.115298	0.135×	3
rational/dense3_d12/b12/qfalse/exact_division	3.365926	1.273408	0.378×	3
rational/dense3_d12/b12/qtrue/multiply	13.155903	0.119096	0.00905×	3
rational/dense3_d12/b12/qtrue/exact_division	3.423173	1.276294	0.373×	3
rational/dense3_d12/b63/qfalse/multiply	2.093660	1.861141	0.889×	3
rational/dense3_d12/b63/qfalse/exact_division	4.493509	3.516311	0.783×	3
rational/dense3_d12/b63/qtrue/multiply	40.839292	3.633703	0.089×	3
rational/dense3_d12/b63/qtrue/exact_division	6.572872	5.105757	0.777×	3
rational/dense3_d12/b127/qfalse/multiply	3.749922	3.911588	1.04×	3
rational/dense3_d12/b127/qfalse/exact_division	6.461512	5.321014	0.823×	3
rational/dense3_d12/b127/qtrue/multiply	52.839028	8.120999	0.154×	3
rational/dense3_d12/b127/qtrue/exact_division	7.102183	7.456523	1.05×	3
rational/dense3_d20/b12/qfalse/multiply	13.470117	1.543459	0.115×	3
rational/dense3_d20/b12/qfalse/exact_division	53.472576	20.300977	0.38×	3
rational/dense3_d20/b12/qtrue/multiply	217.797334	1.492562	0.00685×	3
rational/dense3_d20/b12/qtrue/exact_division	52.787600	20.451531	0.387×	3
rational/dense3_d20/b63/qfalse/multiply	35.314071	31.239265	0.885×	3
rational/dense3_d20/b63/qfalse/exact_division	65.816653	54.834863	0.833×	3
rational/dense3_d20/b63/qtrue/multiply	677.372206	60.126841	0.0888×	3
rational/dense3_d20/b63/qtrue/exact_division	94.155111	84.555114	0.898×	3
rational/dense3_d20/b127/qfalse/multiply	58.163177	61.560994	1.06×	3
rational/dense3_d20/b127/qfalse/exact_division	91.851054	85.905348	0.935×	3
rational/dense3_d20/b127/qtrue/multiply	864.581691	91.711199	0.106×	3
rational/dense3_d20/b127/qtrue/exact_division	104.703602	122.742249	1.17×	3
rational/sparse8_div127/b12/qfalse/multiply	0.013816	0.009137	0.661×	3
rational/sparse8_div127/b12/qfalse/exact_division	0.051818	0.011411	0.22×	3
rational/sparse8_div127/b12/qtrue/multiply	0.027212	0.008486	0.312×	3
rational/sparse8_div127/b12/qtrue/exact_division	0.050205	0.006692	0.133×	3
rational/sparse8_div127/b63/qfalse/multiply	0.007865	0.006843	0.87×	3
rational/sparse8_div127/b63/qfalse/exact_division	0.034175	0.013285	0.389×	3
rational/sparse8_div127/b63/qtrue/multiply	0.014898	0.008846	0.594×	3
rational/sparse8_div127/b63/qtrue/exact_division	0.050575	0.017273	0.342×	3
rational/sparse8_div127/b127/qfalse/multiply	0.015068	0.008717	0.579×	3
rational/sparse8_div127/b127/qfalse/exact_division	0.053702	0.017613	0.328×	3
rational/sparse8_div127/b127/qtrue/multiply	0.042541	0.013465	0.317×	3
rational/sparse8_div127/b127/qtrue/exact_division	0.099458	0.021571	0.217×	3
rational/sparse8_div128/b12/qfalse/multiply	0.007485	0.007053	0.942×	3
rational/sparse8_div128/b12/qfalse/exact_division	0.069791	0.009007	0.129×	3
rational/sparse8_div128/b12/qtrue/multiply	0.013696	0.006953	0.508×	3
rational/sparse8_div128/b12/qtrue/exact_division	0.088036	0.009258	0.105×	3
rational/sparse8_div128/b63/qfalse/multiply	0.007313	0.009608	1.31×	3
rational/sparse8_div128/b63/qfalse/exact_division	0.068820	0.016571	0.241×	3
rational/sparse8_div128/b63/qtrue/multiply	0.016020	0.011321	0.707×	3
rational/sparse8_div128/b63/qtrue/exact_division	0.089209	0.020479	0.23×	3
rational/sparse8_div128/b127/qfalse/multiply	0.014939	0.011171	0.748×	3
rational/sparse8_div128/b127/qfalse/exact_division	0.089759	0.021751	0.242×	3
rational/sparse8_div128/b127/qtrue/multiply	0.036900	0.014567	0.395×	3
rational/sparse8_div128/b127/qtrue/exact_division	0.137510	0.025668	0.187×	3
dense small multiplication	0.188687	0.118935	0.63×	3
dense high multiplication	3.424106	5.086571	1.49×	3
dense large multiplication	10.895287	18.531028	1.7×	3
dense very large multiplication	100.054629	211.228978	2.11×	3
dense high large multiplication	76.538571	129.925464	1.7×	3
sparse separated multiplication	16.737476	13.766889	0.823×	3
sparse large multiplication	330.094152	343.017024	1.04×	3
seven-variable power-minus-one multiplication	45.978391	85.356238	1.86×	3
dense exact division	1.093959	0.460862	0.421×	3
dense large exact division	14.690776	5.828164	0.397×	3
high-height exact division	4.760473	4.418797	0.928×	3
GF(17) dense univariate degree-4912 multiplication	0.291512	0.268468	0.921×	3
GF(17) dense large multiplication	0.191973	0.196571	1.02×	3
GF(17) dense very large multiplication	0.731964	0.721004	0.985×	3
GF(17) five-variable total-degree multiplication	333.371107	346.492264	1.04×	3
GF(17) sparse large multiplication	85.760622	72.204423	0.842×	3
GF(17) seven-variable power-minus-one multiplication	55.504067	77.691621	1.4×	3
GF(18446744073709551557) dense univariate degree-4912 multiplication	0.439551	1.458208	3.32×	3
GF(18446744073709551557) dense large multiplication	4.996541	20.404342	4.08×	3
GF(18446744073709551557) dense very large multiplication	64.468772	118.738003	1.84×	3
GF(18446744073709551557) five-variable total-degree multiplication	376.371022	353.130918	0.938×	3
GF(18446744073709551557) sparse large multiplication	344.583580	329.950926	0.958×	3
GF(18446744073709551557) seven-variable power-minus-one multiplication	72.457471	79.497446	1.1×	3
GF(65000011) dense univariate degrees 128/64 accumulator-bound multiplication	0.007304	0.007534	1.03×	3
GF(500000003) dense univariate degrees 128/64 accumulator-bound multiplication	0.006762	0.007955	1.18×	3
resultant Brown: dense outer degrees 7/6	10.761323	7.517788	0.699×	3
resultant Ducos: dense outer degrees 7/6	4.004825	7.475068	1.87×	3
resultant CRT: dense outer degrees 7/6	26.232875	7.446313	0.284×	3
resultant Brown: lacunary outer degrees 18/11	63.305532	54.393876	0.859×	3
resultant Ducos: lacunary outer degrees 18/11	27.486816	54.633691	1.99×	3
resultant CRT: lacunary outer degrees 18/11	183.159233	54.364311	0.297×	3
resultant Brown: nonunit leading degrees 9/7	5.915428	4.927850	0.833×	3
resultant Ducos: nonunit leading degrees 9/7	2.848758	4.879899	1.71×	3
resultant CRT: nonunit leading degrees 9/7	8.846642	4.863238	0.55×	3
resultant Brown: large high-height degrees 14/10	8408.854547	4623.354291	0.55×	3
resultant Ducos: large high-height degrees 14/10	3040.571170	4683.198087	1.54×	3
resultant CRT: large high-height degrees 14/10	10904.909677	4696.211669	0.431×	3
resultant Brown: dense outer degrees 10/8 CRT crossover	264.699006	151.290499	0.572×	3
resultant Ducos: dense outer degrees 10/8 CRT crossover	90.949853	151.299786	1.66×	3
resultant CRT: dense outer degrees 10/8 CRT crossover	152.952011	155.734623	1.02×	3
resultant Brown: outer-sparse degrees 12/9 CRT crossover	504.090227	397.725231	0.789×	3
resultant Ducos: outer-sparse degrees 12/9 CRT crossover	228.834207	385.685632	1.69×	3
resultant CRT: outer-sparse degrees 12/9 CRT crossover	255.766300	383.833509	1.5×	3
GCD products: dense 7 variables degree 7	93.719030	169.902285	1.81×	3
GCD auto: dense 7 variables degree 7	5023.687657	3362.028771	0.669×	3
generated GCD products: dense 1 variables degree 32	0.008166	0.008005	0.98×	3
generated GCD auto: dense 1 variables degree 32	0.096823	0.069682	0.72×	3
generated GCD products: dense 2 variables degree 5	0.004519	0.010199	2.26×	3
generated GCD auto: dense 2 variables degree 5	0.068569	0.053171	0.775×	3
generated GCD products: dense 3 variables degree 7	0.023895	0.028784	1.2×	3
generated GCD auto: dense 3 variables degree 7	1.088669	4.196737	3.85×	3
generated GCD products: dense 5 variables degree 7	1.829600	3.694847	2.02×	3
generated GCD auto: dense 5 variables degree 7	210.334755	96.443560	0.459×	3
generated GCD products: dense 8 variables degree 5	14.273103	24.219463	1.7×	3
generated GCD auto: dense 8 variables degree 5	175.315778	779.666737	4.45×	3
generated GCD products: dense 8 variables asymmetric cofactor terms 8/45 common terms 165 degrees 1/2/3	0.078839	0.092415	1.17×	3
generated GCD auto: dense 8 variables asymmetric cofactor terms 8/45 common terms 165 degrees 1/2/3	0.934357	2.593755	2.78×	3
generated GCD products: dense 8 variables asymmetric cofactor terms 8/45 common terms 495 degrees 1/2/4	0.222311	0.275902	1.24×	3
generated GCD auto: dense 8 variables asymmetric cofactor terms 8/45 common terms 495 degrees 1/2/4	2.059495	5.724486	2.78×	3
generated GCD products: dense 8 variables asymmetric cofactor terms 8/45 common terms 1287 degrees 1/2/5	0.540301	0.708140	1.31×	3
generated GCD auto: dense 8 variables asymmetric cofactor terms 8/45 common terms 1287 degrees 1/2/5	4.457880	12.243484	2.75×	3
generated GCD products: sparse 5 variables degree 7	0.156937	0.168188	1.07×	3
generated GCD auto: sparse 5 variables degree 7	7.837252	9.030919	1.15×	3
generated GCD products: sparse 8 variables degree 5	0.403293	0.413923	1.03×	3
generated GCD auto: sparse 8 variables degree 5	12.288649	45.335677	3.69×	3
generated GCD products: high-gap 5 variables degree 5 gap 64	0.093577	0.097304	1.04×	3
generated GCD auto: high-gap 5 variables degree 5 gap 64	1.478245	25.525532	17.3×	3
generated GCD products: high-gap 8 variables degree 4 gap 256	0.321618	0.446064	1.39×	3
generated GCD auto: high-gap 8 variables degree 4 gap 256	4.162291	231.571115	55.6×	3
generated GCD products: high-height 5 variables degree 4 coefficient bits 128	1.543468	2.396001	1.55×	3
generated GCD auto: high-height 5 variables degree 4 coefficient bits 128	9.316489	31.525259	3.38×	3
generated GCD products: high-height 5 variables degree 4 coefficient bits 256	4.049348	4.818282	1.19×	3
generated GCD auto: high-height 5 variables degree 4 coefficient bits 256	17.679584	58.130273	3.29×	3
generated GCD products: high-height 5 variables degree 4 coefficient bits 512	11.818832	12.707612	1.08×	3
generated GCD auto: high-height 5 variables degree 4 coefficient bits 512	40.404010	123.895244	3.07×	3
generated GCD products: high-height 5 variables degree 4 coefficient bits 1024	34.830199	35.904751	1.03×	3
generated GCD auto: high-height 5 variables degree 4 coefficient bits 1024	107.413134	286.567551	2.67×	3
generated GCD products: high-height 8 variables degree 3 coefficient bits 256	4.494309	5.350028	1.19×	3
generated GCD auto: high-height 8 variables degree 3 coefficient bits 256	33.042899	117.539083	3.56×	3
generated GCD products: high-height 8 variables asymmetric cofactor terms 8/45 common terms 165 degrees 1/2/3 coefficient bits 256	0.642425	0.669135	1.04×	3
generated GCD auto: high-height 8 variables asymmetric cofactor terms 8/45 common terms 165 degrees 1/2/3 coefficient bits 256	8.343349	10.554260	1.26×	3
rational GCD control: divisible, power 2	0.008326	0.008075	0.97×	3
rational GCD control: proportional, power 2	0.003506	0.003336	0.952×	3
rational GCD control: coprime, power 2	0.012613	0.004299	0.341×	3
rational GCD control: shared factor, power 2	0.120398	0.080382	0.668×	3
rational GCD control: divisible, power 3	0.014528	0.014227	0.979×	3
rational GCD control: proportional, power 3	0.009278	0.007565	0.815×	3
rational GCD control: coprime, power 3	0.014237	0.008456	0.594×	3
rational GCD control: shared factor, power 3	0.195760	0.135486	0.692×	3
rational GCD control: divisible, power 4	0.025448	0.023645	0.929×	3
rational GCD control: proportional, power 4	0.016631	0.014277	0.858×	3
rational GCD control: coprime, power 4	0.026600	0.014738	0.554×	3
rational GCD control: shared factor, power 4	0.297813	0.190049	0.638×	3
generated factor product: dense 1-variable degrees 32/31	0.004398	0.004679	1.06×	3
generated factorization: dense 1-variable degrees 32/31	6.544948	2.794685	0.427×	3
generated factor product: dense 2-variable degrees 10/9	0.008286	0.006242	0.753×	3
generated factorization: dense 2-variable degrees 10/9	3.527320	9.970035	2.83×	3
generated factor product: dense 3-variable degrees 6/5	0.006973	0.007093	1.02×	3
generated factorization: dense 3-variable degrees 6/5	1.167989	3.962985	3.39×	3
generated factor product: dense high-height 1-variable degrees 17/16 total 33	0.008536	0.010490	1.23×	3
generated factorization: dense high-height 1-variable degrees 17/16 total 33	1.288667	1.258260	0.976×	3
generated factor product: dense 1-variable degrees 33/31 total 64	0.004909	0.004359	0.888×	3
generated factorization: dense 1-variable degrees 33/31 total 64	2.830984	2.521418	0.891×	3
generated factor product: dense 1-variable degrees 33/32 total 65	0.004689	0.004809	1.03×	3
generated factorization: dense 1-variable degrees 33/32 total 65	3.893394	2.916475	0.749×	3
polybench GCD products: polybench 5v uniform nontrivial GCD #11	0.129394	0.168248	1.3×	3
polybench GCD: polybench 5v uniform nontrivial GCD #11	7.084999	12.092959	1.71×	3
polybench GCD products: polybench 5v sharp nontrivial GCD #11	0.079169	0.087896	1.11×	3
polybench GCD: polybench 5v sharp nontrivial GCD #11	1.995003	5.049429	2.53×	3
polybench GCD products: polybench 8v uniform nontrivial GCD #11	0.133312	0.164831	1.24×	3
polybench GCD: polybench 8v uniform nontrivial GCD #11	5.050070	9.427429	1.87×	3
polybench GCD products: polybench 8v uniform nontrivial GCD #55	0.118314	0.134885	1.14×	3
polybench GCD: polybench 8v uniform nontrivial GCD #55	5.437924	9.051618	1.66×	3
polybench GCD products: polybench 8v uniform nontrivial GCD #56	0.100730	0.119566	1.19×	3
polybench GCD: polybench 8v uniform nontrivial GCD #56	2.962943	8.617557	2.91×	3
polybench GCD products: polybench 8v uniform nontrivial GCD #168	0.116189	0.138803	1.19×	3
polybench GCD: polybench 8v uniform nontrivial GCD #168	3.817670	8.812235	2.31×	3
polybench GCD products: polybench 8v uniform nontrivial GCD #188	0.101983	0.114456	1.12×	3
polybench GCD: polybench 8v uniform nontrivial GCD #188	2.908911	7.422547	2.55×	3
polybench GCD products: polybench 8v sharp nontrivial GCD #11	0.081343	0.082165	1.01×	3
polybench GCD: polybench 8v sharp nontrivial GCD #11	2.334775	7.204655	3.09×	3
polybench GCD products: polybench 8v sharp nontrivial GCD #53	0.086905	0.082987	0.955×	3
polybench GCD: polybench 8v sharp nontrivial GCD #53	2.182057	6.491988	2.98×	3
polybench GCD products: polybench 8v sharp nontrivial GCD #35	0.097956	0.109707	1.12×	3
polybench GCD: polybench 8v sharp nontrivial GCD #35	2.251798	7.551281	3.35×	3
polybench GCD products: polybench 8v sharp nontrivial GCD #140	0.087104	0.117802	1.35×	3
polybench GCD: polybench 8v sharp nontrivial GCD #140	2.212775	6.483231	2.93×	3
polybench GCD products: polybench 8v uniform trivial GCD #11	0.137651	0.161696	1.17×	3
polybench GCD: polybench 8v uniform trivial GCD #11	0.493914	0.710854	1.44×	3
polybench factor product: polybench 5v uniform nontrivial factor #32	0.067678	0.080332	1.19×	3
polybench factorization: polybench 5v uniform nontrivial factor #32	8.796085	90.729883	10.3×	3
polybench factor product: polybench 5v uniform nontrivial factor #159	0.050185	0.058711	1.17×	3
polybench factorization: polybench 5v uniform nontrivial factor #159	9.701908	95.595434	9.85×	3
polybench factor product: polybench 5v uniform nontrivial factor #163	0.065193	0.077477	1.19×	3
polybench factorization: polybench 5v uniform nontrivial factor #163	8.362635	45.060716	5.39×	3
polybench factor product: polybench 5v uniform nontrivial factor #131	0.046397	0.055885	1.2×	3
polybench factorization: polybench 5v uniform nontrivial factor #131	7.196851	42.763722	5.94×	3
polybench factor product: polybench 8v sharp nontrivial factor #178	0.045927	0.050225	1.09×	3
polybench factorization: polybench 8v sharp nontrivial factor #178	3.228276	10.461945	3.24×	3
polybench factor product: polybench 8v sharp nontrivial factor #92	0.049814	0.053651	1.08×	3
polybench factorization: polybench 8v sharp nontrivial factor #92	6.933533	13.893123	2×	3
polybench factor product: polybench 8v sharp nontrivial factor #84	0.042681	0.046688	1.09×	3
polybench factorization: polybench 8v sharp nontrivial factor #84	6.028520	20.768967	3.45×	3
polybench factor product: polybench 8v sharp nontrivial factor #176	0.063390	0.061557	0.971×	3
polybench factorization: polybench 8v sharp nontrivial factor #176	20.830033	14.741357	0.708×	3
polybench factor product: polybench 8v uniform nontrivial factor #44	0.047550	0.053401	1.12×	3
polybench factorization: polybench 8v uniform nontrivial factor #44	13.774660	202.301220	14.7×	3
polybench factor product: polybench 8v uniform nontrivial factor #105	0.062258	0.071435	1.15×	3
polybench factorization: polybench 8v uniform nontrivial factor #105	16.651610	34.794081	2.09×	3
polybench factor product: polybench 8v uniform nontrivial factor #159	0.052059	0.061466	1.18×	3
polybench factorization: polybench 8v uniform nontrivial factor #159	13.670001	729.897789	53.4×	3
polynomial/resultant/g33D-res-01-1715cd542f	0.102173	0.098616	0.965×	3
polynomial/discriminant/g33D-dis-05-eaa3ad4f21	0.795504	0.859956	1.08×	3
polynomial/resultant/g30base-res-01-81738fb00b	0.249461	0.256815	1.03×	3
polynomial/resultant/g30base-res-04-526f9a9d80	0.018234	0.025809	1.42×	3
polynomial/discriminant/g30base-dis-07-c740dd2c56	0.133793	0.136859	1.02×	3
polynomial/resultant/fixedA377-res-00-c3f6f2e500	0.685206	0.806606	1.18×	3
polynomial/resultant/fixedA377-res-01-db885cb0b3	0.433169	0.502861	1.16×	3
polynomial/resultant/fixedA377-res-02-a5fe517439	0.220527	0.229554	1.04×	3
polynomial/resultant/fixedA377-res-03-8f93b7ef68	0.228191	0.248981	1.09×	3
polynomial/discriminant/fixedA377-dis-06-0b154e382f	1.017885	1.256556	1.23×	3
polynomial/discriminant/fixedA377-dis-07-959550185e	0.229845	0.228081	0.992×	3
polynomial/discriminant/fixedA377-dis-08-cf10338d93	0.091603	0.085272	0.931×	3
polynomial/resultant/symA5_31-res-00-dee965d81f	0.009729	0.015178	1.56×	3
polynomial/resultant/symA5_31-res-01-9aba7cdcc3	0.010961	0.019638	1.79×	3
polynomial/resultant/symA5_31-res-02-ba5210d195	0.007384	0.014317	1.94×	3
polynomial/resultant/symA5_31-res-03-fc97cfbbef	0.010309	0.018996	1.84×	3
polynomial/resultant/g33S2-res-12-0bcabfb761	87.792417	142.555943	1.62×	3
polynomial/resultant/g33S2-res-13-9b5ef38ded	20.988072	32.878448	1.57×	3
polynomial/resultant/g33S2-res-17-fc148e65de	0.146236	0.172586	1.18×	3
polynomial/resultant/g33S2-res-19-21abb3fee9	0.017704	0.036599	2.07×	3
polynomial/discriminant/g33S2-dis-27-60776320bd	0.020679	0.023203	1.12×	3
polynomial/resultant/symA317-res-03-db3a4322d9	34.840118	56.043726	1.61×	3
polynomial/resultant/symA317-res-06-ff0b838eb1	0.074351	0.094279	1.27×	3
polynomial/resultant/symA317-res-07-c1b2d66c5c	0.042641	0.055966	1.31×	3
polynomial/resultant/symA317-res-08-4dd94dfa1e	0.002314	0.014528	6.28×	3
polynomial/resultant/g33S2-res-02-41f64e7684-segenc	10754.930289	11758.085403	1.09×	3
polynomial/factor/g33D-fac-08-967b2265b1	1.291292	2.364812	1.83×	3
polynomial/gcd/g33D-fac-08-967b2265b1	0.275811	0.958843	3.48×	3
polynomial/factor/g33D-fac-09-a4744cff24	0.123163	0.222461	1.81×	3
polynomial/factor/g33D-fac-10-abe6a8ebdd	0.285730	0.426837	1.49×	3
polynomial/gcd/g33D-fac-10-abe6a8ebdd	0.080562	0.243260	3.02×	3
polynomial/factor/g33D-fac-11-d2c3440e52	0.170844	0.622708	3.64×	3
polynomial/gcd/g33D-fac-11-d2c3440e52	0.014097	0.021241	1.51×	3
polynomial/factor/g33D-fac-12-124c41fcf7	0.054363	0.054303	0.999×	3
polynomial/factor/g30base-fac-10-de301fb47b	0.709812	0.415366	0.585×	3
polynomial/factor/g30base-fac-11-7421067c62	0.452656	0.263649	0.582×	3
polynomial/factor/g30base-fac-12-6a00291643	0.271173	0.353117	1.3×	3
polynomial/factor/g30base-fac-13-399b6b3505	0.247928	0.211891	0.855×	3
polynomial/factor/g30base-fac-14-45bf0db98e	0.065604	0.115629	1.76×	3
polynomial/factor/g30base-fac-15-a0ed6c27bf	0.018695	0.041999	2.25×	3
polynomial/factor/fixedA377-fac-10-ed80d981cd	0.639349	0.515725	0.807×	3
polynomial/factor/fixedA377-fac-11-79a6584b20	0.246536	0.266484	1.08×	3
polynomial/factor/fixedA377-fac-12-09df769d8f	0.138682	0.156887	1.13×	3
polynomial/factor/fixedA377-fac-13-86542ce6d5	0.930770	0.673463	0.724×	3
polynomial/factor/fixedA377-fac-14-ead5d84725	0.080582	0.109027	1.35×	3
polynomial/factor/fixedA377-fac-15-07877f0571	0.010129	0.021510	2.12×	3
polynomial/factor/symA5_31-fac-10-b7e40fd7bb	0.289978	0.641543	2.21×	3
polynomial/gcd/symA5_31-fac-10-b7e40fd7bb	0.009448	0.016000	1.69×	3
polynomial/factor/symA5_31-fac-11-b93dca9a03	0.175121	0.787930	4.5×	3
polynomial/gcd/symA5_31-fac-11-b93dca9a03	0.047720	0.369388	7.74×	3
polynomial/factor/symA5_31-fac-12-4262afaab8	0.102815	0.108035	1.05×	3
polynomial/factor/symA5_31-fac-13-b08fda434b	0.184539	0.265963	1.44×	3
polynomial/gcd/symA5_31-fac-13-b08fda434b	0.154913	0.808189	5.22×	3
polynomial/factor/symA5_31-fac-14-9ed62b27e5	0.008636	0.026851	3.11×	3
polynomial/factor/symA5_31-fac-15-7db2f4e614	0.013015	0.023925	1.84×	3
polynomial/factor/g33S2-fac-28-3a75cafd52	289.446154	1536.940330	5.31×	3
polynomial/gcd/g33S2-fac-28-3a75cafd52	266.477960	472.163566	1.77×	3
polynomial/factor/g33S2-fac-35-b4319ee36c	168.782971	299.600706	1.78×	3
polynomial/gcd/g33S2-fac-35-b4319ee36c	132.354732	228.783757	1.73×	3
polynomial/factor/g33S2-fac-36-b41f46e3ca	15.889087	16.547252	1.04×	3
polynomial/gcd/g33S2-fac-36-b41f46e3ca	7.933443	9.696256	1.22×	3
polynomial/factor/g33S2-fac-37-12e6b014e4	4.160629	6.619368	1.59×	3
polynomial/gcd/g33S2-fac-37-12e6b014e4	1.722207	4.814735	2.8×	3
polynomial/factor/g33S2-fac-38-fd912ec349	2.712078	2.566344	0.946×	3
polynomial/gcd/g33S2-fac-38-fd912ec349	0.975666	1.350765	1.38×	3
polynomial/factor/g33S2-fac-39-035774c28d	1.416438	1.644630	1.16×	3
polynomial/gcd/g33S2-fac-39-035774c28d	0.429042	0.842784	1.96×	3
polynomial/factor/g33S2-fac-40-2e7bf33ca7	0.657042	0.543618	0.827×	3
polynomial/gcd/g33S2-fac-40-2e7bf33ca7	0.242508	0.230346	0.95×	3
polynomial/factor/g33S2-fac-41-a2d1fd70cf	0.140896	0.184890	1.31×	3
polynomial/gcd/g33S2-fac-41-a2d1fd70cf	0.011511	0.021180	1.84×	3
polynomial/factor/symA317-fac-15-83da5119a0	2.485400	4.183572	1.68×	3
polynomial/gcd/symA317-fac-15-83da5119a0	1.998830	4.930143	2.47×	3
polynomial/factor/symA317-fac-16-37628a37c1	1.538278	6.071693	3.95×	3
polynomial/gcd/symA317-fac-16-37628a37c1	1.839368	7.651950	4.16×	3
polynomial/factor/symA317-fac-17-4521c9cb95	0.456152	1.147790	2.52×	3
polynomial/gcd/symA317-fac-17-4521c9cb95	0.054664	0.079831	1.46×	3
polynomial/factor/symA317-fac-18-b3e6617037	0.486460	1.750210	3.6×	3
polynomial/gcd/symA317-fac-18-b3e6617037	0.166174	0.914128	5.5×	3
polynomial/factor/symA317-fac-19-593adb827e	0.033814	0.064221	1.9×	3

Measurement and reproduction

Download the timings (CSV) · Download the plot (SVG) · Run metadata

Measured on 20 October 2026, on an AMD Ryzen 9 5900X running Linux x86-64. Both libraries use one thread, pinned to the same logical CPU. Each operation gets one untimed warmup and three measured samples; the plot shows the median. Symbolica-first and FLINT-first measurements alternate. Input construction, parsing, correctness checks, and dropping the output are outside the timer; allocation of the output is included.

Symbolica uses the native Rust workspace at 3bbffc2e (package version 3.0.1), compiled with Rust 1.99.0 and native CPU optimizations. FLINT is 3.6.0, linked through flint3-sys 3.6.1.

The suite covers generated dense and sparse inputs, integer and rational coefficients, finite fields, and fixtures from polybench (seed 42, generator revision f3a25498).

The polynomial fixtures cover 36 factorizations, 18 derivative-based GCDs, 25 multivariate resultants and discriminants, and one large segmentation-encoded resultant. Their inputs and source hashes are recorded in the committed fixture manifest. Fixture factorization and GCD, and the 12 rational GCD controls, use rational polynomials in Symbolica and integer polynomials in FLINT; GCDs are compared after monic normalization. Resultants and discriminants use integer inputs. The large encoded case uses FLINT’s univariate Euclidean resultant. All inputs, factor reconstructions, and exact outputs are checked before timing.

To reproduce, check out the revision above and use the benchmark source:

SYMBOLICA_FLINT_BENCH_PAIRED=1 \
SYMBOLICA_FLINT_BENCH_CSV=1 \
SYMBOLICA_FLINT_BENCH_SAMPLES=3 \
cargo bench --bench flint_comparison --features flint_benchmarks \
  > flint-comparison-results.csv

The flint_benchmarks feature builds its bundled FLINT. To match this run’s installed FLINT 3.6.0, use --features flint_system_benchmarks with that version available through pkg-config. On Linux, pin the benchmark process to a CPU with taskset -c 0. The complete paired results are retained, including cases where FLINT is faster; this workload selection is not an overall library ranking.

Numerical evaluation

Symbolica can turn large symbolic expressions into fast numerical code. These benchmarks compare the resulting native programs on two inputs: the F13 polynomial and a nested analytic expression containing polynomials, exponentials, sines, and cosines.

Evaluation time measures how fast the generated program runs. Generation time and peak memory measure the cost of building that program. Lower is better in all three plots. Generation time uses a logarithmic scale to show results ranging from fractions of a second to many minutes. Hatched memory bars marked ≥ show lower bounds observed before a build failed.

F13 polynomial

105,114 terms in 29 variables. Symbolica evaluates F13 1.39× as fast as FORM O4.

Exact measurements

Native program	Evaluation / point	Generate code	Peak build memory
Symbolica	5.64 µs	5.62 s	173 MiB
FORM O4	7.87 µs	3.3 min	813 MiB
FORM O2	8.96 µs	2.1 min	822 MiB
Mathematica	GCC compilation failed	GCC compilation failed	≥ 4,892 MiB

Mathematica’s generated C is very large and triggers an internal compiler error in GCC.

Nested analytic expression

A large expression in seven variables, containing polynomial products, exponentials, sines, and cosines.

Exact measurements

Native program	Evaluation / point	Generate code	Peak build memory
Symbolica · nested	3.63 µs	0.68 s	148 MiB
Mathematica · nested	33.92 µs	1.9 min	1,922 MiB
Symbolica · expanded	46.24 µs	1.1 min	2,233 MiB
Mathematica · expanded	GCC compilation failed	GCC compilation failed	≥ 15,674 MiB
FORM O4	61.66 µs	34.2 min	3,019 MiB
FORM CSEGreedy	72.38 µs	14.6 min	3,534 MiB

FORM cannot represent nested expressions and does not optimize arithmetic inside. The above demonstration shows that not being able to represent nested expressions comes at a large performance cost. For a more direct comparison, we expanded the Symbolica output before optimizing, and the resulting code is still faster than FORM’s output.

Mathematica’s original nested input compiled successfully, however the fully expanded input triggered a GCC internal compiler error after reaching ≥ 15,674 MiB of process-tree memory.

Arithmetic counts and generation stages

Input	Native program	Multiplications	Additions	Function calls	Evaluation / point
F13 polynomial	Symbolica	18,538	26,965	0	5.64 µs
F13 polynomial	FORM O4	17,536	27,314	0	7.87 µs
F13 polynomial	FORM O2	27,421	29,385	0	8.96 µs
Nested analytic expression	Symbolica	3,220	22,243	24	3.63 µs
Nested analytic expression	FORM O4	181,610	190,520	24	61.66 µs
Nested analytic expression	FORM CSEGreedy	199,019	193,707	24	72.38 µs
Nested analytic expression	Mathematica	92,007	27,622	24	33.92 µs
Nested analytic expression · expanded	Symbolica	162,371	193,916	24	46.24 µs
Nested analytic expression · expanded	FORM O4	181,610	190,520	24	62.35 µs
Nested analytic expression · expanded	FORM CSEGreedy	199,019	193,707	24	73.08 µs

FORM O4 uses fewer arithmetic operations on F13, while Symbolica’s native program evaluates faster in this measurement. Operation counts alone do not determine execution time. Multiplication counts include negations (sign changes) for every system. Integer powers also count as multiplications using repeated squaring: for example, \(x^4\) counts as two multiplications. Function calls are counted separately; a divisions column appears only when needed.

Symbolica input	Parse	Expand	Optimize	Native compile	Total build
F13 polynomial	0.23 s	0.00 s	5.51 s	0.11 s	5.86 s
Nested analytic expression	0.07 s	0.00 s	0.55 s	0.05 s	0.68 s
Nested analytic expression · expanded	0.07 s	6.26 s	59.71 s	0.90 s	66.91 s

Complete results · Build memory measurements · Additional constructor memory measurements, including 128 non-inlined functions with 90.5 MiB of expression bodies.

Run the examples

Download F13 or the nested analytic expression, the Python runner, and its reference points into one folder. Use the Symbolica source build identified above, with NumPy installed:

python run-output-example.py --case f13
python run-output-example.py --case nested
python run-output-example.py --case nested-expanded

The runner builds the native evaluator and checks it against 32 independent reference points. To use the nested expression directly:

from symbolica import E, S

expression = E(open("large_nested.txt").read())
parameters = [S(f"x{i}") for i in range(7)]
evaluator = expression.evaluator(
    parameters, iterations=10, n_cores=1,
    horner_use_cpe=True, jit_compile=False,
)
evaluator.jit_compile(True, direct_translation=True, optimization_level=3)
print(evaluator.evaluate([[0.125] * 7]))

For FORM, this complete input file generates the optimized arithmetic for the nested expression; save it beside large_nested.txt and run form benchmark.frm:

Symbols x0,x1,x2,x3,x4,x5,x6;
CFunctions exp,sin,cos;
Local F =
#include large_nested.txt
;
.sort
ExtraSymbols,array,w;
Format C;
Format O4;
#optimize F
#write <generated.inc> "double w[1+`optimmaxvar_'];"
#write <generated.inc> "%O"
#write <generated.inc> "double result = %e",F
.end

The full nested harness supplies the C++ wrapper and Mathematica Compile route; keep its shared helper beside it. Mathematica’s input uses the same expression with function-call syntax translated.

To reproduce the source build, apply the source patch to Symbolica commit 3bbffc2ecb32, use the supplied Cargo lockfile, and build a release wheel with maturin build --release --locked. The horner_use_cpe option ranks each Horner candidate after common-pair elimination; final pair elimination runs to completion. Ten candidates are tried per algebraic region.

Method, platform limitations, and raw data

Symbolica and paired FORM evaluation timings were measured on 20 October 2026, AMD Ryzen 9 5900X, Linux x86-64, pinned to logical CPU 10. Python 3.12.13, NumPy 2.5.4, and Symbolica development build 16e6fcfb. Symbolica uses scalar real SymJIT at optimization level three. The retained FORM 5.0.1 programs were compiled with GCC 16.2.1, -O3 -march=native, without fast-math. O4 uses its default 1,000 simulated-annealing iterations; O2 and CSEGreedy use their recorded defaults without a greedy time cutoff.

Each timing session warms up all programs and alternates their execution order. Batches contain 4,096 points for F13 and 1,024 for the nested expression. Output allocation is included; numerical verification is outside the timer. Every program passes the same independent 32-point check before and after timing, and Symbolica’s evaluator is also checked after serialization and reload. Speed ratios are medians of paired rounds; they can differ slightly from ratios of the plotted medians. These workloads are examples, not an overall system ranking.

Mathematica uses Wolfram Engine 13.2.0, Compile with expression optimization, the C target, listable real arguments, and no callbacks to the interpreter. Its original nested results are from 19 October on the same machine, measured through the native compiled function in the Wolfram kernel; they were not retimed in the interleaved Python session. FORM and Mathematica build costs are from the original 17–19 October runs. Raw records retain each run’s date, CPU affinity, compiler commands, and sampling details.

FORM expands products, so it cannot retain the original nested products in an ordinary expression. Its output optimizer also leaves arithmetic inside function arguments unoptimized, as demonstrated by this small example and its output. The expanded Symbolica case expands the outer expression and every elementary-function argument, without applying trigonometric identities.

Mathematica-generated C has repeatedly triggered GCC internal compiler errors on these large inputs. The F13 attempts failed with expression optimization off and on; the expanded nested attempt also failed. These are compilation failures, so no runtime is assigned to them. The completed original nested Mathematica runs are included above and in the original measurements.

F13 polynomial · Nested analytic expression · Nested analytic expression · expanded

The complete summary links the three builds for each input. All timing samples, program hashes, compiler settings, and validation results are retained in those files. Timing driver · Shared timing helper · Memory profiler · Arithmetic counter.

Memory efficiency

Compare the same expression before and after expansion:

\[ P_n = \prod_{i=1}^{n}(1+a+b+c+f(i)). \]

The factored comparison uses 100,000 factors, each with a distinct function argument. The expanded comparison uses 10 factors, producing 66,304 terms. These are separate workloads, not a before/after comparison at the same value of \(n\).

Factored product100,000 factors · sums remain nested

Expanded product10 factors · 66,304 expanded terms

For the expanded case, Symbolica’s specialized Polynomial representation is smaller and supports faster polynomial arithmetic than working with a general expression.

Mathematica and SymPy can share repeated subexpressions, recycling common parts instead of storing every occurrence separately. FORM’s native term representation stores repeated parts inline. Symbolica’s general expression representation also stores its packed payload directly, but the Rust API’s AliasedAtom lets you introduce explicit aliases for common subexpressions.

FORM does not support factored expressions, and therefore each factor had to be wrapped in a function. Due to maximum term limits, FORM 5.0.1 failed to construct the factored product with 100,000 factors. Generating the expanded product required careful treatment: a factor must be expanded one at a time with a .sort after every multiplication, to prevent a rapid growth of intermediate terms.

Measurement and reproduction

Factored-product measurements · Expanded-product measurements · Complete benchmark script

Measured on 16 October 2026, on an AMD Ryzen 9 5900X running Linux x86-64, pinned to one logical CPU. Software: Symbolica 3.0.2, FORM 5.0.1, Wolfram Engine 13.2.0, SymPy 1.14.0, and Python 3.12.13. These results apply to these workloads and versions, including the older Wolfram Engine available for this run.

The factored validator checks every sum and requires exactly the distinct indices \(1,\ldots,n\). FORM’s internal wrappers must preserve those same sums. The expanded validator checks the term count and an exact evaluation with a different value for every function call.

FORM’s 10,000-factor construction attempts failed, including attempts with larger buffers and a balanced construction. No FORM value is plotted for the 100,000-factor case; the raw data records the failed attempts.

The engine-specific counters are:

• Symbolica: expression.get_byte_size(), its packed expression payload, excluding the Python wrapper and global symbol definitions.
• FORM: uncompressed Bytes used with Off Compress;, cross-checked against sizeof_(Expression) and an in-memory dollar variable. FORM words occupy four bytes in this build.
• Mathematica: live kernel bytes released when deleting the expression, after ClearSystemCache[], with history disabled and an empty-expression control. Normal construction and construction followed by Share use separate fresh kernels. Shared branches count once. ByteCount, which ignores sharing, is recorded only as a diagnostic.
• SymPy: live Python allocations released after clearing SymPy caches, deleting the expression, and collecting garbage, measured with tracemalloc. Tracing starts before construction; an empty control corrects measurement overhead.

These are representation-storage counters, not total process RAM or peak memory. Startup, unused allocator reserves, construction workspaces, and external caches are excluded. All individual samples and failed FORM attempts are retained in the raw data.

Install uv and make an activated wolframscript and form available on your path. From the folder containing the downloaded script:

uv run expression-storage.py --case factored_indexed_functions --factors 100000 --engines symbolica mathematica sympy --runs 3 --output factored-product-storage-results.json
uv run expression-storage.py --case indexed_functions --factors 10 --runs 3 --output indexed-product-storage-results.json

The script installs pinned Python packages in an isolated environment. FORM construction limits are included in the script; the measured expression counter excludes those workspaces.