Erable and alg48.

Erable is partially derived from the alg48 package. The arithmetic functions of Erable are derived from those of alg48. Erable and alg48 have some other common features like simplifications, partial fraction expansion or rational integration. The main differences are:

• Erable is most of the time slower (about 2 times I would say, this is only a mean, sometimes alg48 is 3 times faster than Erable, sometimes Erable is 1 to 2 faster than alg48). Why? Because Erable handles complexes and square roots natively: e.g. Erable simplifies expressions like $(1+i)/(1+2i)=\frac{3-i}{5}$, $(1+\sqrt{2})/(1-2\sqrt{2})=-\frac{5}{7}+\frac{-3}{7}\sqrt{2}$, $e^{5i\pi/6}=-\frac{1}{2}\sqrt{3} +\frac{i}{2}$. Hence arithmetic operations in Erable must be generic, that's why they are slower. On the other hand, Erable treats matrices as global objects for simplifications, as alg48 simplifies matrices element by element, hence Erable may be faster if matrices are involved.
• Erable accepts strings embedded in symbolics, this means that if you EXPAnd (5x+12)¹⁶ with Erable you'll get the exact answer. You may also do arithmetic operations on strings representing integers (e.g. try "123456789123456789" DUP MULT)
• Erable handles numerical data (reals which are not integers and complexes with non integral real and/or imaginary parts)
• Erable has a partial implementation of the Risch integration algorithm: it handles most of the common integrals, not only rational fractions.
• alg48 (version 4) implements the complete factorization algorithm over the rationals, Erable finds only first order factors of the square-free factorization and then, for 1-variable polynomials, calls the numeric solver if necessary and tries to rebuild 2nd order factor. Hence, alg48 factors the expanded form of (x⁴+x³+1)(x⁴+x+1) but not x⁴+1 and Erable does not factor the first example but factors the second one.
• The main specific feature of alg48 is the Gröbner base computation. The main specifics features of Erable are eigenvalues and eigenvectors of matrices, differential equations (first order: linear, Bernouilli, homogenous; linear with constant coefficients), limits and Taylor series, quadratic forms, permutations, variable substitutions.

If you have enough memory, do like me: keep both on your calc and choose the right instruction!