http://www.mitchr.me/SS/mjrcalc/

What kinds things can all of these LISP functions do? That's a hard question to answer directly, so I'll just list some of the things I have used it for over the years:

- Development and prototyping of deeply algorithmic and/or mathematical software
- Numerical analysis
- linear algebra
- integration
- ODEs
- PDEs
- Optimization
- Root finding

- Dynamical systems modeling and simulation (ODEs, PDEs, and algebraic equations)
- Efficient solutions to Kepler's equation
- High speed orbital dynamics (finding earth satellites and planets with my telescope in real time)
- High accuracy integration of planetary systems
- Large scale particle simulation (Galactic dynamics, fungal growth, fountains, ...)
- Diffusion simulations
- Mathematical Population dynamics

- Symbolic algebra
- Computational commutative algebra (solving polynomial systems, Grobner basis, etc...)
- Non-numeric root localization
- Rational and integer roots and critical points
- Combined symbolic and numerical algorithms
- Factorization of polynomials over the integers and prime order fields
- Symbolic differentiation of LISP forms
- Analytical solution of differential equations

- Combinatorial enumeration and counting algorithms
- Most of the 12-fold way
- Several combinatorial functions
- Generation of combinatorial objects (sets, cross products, permutations, combinations)

- Computational group theory
- Electron orbital mechanics
- Crystal lattice formation

- Probabilistic modeling and simulation (complex systems or statistical tests)
- Sophisticated data visualization problems
- Project Euler problems