Computer arithmetic

Computer arithmetic is the scientific field that deals with representation of numbers on computers and corresponding implementations of the arithmetic operations.

It includes:

• Fixed-point arithmetic
• Floating-point arithmetic
• Interval arithmetic
• Arbitrary-precision arithmetic
• Modular arithmetic
  • Multi-modular arithmetic
  • p-adic arithmetic, consisting of computing modulo a single prime number and retrieving the integer or rational result by using Hensel lifting
  • Finite field arithmetic
• Matrix arithmetic

In the cases where the size of the representation of a number is fixed (fixed-point, floating-point and interval arithmetic), the main concern is the control the computational error, as far as possible; see, for example IEEE 754.

In the other cases, where an exact result should be provided, the main concern is the practical efficiency, which is optimized by combining improvements of computational complexity with hardware specificities.