The hardware used in ordinary deep learning uses a data format called 32-bit

floating-point representation for processing calculations.

Section 3, presents a detailed description of our algorithm and a brief discussion about the floating-point representation.

n], a binary64 floating-point representation is used as shown in Figure 1 The algorithmic principle is simple and consists at each iteration, to apply a xor operation on the 32 bits of mantissa 1 of the three output elements [X.

Among the topics are mathematical preliminaries and

floating-point representation, numerical integration, least squares methods and Fourier series, boundary-value problems, and linear programming problems.

If students are not aware of 64-bit floating-point representation at this point in their study, then the next step usually involves the default 32-bit floating-point representation.

When the programming assignment was returned, the 64-bit floating-point representation was discussed and the correct values revealed.

Floating-point representation of data has a smaller amount of probable error and noise.

Not infrequently in that era, hardware (and, especially, double-precision software) was sloppy enough to require penalizing by several bits from what one would naively guess by counting bits in the

floating-point representation.

The concern is about scientific applications using

floating-point representation, and justifiably so.

2 A common definition of the relative machine precision, or unit roundoff, is the smallest positive floating-point value, [Epsilon], such that fl(1 + [Epsilon]) [greater than] 1, where fl(x) is the

floating-point representation of x.

If you are not a floating-point expert, Micromega provides a utility that converts back and forth between hexadecimal and

floating-point representations.

Some specific topics include FPGA particle graphics hardware, hardware factorization based on the elliptic curve method, higher radix

floating-point representations for FPGA-based arithmetic, and interleaving behavioral and cycle-accurate descriptions for reconfigurable hardware compilation.