Detailed explanation of floating-point numbers in Z/OS: HFP, BFP and DFP

In the z/OS operating system, the floating-point number type is divided into binary floating-point number schema BFP, hexadecimal floating-point architecture hfp and decimal floating-point architecture DFP three categories. IBM has established HFP as its base floating point architecture since 1964 s/360 system, 1998 S/390 system formally introduced BFP, that is ieee754-1985 Standard, to 2007 Z9 system began to support DFP, namely ieee754-2008 standard. This paper introduces the coding principles and properties of these three kinds of floating point numbers, helps readers to choose the appropriate floating-point number types in the application in Z/OS, and reads binary coded floating-point numbers in debugging.

Basic knowledge of floating-point architecture

In computer systems, floating-point numbers are a technical scheme for representing real numbers in a larger range. Usually a floating-point number is represented as the mantissa m multiplied by several integer power E of base B, which is very similar to the scientific notation commonly used in mathematics.

M * be

The specific floating-point architecture specifies the numbering scheme of mantissa and exponent. Cardinality is implied in the schema, for example, the cardinality of the IEEE-754 binary floating-point number format is 2,IBM, the base of the hexadecimal floating-point number format is 16. The mantissa is a fixed number of digits (in cardinality) and the exponent is generally encoded using an offset method. Below in the specific floating-point architecture, we use m,e to separate the m,e after the encoding. ,... with M0,M1,M2 Represents each bit from left to right in M, with | M| represents the number of bits m. e does the same.

In addition to the coding scheme, the floating-point architecture also prescribes the following, the basic operation of the calculation, the approximation of the computation under the finite precision, the special floating-point type, and the exception handling in the calculation. In this paper, we will focus on the coding scheme of floating point numbers for three kinds of floating-point number architectures in Z/OS. Floating-point calculation instructions and exception handling will be outside the scope of this article.

Hexadecimal floating-point number HFP

A HFP floating-point number consists of a bit of sign bit, a binary exponent of the seven bits, and the remainder is a 16-digit tail part. The exponent, which is called characteristic, is the index encoded according to the migration method, and the values 0 to 127 represent 16-64 to 16+63 respectively. In the hexadecimal mantissa, the decimal point is at the far left of the mantissa, representing a number of the mantissa, from the highest to the lowest bit, each four bits.

The HFP consists of 32-bit, 64-bit, and 128-bit three formats, as shown in Figure 1:

Figure 1:HFP floating-point numbering format [2]

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