High Performance Computing - Charles Severance [22]
The solution is to add extra guard digits which are maintained during the interim steps of the computation. In our case, if we maintained six digits of accuracy while aligning operands, and rounded before normalizing and assigning the final value, we would get the proper result. The guard digits only need to be present as part of the floating-point execution unit in the CPU. It is not necessary to add guard digits to the registers or to the values stored in memory.
It is not necessary to have an extremely large number of guard digits. At some point, the difference in the magnitude between the operands becomes so great that lost digits do not affect the addition or rounding results.
History of IEEE Floating-Point Format*
History of IEEE Floating-Point Format
Prior to the RISC microprocessor revolution, each vendor had their own floating- point formats based on their designers’ views of the relative importance of range versus accuracy and speed versus accuracy. It was not uncommon for one vendor to carefully analyze the limitations of another vendor’s floating-point format and use this information to convince users that theirs was the only “accurate” floating- point implementation. In reality none of the formats was perfect. The formats were simply imperfect in different ways.
During the 1980s the Institute for Electrical and Electronics Engineers (IEEE) produced a standard for the floating-point format. The title of the standard is “IEEE 754-1985 Standard for Binary Floating-Point Arithmetic.” This standard provided the precise definition of a floating-point format and described the operations on floating-point values.
Because IEEE 754 was developed after a variety of floating-point formats had been in use for quite some time, the IEEE 754 working group had the benefit of examining the existing floating-point designs and taking the strong points, and avoiding the mistakes in existing designs. The IEEE 754 specification had its beginnings in the design of the Intel i8087 floating-point coprocessor. The i8087 floating-point format improved on the DEC VAX floating-point format by adding a number of significant features.
The near universal adoption of IEEE 754 floating-point format has occurred over a 10-year time period. The high performance computing vendors of the mid 1980s (Cray IBM, DEC, and Control Data) had their own proprietary floating-point formats that they had to continue supporting because of their installed user base. They really had no choice but to continue to support their existing formats. In the mid to late 1980s the primary systems that supported the IEEE format were RISC workstations and some coprocessors for microprocessors. Because the designers of these systems had no need to protect a proprietary floating-point format, they readily adopted the IEEE format. As RISC processors moved from general-purpose integer computing to high performance floating-point computing, the CPU designers found ways to make IEEE floating-point operations operate very quickly. In 10 years, the IEEE 754 has gone from a standard for floating-point coprocessors to the dominant floating-point standard for all computers. Because of this standard, we, the users, are the beneficiaries of a portable floating-point environment.
IEEE Floating-Point Standard
The IEEE 754 standard specified a number of different details of floating-point operations, including:
Storage formats
Precise specifications of the results of operations
Special values
Specified runtime behavior on illegal operations
Specifying the floating-point format to this level of detail insures that when a computer system is compliant with the standard, users can expect repeatable execution from one hardware platform to another when operations are executed in the same order.
IEEE Storage Format
The two most common IEEE floating-point formats