User's Guide

Hardware-Specific Floating-Point Overview

Single- and Double-Precision Values

The PowerPC floating-point hardware performs calculations in either IEEE single-precision (equivalent to REAL(4) in Fortran programs) or IEEE double-precision (equivalent to REAL(8) in Fortran programs).

Keep the following considerations in mind:

• Double precision provides greater range (approximately 10**(-308) to 10**308) and precision (about 15 decimal digits) than single precision (approximate range 10**(-38) to 10**38, with about 7 decimal digits of precision).

• Computations that mix single and double operands are performed in double precision, which requires conversion of the single-precision operands to double-precision. These conversions do not affect performance.

• Double-precision values that are converted to single-precision (such as when you specify the SNGL intrinsic or when a double-precision computation result is stored into a single-precision variable) require rounding operations. A rounding operation produces the correct single-precision value which is based on the IEEE rounding mode in effect. The value may be less precise than the original double-precision value, as a result of rounding error. Conversions from double-precision values to single-precision values may reduce the performance of your code.

• Programs that manipulate large amounts of floating-point data may run faster if they use REAL(4) rather than REAL(8) variables. (You need to ensure that REAL(4) variables provide you with acceptable range and precision.) The programs may run faster because the smaller data size reduces memory traffic, which can be a performance bottleneck for some applications.

The floating-point hardware also provides a special set of double-precision operations that multiply two numbers and add a third number to the product. These combined multiply-add (MAF) operations are performed at the same speed as either a multiply or an add operation alone is performed. The MAF functions provide an extension to the IEEE standard because they perform the multiply and add with one (rather than two) rounding errors. The MAF functions are faster and more accurate than the equivalent separate operations.

Extended-Precision Values

XL Fortran extended precision is not in the format suggested by the IEEE standard, which suggests extended formats using more bits in both the exponent (for greater range) and the fraction (for greater precision).

XL Fortran extended precision, equivalent to REAL(16) in Fortran programs, is implemented in software. Extended precision provides the same range as double precision (about 10**(-308) to 10**308), but more precision (a variable amount, about 31 decimal digits or more). The software support is restricted to round-to-nearest mode. Programs that use extended precision must ensure that this rounding mode is in effect when extended-precision calculations are performed. See Selecting the Rounding Mode for the different ways you can control the rounding mode.

Programs that specify extended-precision values as hexadecimal, octal, binary, or Hollerith constants must follow these conventions:

• Extended-precision numbers are composed of two double-precision numbers with different magnitudes that do not overlap. That is, the binary exponents differ by at least the number of fraction bits in a REAL(8). The high-order double-precision value (the one that comes first in storage) must have the larger magnitude. The value of the extended-precision number is the sum of the two double-precision values.

• For a value of NaN or infinity, you must encode one of these values within the high-order double-precision value. The low-order value is not significant.

Because an XL Fortran extended-precision value can be the sum of two values with greatly different exponents, leaving a number of assumed zeros in the fraction, the format actually has a variable precision, with a minimum of about 31 decimal digits. You get more precision in cases where the exponents of the two double values differ in magnitude by more than the number of digits in a double-precision value. This encoding allows an efficient implementation intended for applications requiring more precision but no more range than double precision.

Notes:

1. In the discussions of rounding errors because of compile-time folding of expressions, keep in mind that this folding produces different results for extended-precision values more often than for other precisions.
2. Special numbers, such as NaN and infinity, are not fully supported for extended-precision values. Arithmetic operations do not necessarily propagate these numbers in extended precision.
3. XL Fortran does not always detect floating-point exception conditions (see Detecting and Trapping Floating-Point Exceptions) for extended-precision values. If you turn on floating-point exception trapping in programs that use extended precision, XL Fortran may also generate signals in cases where an exception condition does not really occur.