C166 User's GuideC166 Introduction Compiling Programs Language Extensions Preprocessor Advanced Programming Sections, Classes, and Groups Special Sections Locating Sections Customization Files Startup Code STARTUP.A66 START167.A66 START_V2.A66 START_V3.A66 Basic I/O Memory Allocation TRAPS.C Optimizer Optimizer Options General Optimizations 80C166-Specific Optimizations Global Register Coloring DPP Registers Stacks System Stack User Stack Stack Sizes System Stack Size User Stack Size XBUS Memory and Peripherals On-chip XRAM XBUS Peripherals Interfacing C to Assembler Function Parameters Function Return Values Using the SRC Directive Register Usage Example Routines TINY Model SMALL Model COMPACT Model MEDIUM Model LARGE Model Data Storage Formats Bit Variables 1-byte Scalars 2-byte Scalars 4-byte Scalars float Scalars float Errors double Scalars double Errors near Pointers far Pointers huge Pointers xhuge Pointers Absolute Memory Access Absolute Access Macros Linker Location Controls Debugging Error Messages Library Reference Appendix
Scalars of type double are stored using eight bytes (64 bits). The format used corresponds to that of the IEEE-754 standard and is stored little endian or LSB first.
A double precision floating-point number is expressed as the product of two parts: the mantissa and a power of two. For example:
±mantissa × 2exponent
The mantissa represents the actual binary digits of the floating-point number.
The power of two is represented by the exponent. The stored form of the exponent is an 11-bit value from 0 to 2047. The actual value of the exponent is calculated by subtracting 1023 from the stored value (0 to 2047) giving a range of –1023 to +1024.
The mantissa is a 53-bit value (representing about thirteen decimal digits) whose most significant bit (MSB) is always 1 and is, therefore, not stored. There is also a sign bit that indicates whether the floating-point number is positive or negative.
Double precision floating-point numbers are stored on word boundaries in the following format:
Zero is a special value denoted with an exponent field of 0 and a mantissa of 0.
Using the above format, the floating-point number -12.5 is stored as a hexadecimal value of 0xC029000000000000. In memory, this value appears as follows:
It is fairly simple to convert floating-point numbers to and from their hexadecimal storage equivalents. The following example demonstrates how this is done for the value -12.5 shown above.
The floating-point storage representation is not an intuitive format. To convert this to a floating-point number, the bits must be separated as specified in the floating-point number storage format table shown above. For example:
From this illustration, you can determine the following:
There is an understood binary point at the left of the mantissa that is always preceded by a 1. This digit is omitted from the stored form of the floating-point number. Adding 1 and the binary point to the beginning of the mantissa gives the following value:
To adjust the mantissa for the exponent, move the decimal point to the left for negative exponent values or right for positive exponent values. Since the exponent is three, the mantissa is adjusted as follows:
The result is a binary floating-point number. Binary digits to the left of the decimal point represent the power of two corresponding to their position. For example, 1100 represents (1 × 23) + (1 × 22) + (0 × 21) + (0 × 20), which is 12.
Binary digits to the right of the decimal point also represent the power of two corresponding to their position. However, the powers are negative. For example, .100... represents (1 × 2-1) + (0 × 2-2) + (0 × 2-3) + ... which equals .5.
The sum of these values is 12.5. Because the sign bit was set, this number should be negative.
So, the hexadecimal value 0xC029000000000000 is -12.5.
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