In 1980, Intel introduced the 8087, a chip that made floating-point operations much faster in the IBM PC and other systems. Although the 8087 supports advanced operations such as tangents, square roots, and logarithms, this article focuses on basic arithmetic operations: addition and subtraction. The 8087 provided a spectacular speed improvement over the 8086 processor, speeding up addition by a factor of almost 100, and complicated operations by even more.
To explore the 8087's circuitry, I popped the lid off a chip with a chisel and used a microscope to create the high-resolution image below. The microcode ROM is the large rectangular region in the center of the die, holding the 1648 micro-instructions that control the chip. The bottom half of the chip (red box) is the datapath, the circuitry that performs floating-point calculations on 80-bit values.
A close-up of the 8087's datapath shows the functional blocks that are used for addition and subtraction. The key component is the integer adder, located in the center. It receives one argument from the B register and another argument from a component such as the temporary registers or the stack registers. The adder's output is stored in the sum register. The shifter can shift the bits in a value to the left or the right. The shifter occupies a lot of space on the die because its circuitry allows a value to be rapidly shifted by an arbitrary number of bits. Floating-point operations make heavy use of the shifter to align values; the leading zero circuitry counts the number of zero bits or bytes on the left of a value so they can be shifted away. The exponent converter processes exponents, detecting overflows and underflows. These specialized circuits helped the 8087 achieve its high performance.
Most computers use integers as their basic representation for numbers. But many applications need to use numbers with a fractional part, such as 1.23. Moreover, integers have a very limited range: a 32-bit integer can express numbers from about -2 billion to 2 billion. To solve these problems, computers usually implement floating-point numbers, which provide fractional numbers over an enormous range. In most cases, a floating-point number is represented by sign × significand × 2^exponent. The significand is a binary number of the form 1.bbb…: a leading 1, followed by the binary point (the binary equivalent of the decimal point) and the rest of the bits.
In a binary number, each bit represents a power of 2. For the fractional part, the part after the binary point, the powers of 2 are negative: 2^-1, 2^-2, and so forth. So the binary number 1.1 is 1 + 1/2, or 1.5 in decimal. Similarly, binary 1.0011 is 1 + 1/8 + 1/16 or 1.1875. The exponent allows numbers to escape the limited range of the significand. For instance, 1.1 (binary) × 2^8 is 384. With larger exponents, floating-point numbers can cover a huge range; even a 32-bit short floating-point number covers a range exceeding 10^-36 to 10^38.
Floating-point addition and subtraction are more complicated than integer operations. Before the numbers can be added (or subtracted), their binary points must be aligned, the same as aligning the decimal points for decimal numbers. After the operation, the exponent of the result must be adjusted, normalizing the significand so it starts with a 1 bit. The steps for a simple floating-point addition. The diagram above illustrates a simple floating-point addition: 13.5 + 3.5. The first two lines show the two arguments expressed with an exponent and significand. Both arguments are normalized, so the leftmost bit is a 1. Before adding, the second argument must be aligned—ma...
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