● Covered by 1 source · 1 reportLow impact1 neutral
Intel 8087 Floating-Point Chip's Tangent Algorithm Explained
Key points
- Intel 8087's FPTAN algorithm uses CORDIC and polynomial approximation.
- The 8087 computed tangent in 90 microseconds, 8086 took 13,000 microseconds.
- Reverse-engineering involved examining chip circuitry and microcode.
- CORDIC algorithm uses shifts, adds, and table lookups without multiplication.
8087's Tangent Algorithm Revealed
The Intel 8087 floating-point chip, introduced in 1980, significantly improved floating-point operation speeds. Its tangent instruction, FPTAN, utilizes a hybrid algorithm combining CORDIC and polynomial approximation. This approach allowed the 8087 to achieve both high accuracy and performance in trigonometric calculations.
The 8087 could compute a tangent in 90 microseconds, a substantial improvement over the 13,000 microseconds required by the 8086 microprocessor.
Reverse-Engineering Methodology
The algorithm was uncovered by reverse-engineering the 8087's circuitry and microcode. This process involved physically opening a chip and creating a high-resolution image using a microscope. The microcode ROM, containing 1648 micro-instructions, and the datapath, responsible for 80-bit floating-point calculations, were key areas of examination.
Internal Components and CORDIC
Key functional units identified in the datapath include the exponent ROM, constant ROM (holding CORDIC constants), a shifter, an adder, and various registers. The adder is central to calculations, performing addition, subtraction, and iterative operations for multiplication, division, and square roots. A shift register holds 16 status bits specifically for CORDIC calculations.
The CORDIC algorithm, developed in 1956, is efficient for transcendental functions, relying on shifts, additions, and table lookups without requiring multiplication or division hardware.
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Primary sources
GitHub a-mcego/graniteReporting from
Analysis of the Intel 8087 floating-point chip reveals its tangent instruction (FPTAN) algorithm combines CORDIC and polynomial approximation for accuracy and performance. This reverse-engineering effort involved examining the chip's circuitry and microcode to understand its internal operations. The 8087 significantly accelerated floating-point calculations compared to the 8086 microprocessor.