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How do computers calculate complicated trig functions such as tangent? I took a look at Intel's 8087 floating-point chip under the microscope to find out. The chip combines two techniques to evaluate the tan function quickly and accurately: polynomials and an algorithm called CORDIC...

Back in 1956, the Mach-2 B-58 Hustler had an analog navigation computer. It wasn't accurate enough, so they built a digital replacement. Sines and cosines are easy in analog, but needed a new digital algorithm. They invented the CORDIC algorithm and built the 134-pound CORDIC computer.

Usually, rotating a vector by an angle requires complicated math. But for some special angles, the calculation requires just shifts and adds, easy to do in hardware. The CORDIC algorithm breaks down an angle into special angles (using a table), does quick vector rotations, producing the trig answer.

The 8087 chip implements its algorithms in microcode, specialized low-level code inside the chip. By examining the microcode ROM under a microscope, the Opcode Collective extracted the bits and I reverse-engineered the algorithm.

For details on how the 8087 calculates tangents by using the CORDIC algorithm combined with polynomials, see my latest article: www.righto.com/2026/09/8087...

Credits: I've been working on this with the members of the "Opcode Collective", especially Smartest Blob and @gloriouscow.bsky.social, who used ML to convert ROM images to microcode data and extensively analyzed the contents. 7/7