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Reply to @kenshirriff@oldbytes.space
Ken Shirriff@kenshirriff@oldbytes.space (2026-09-27 02:04:21)
CORDIC became popular, especially in scientific calculators. The 8087 chip uses CORDIC, but goes further, also using a polynomial ratio to get more accuracy. The simple function 3x/(3-x^2) (blue) is surprisingly close to the tangent (red), better than a Taylor series (green).
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Ken Shirriff@kenshirriff@oldbytes.space (2026-09-27 02:05:06)
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.
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image: https://assets.oldbytes.space/assets.oldbytes.space/media_attachments/files/117/338/425/014/611/055/original/5ce63607afd9ba14.png
image: https://assets.oldbytes.space/assets.oldbytes.space/media_attachments/files/117/338/426/271/805/716/original/59e1d6ad620678a6.png

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Ken Shirriff@kenshirriff@oldbytes.space (2026-09-27 02:05:29)
For details on how the 8087 calculates tangents by using the CORDIC algorithm combined with polynomials, see my latest article:
https://www.righto.com/2026/09/8087-tangent-cordic.html