Going on a Tangent with the Intel 8087’s Hybrid CORDIC Algorithm
Ken Shirriff and colleagues reverse-engineered the Intel 8087 FPU to analyze its hybrid CORDIC algorithm for trigonometric functions. The implementation combines CORDIC iterations with Padé approximants to achieve 64-bit accuracy efficiently. This approach allowed the 8087 to balance speed and precision without requiring large lookup tables or excessive computation time.
- ▪The Intel 8087 FPU uses a hybrid algorithm that calculates the first 16 bits of trigonometric results using CORDIC.
- ▪The remaining bits are computed using Padé approximants, which are fast and accurate for small residual values.
- ▪Analysis shows that FPTAN operations spend approximately 33% of time on CORDIC pseudo-division and 47% on pseudo-multiplication.
- ▪Intel later abandoned CORDIC in the Pentium series due to scaling difficulties for high-precision calculations.
- ▪The full microcode listing for the 8087's trigonometric functions has been documented with comments by Ken Shirriff.
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Going On A Tangent With The Intel 8087’s Hybrid CORDIC Algorithm No comments by: Maya Posch September 27, 2026 Title: Copy Short Link: Copy Continuing their reverse-engineering of Intel’s 8087 FPU, [Ken Shirriff] and friends took a look at one of the trigonometric functions, specifically FPTAN. The most exciting part with such reverse-engineering is probably figuring out which algorithm was used in the implementation, while trying to determine the reasoning behind the final hardware design. If you’re running a simple MCU or MPU like the 6502 or Z80 without hardware functions you’d likely use an algorithm such as CORDIC or similar, as this requires only basic hardware features like addition, subtraction, bitshift, and look-up tables.
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