Ken Shirriff reverse-engineers the Intel 8087 tangent instruction and finds CORDIC combined with a Pade approximant

Sources: primary, discussion
Summary: The post explains why FPTAN uses two algorithms rather than one. Sixteen CORDIC terms give about sixteen bits, so the 8087 covers the tiny residual angle with the rational approximation 3x/(3-x^2), whose error is proportional to x^4 and therefore under 2^-64 for a residual below 2^-16, reaching 64-bit accuracy without 64 CORDIC terms. The account is grounded in the decapped die and the microcode listing, which shows the pseudo-division phase recording decision bits into a 16-bit shift register and the pseudo-multiplication phase shifting them out in reverse so the smallest rotation is applied first to limit rounding error.
Why it matters: The division in that ratio costs nothing because FPTAN returns a separate numerator and denominator rather than the tangent, and the chip computed a tangent in 90 microseconds against 13,000 on the 8086.