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Comment by simonbyrne | Hacker News Reader

I'd be keen to know more about how the transcendental instructions worked.

I plan to write about that, but the short answer is that the 8087 computes transcendental functions using CORDIC. The chip contains constants atan(2^0), ..., atan(2^-15). It operates a bit at a time using these constants to compute tan and arctan. It doesn't natively support sine and cosine, but these can be computed from the tangent.

Likewise, the constants log2(1+2^(-2)), ..., log2(1+2^(-15)) are used for logarithm and exponentiation.

https://en.wikipedia.org/wiki/CORDIC

Edit: the constant ROM is the vertical stripe in the lower left in the die photo. You can see each bit in the ROM as a green or purple rectangle: https://lh3.googleusercontent.com/-7S_Q0bzu-7A/W3RDsZwprvI/A...

> It doesn't natively support sine and cosine, but these can be computed from the tangent.

I'd not ever looked into it, but that neatly explains the existence of the sincos instruction.

sincos is useful in general. Rotations for example often need sine and cosine of the same angle.

Because transcendental functions are implemented as a microcode loop, a sincos instruction lets you cut in half the loop overhead. If the overhead were 50% (e.g. 100 cycles arithmetic and 50 cycles overhead) you could cut sincos to 250 cycles, for an overhead of only 25%.

CORDIC is itself a very interesting algorithm due to its conceptual simplicity: a vector rotation using successive approximations. When stripped down to its essentials, it forms the basis of a surprisingly simple circle-drawing algorithm:

https://nbickford.wordpress.com/2011/04/03/the-minsky-circle...

(Previously also discussed here at https://news.ycombinator.com/item?id=15266331 )

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