x87 trigonometric instructions

Question

The 8087 has instructions FPTAN and FPATAN, which are Partial Tangent and Partial Arctangent. The "partial" is presumably is to do with the range of the operands. For FPTAN the operand must be less than pi / 4 in absolute value.

Why was there this restriction, so why is it not pi / 2?

With the 80387, the FPTAN has to have an absolute value less than 2^63, but what is the reason for this number?

Finally, was there a reason that the 8087 did not include FCOS and FSIN?

Answer 1

The tan(x) function has a period of π radians, with asymptotes at ±π/2; it effectively calculates sin(x)/cos(x), and the latter goes to zero at those points. So a function which evaluates properly over that interval can be used for any angle, by first reducing the angle to the range supported.

However, accurately calculating the tangent function near the asymptotes is more difficult than near the origin. It can still be done by further reducing the input angle to the ±π/4 range and taking the reciprocal of the result. This works because sin(x) == cos(x - π/2) and both functions are periodic, which you can substitute into the identity above.

The 8087 FPTAN instruction was evidently designed to cover only the half-period immediately surrounding the origin, making the assumption that the programmer would reduce angles to that range before executing it. This is common with polynomial approximations, which are much easier to implement in an FPU than a fully accurate calculation.

The 80387 runs into a different limitation, as it clearly implements a reduction step to permit an extended range of inputs. But the double-extended format has a 63-bit significand, so reducing a value larger than 2^63 would leave no significant precision behind. Even as you approach that limit, the available precision reduces substantially in absolute terms.

While the 8087 did not include sine and cosine calculations, it is possible to derive them using the following identities:

1+Tan^2(a) = 1/Cos^2(a)
1+Cot^2(a) = 1/Sin^2(a)

Hence something like the following sequence would apply:

FPTAN
FMUL ST(0)  ; square the tangent
FLD1
FADDP       ; add 1
FSQRT       ; square root
FLD1
FDIVP       ; reciprocal

The above would produce the cosine, but not necessarily with the correct sign. This would have to be inferred from the quadrant of the original input (before reduction).

Obviously, obtaining the correct result from a single instruction is much faster and more convenient, which is why it was added to the 80387. In the earlier 8087, the space for microcode was decidedly limited, so functions that weren't strictly necessary were simply left out.

Full implementations of trigonometric functions usually start with a reduction step anyway, to improve the precision of the result by avoiding large values internal to the computation. These can still be implemented on the 8087 using the five elementary operations - addition, subtraction, multiplication, division, and square root.

Answer 2

The restrictions on the range of arguments the transcendental instructions are able to handle is a direct result of hardware resource limitations in these early floating-point units. The primary source for the implementation details of the transcendental instructions in the 8087 is:

Rafi Nave, "Implementation of transcendental functions on a numerics processor". Microprocessing and Microprogramming, Vol. 11, No. 3-4, March-April 1983, pp. 221-225

It states that the microcode size was limited to about 500 lines for all of the transcendentals combined. Because of limited hardware, the algorithms used are based on CORDIC for the initial steps, followed by rational approximation once the partial remainder has become sufficiently small.

To fit the ~30kbit microcode ROM into the chip at all with the transistor densities and die sizes available at the time (the 8087 contained more transistors than the 8086), Intel had to resort to a special four-state ROM, as described in

Rafi Nave and John Palmer, "A Numeric Data Processor". In 1980 IEEE International Solid-State Circuits Conference, Philadelphia, PA, USA, February 13-15, 1980, pp. 108-109

A second source, that pertains to the 80387 which relaxed several range restrictions on the transcendental instructions describes the same combination of CORDIC and rational approximation as was used in the 8087:

Alan K. Yuen, "Intel's Floating-Point Processors". Electro/88 Conference Record, Boston, MA, USA, May 10-12, 1988, pp. 48/5/1-7.

Palmer and Morse, who were architects of the 8086/8087, published a book on the math coprocessor:

John F. Palmer and Stephen P. Morse, "The 8087 Primer", John Wiley & Sons, 1984

In the chapter on the transcendental functions, they likewise cite the severe restrictions on microcode size as the reason for the range limitations of the built-in transcendental instructions of the 8087. The fundamental design idea was to support only the most time-consuming part of these computations in hardware and let programmers handle the argument reduction in software.

All trigonometric functions can be computed via FPTANand all inverse trigonometric functions can be computed via FPATAN, as shown in "The 8087 Primer", alleviating the need for direct hardware support given the severe hardware resource limitations in the 8087. For the 80387, more hardware resources were available, which allowed support for the FSINCOS instruction.

Answer 3

I remember that FPTAN didn't give the actual tg(x), but two results and you had to divide one by another in order to get tg(x) (and they were not sin(x) and cos(x) as one would hope).

Probably that's why it was "Partial".