How to compute correctly rounded trigonometric functions in degrees?

Question

How could I define trigonometric functions that take arguments in degrees instead of the usual radians, and compute correctly rounded results for these arguments?

Multiplying the argument by M_PI/180.0 before passing it to the corresponding function in radians does not work, because M_PI/180.0 is not π/180. Section 5.5 of the Handbook of Floating-Point Arithmetic offers a method to compute the correctly rounded product of the argument by π/180, but some arguments will still be such that this product is close to the midpoint between two consecutive representable floats, and then applying even a correctly rounded function in radians can produce the wrong final result.

Two strategies that may work alone or in combination are using higher precision and using the sinpi,cospi, tanpi trigonometric functions from CRlibm, that compute respectively sin(πx), cos(πx) and tan(πx).

For the latter strategy, there remains the problem of the division by 180, which is not exact for many arguments.

Regarding the higher-precision strategy (multiplying the argument by an extended-precision representation of π/180, then applying the extended-precision function in radians), there may remain a problem with “exact” cases. The theorem that states that the only rational results of sin, cos and tan for a rational argument are obtained in 0 only applies to the radian versions. It obviously does not apply to the degree versions, and if for some floating-point input x, sindeg(x) is exactly the midpoint between two consecutive representable floating-point numbers, then no amount of intermediate precision is enough to guarantee that the final result is correctly rounded.

Answer 1

The only rationals q for which cosdeg(360q) is rational have 1, 2, 3, 4, or 6 as the denominator. This paper by Joerg Jahnel contains a short and beautiful proof using field theory in section 6. (Indeed, the author characterises the degree of the algebraic number cosdeg(360q) using Euler's totient function.) So there is no floating-point q such that cosdeg(360q) is halfway between two adjacent floating-point numbers.

So I guess the answer is "about the same way you implement sin and friends for radians," though @gnasher729 makes the excellent point that argument reduction for degrees is much, much nicer.