不要用泰勒级数。切比雪夫多项式更快更准确，正如上面几个人指出的那样。下面是一个实现(最初来自ZX Spectrum ROM): https://albertveli.wordpress.com/2015/01/10/zx-sine/

OK kiddies, time for the pros.... This is one of my biggest complaints with inexperienced software engineers. They come in calculating transcendental functions from scratch (using Taylor's series) as if nobody had ever done these calculations before in their lives. Not true. This is a well defined problem and has been approached thousands of times by very clever software and hardware engineers and has a well defined solution. Basically, most of the transcendental functions use Chebyshev Polynomials to calculate them. As to which polynomials are used depends on the circumstances. First, the bible on this matter is a book called "Computer Approximations" by Hart and Cheney. In that book, you can decide if you have a hardware adder, multiplier, divider, etc, and decide which operations are fastest. e.g. If you had a really fast divider, the fastest way to calculate sine might be P1(x)/P2(x) where P1, P2 are Chebyshev polynomials. Without the fast divider, it might be just P(x), where P has much more terms than P1 or P2....so it'd be slower. So, first step is to determine your hardware and what it can do. Then you choose the appropriate combination of Chebyshev polynomials (is usually of the form cos(ax) = aP(x) for cosine for example, again where P is a Chebyshev polynomial). Then you decide what decimal precision you want. e.g. if you want 7 digits precision, you look that up in the appropriate table in the book I mentioned, and it will give you (for precision = 7.33) a number N = 4 and a polynomial number 3502. N is the order of the polynomial (so it's p4.x^4 + p3.x^3 + p2.x^2 + p1.x + p0), because N=4. Then you look up the actual value of the p4,p3,p2,p1,p0 values in the back of the book under 3502 (they'll be in floating point). Then you implement your algorithm in software in the form: (((p4.x + p3).x + p2).x + p1).x + p0 ....and this is how you'd calculate cosine to 7 decimal places on that hardware.

请注意，在FPU中大多数硬件实现的超越操作通常涉及一些微码和类似的操作(取决于硬件)。 切比雪夫多项式用于大多数先验多项式，但不是全部。例:使用Newton raphson方法的两次迭代，首先使用查询表，使用平方根更快。 同样，《计算机逼近》这本书会告诉你。

If you plan on implmementing these functions, I'd recommend to anyone that they get a copy of that book. It really is the bible for these kinds of algorithms. Note that there are bunches of alternative means for calculating these values like cordics, etc, but these tend to be best for specific algorithms where you only need low precision. To guarantee the precision every time, the chebyshev polynomials are the way to go. Like I said, well defined problem. Has been solved for 50 years now.....and thats how it's done.

Now, that being said, there are techniques whereby the Chebyshev polynomials can be used to get a single precision result with a low degree polynomial (like the example for cosine above). Then, there are other techniques to interpolate between values to increase the accuracy without having to go to a much larger polynomial, such as "Gal's Accurate Tables Method". This latter technique is what the post referring to the ACM literature is referring to. But ultimately, the Chebyshev Polynomials are what are used to get 90% of the way there.

享受。

使用泰勒级数，试着找出级数项之间的关系这样你就不用一遍又一遍地计算了

下面是一个关于余窦的例子:

double cosinus(double x, double prec)
{
    double t, s ;
    int p;
    p = 0;
    s = 1.0;
    t = 1.0;
    while(fabs(t/s) > prec)
    {
        p++;
        t = (-t * x * x) / ((2 * p - 1) * (2 * p));
        s += t;
    }
    return s;
}

使用这个，我们可以得到新的和项使用已经使用的和项(我们避免阶乘和x2p)

像正弦和余弦这样的函数是在微处理器内部的微码中实现的。例如，英特尔芯片就有相应的组装指令。C编译器将生成调用这些汇编指令的代码。(相反，Java编译器不会。Java在软件而不是硬件中计算三角函数，因此运行速度要慢得多。)

芯片不使用泰勒级数来计算三角函数，至少不完全是这样。首先，他们使用CORDIC，但他们也可能使用一个短的泰勒级数来优化CORDIC的结果，或者用于特殊情况，例如在非常小的角度下以相对较高的精度计算正弦。有关更多解释，请参阅StackOverflow的回答。

如果你想要一个软件实现，而不是硬件实现，可以在《数值公式》的第5章中找到这个问题的明确答案。我的副本在一个盒子里，所以我不能给出细节，但简短的版本(如果我没记错的话)是你把tan(theta/2)作为你的基本操作，然后从那里计算其他的。计算是用级数近似完成的，但它比泰勒级数收敛得快得多。

抱歉，我没拿到书就想不起来了。

正如许多人指出的那样，它依赖于实现。但就我对你的问题的理解而言，你对数学函数的真正软件实现感兴趣，但只是没有找到一个。如果是这样的话，那么你是这样的:

从http://ftp.gnu.org/gnu/glibc/下载glibc源代码 查看位于解包的glibc根\sysdeps\ieee754\dbl-64文件夹中的文件dosincosc 类似地，您可以找到其余数学库的实现，只需查找具有适当名称的文件

您也可以看看扩展名为.tbl的文件，它们的内容只不过是以二进制形式的不同函数的预计算值的巨大表格。这就是为什么实现如此之快:而不是计算他们使用的任何级数的所有系数，他们只是做一个快速查找，这要快得多。顺便说一下，他们确实用裁缝级数来计算正弦和余弦。

我希望这能有所帮助。