是的，也有计算罪恶的软件算法。基本上，用数字计算机计算这些东西通常是用数值方法来完成的，比如近似表示函数的泰勒级数。

数值方法可以将函数近似到任意精度，因为浮点数的精度是有限的，所以它们非常适合这些任务。

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/

计算正弦/余弦/正切其实很容易通过代码使用泰勒级数来实现。自己写一个只需5秒钟。

整个过程可以用这个方程来概括:

下面是我为C语言写的一些例程:

double _pow(double a, double b) {
    double c = 1;
    for (int i=0; i<b; i++)
        c *= a;
    return c;
}

double _fact(double x) {
    double ret = 1;
    for (int i=1; i<=x; i++) 
        ret *= i;
    return ret;
}

double _sin(double x) {
    double y = x;
    double s = -1;
    for (int i=3; i<=100; i+=2) {
        y+=s*(_pow(x,i)/_fact(i));
        s *= -1;
    }  
    return y;
}
double _cos(double x) {
    double y = 1;
    double s = -1;
    for (int i=2; i<=100; i+=2) {
        y+=s*(_pow(x,i)/_fact(i));
        s *= -1;
    }  
    return y;
}
double _tan(double x) {
     return (_sin(x)/_cos(x));  
}

如果你想犯罪

 __asm__ __volatile__("fsin" : "=t"(vsin) : "0"(xrads));

如果你想的话，因为

 __asm__ __volatile__("fcos" : "=t"(vcos) : "0"(xrads));

如果你想要根号方根

 __asm__ __volatile__("fsqrt" : "=t"(vsqrt) : "0"(value));

那么，既然机器指令可以做到，为什么还要使用不准确的代码呢?

库函数的实际实现取决于特定的编译器和/或库提供程序。不管它是用硬件还是软件，不管它是不是泰勒展开，等等，都会有所不同。

我意识到这完全没有帮助。

切比雪夫多项式，正如在另一个答案中提到的，是函数和多项式之间的最大差异尽可能小的多项式。这是一个很好的开始。

在某些情况下，最大误差不是你感兴趣的，而是最大相对误差。例如，对于正弦函数，x = 0附近的误差应该比较大的值小得多;你想要一个小的相对误差。所以你可以计算sinx / x的切比雪夫多项式，然后把这个多项式乘以x。

Next you have to figure out how to evaluate the polynomial. You want to evaluate it in such a way that the intermediate values are small and therefore rounding errors are small. Otherwise the rounding errors might become a lot larger than errors in the polynomial. And with functions like the sine function, if you are careless then it may be possible that the result that you calculate for sin x is greater than the result for sin y even when x < y. So careful choice of the calculation order and calculation of upper bounds for the rounding error are needed.

例如，sinx = x - x^3/6 + x^5 / 120 - x^7 / 5040…如果你天真地计算sinx = x * (1 - x^2/6 + x^4/120 - x^6/5040…)，那么括号中的函数是递减的，如果y是x的下一个大的数字，那么有时siny会小于sinx。相反，计算sinx = x - x^3 * (1/6 - x^2/ 120 + x^4/5040…)，这是不可能发生的。

例如，在计算切比雪夫多项式时，通常需要将系数四舍五入到双倍精度。但是，虽然切比雪夫多项式是最优的，但系数舍入为双精度的切比雪夫多项式并不是具有双精度系数的最优多项式!

For example for sin (x), where you need coefficients for x, x^3, x^5, x^7 etc. you do the following: Calculate the best approximation of sin x with a polynomial (ax + bx^3 + cx^5 + dx^7) with higher than double precision, then round a to double precision, giving A. The difference between a and A would be quite large. Now calculate the best approximation of (sin x - Ax) with a polynomial (b x^3 + cx^5 + dx^7). You get different coefficients, because they adapt to the difference between a and A. Round b to double precision B. Then approximate (sin x - Ax - Bx^3) with a polynomial cx^5 + dx^7 and so on. You will get a polynomial that is almost as good as the original Chebyshev polynomial, but much better than Chebyshev rounded to double precision.

Next you should take into account the rounding errors in the choice of polynomial. You found a polynomial with minimum error in the polynomial ignoring rounding error, but you want to optimise polynomial plus rounding error. Once you have the Chebyshev polynomial, you can calculate bounds for the rounding error. Say f (x) is your function, P (x) is the polynomial, and E (x) is the rounding error. You don't want to optimise | f (x) - P (x) |, you want to optimise | f (x) - P (x) +/- E (x) |. You will get a slightly different polynomial that tries to keep the polynomial errors down where the rounding error is large, and relaxes the polynomial errors a bit where the rounding error is small.

所有这些将使您轻松地获得最多0.55倍于最后一位的舍入误差，其中+，-，*，/的舍入误差最多为0.50倍于最后一位。