消除乘法运算 - 利用Cordic算法来计算三角函数的值

消除乘法运算
我们已经成功的将乘法的次数减少了一半，还有没有可能进一步降低运算量呢？还要从计算式入手。

第一次循环时，tan(45)=1，所以第一次循环实际上是不需要乘法运算的。第二次运算呢？
Tan(22.5)=0.4142135623731,很不幸，第二次循环乘数是个很不整的小数。是否能对其改造一下呢？答案是肯定的。第二次选择22.5度是因为二分查找法的查找效率最高。如果选用个在22.5到45度之间的值，查找的效率会降低一些。如果稍微降低一点查找的效率能让我们有效的减少乘法的次数，使最终的计算速度提高了，那么这种改进就是值得的。

我们发现tan(26.565051177078)=0.5，如果我们第二次旋转采用26.565051177078度，那么乘数变为0.5，如果我们采用定点数运算的话（没有浮点协处理器时为了加速计算我们会大量的采用定点数算法）乘以0.5就相当于将乘数右移一位。右移运算是很快的，这样第二次循环中的乘法运算也被消除了。

类似的方法，第三次循环中不用11.25度，而采用 14.0362434679265 度。
Tan(14.0362434679265)= 1/4

乘数右移两位就可以了。剩下的都以此类推。
tan(45)= 1
tan(26.565051177078)= 1/2
tan(14.0362434679265)= 1/4
tan(7.1250163489018)= 1/8
tan(3.57633437499735)= 1/16
tan(1.78991060824607)= 1/32
tan(0.8951737102111)= 1/64
tan(0.4476141708606)= 1/128
tan(0.2238105003685)= 1/256

还是给出C语言的实现代码，我们采用循序渐进的方法，先给出浮点数的实现（因为用到了浮点数，所以并没有减少乘法运算量，查找的效率也比二分查找法要低，理论上说这个算法实现很低效。不过这个代码的目的在于给出算法实现的示意性说明，还是有意义的）。
double my_atan4(double x, double y)
{
const double tangent[] = {1.0, 1 / 2.0, 1 / 4.0, 1 / 8.0, 1 / 16.0,
1 / 32.0, 1 / 64.0, 1 / 128.0, 1 / 256.0, 1 / 512.0,
1 / 1024.0, 1 / 2048.0, 1 / 4096.0, 1 / 8192.0, 1 / 16384.0
};
const double angle[] = {45.0, 26.565051177078, 14.0362434679265, 7.1250163489018, 3.57633437499735,
1.78991060824607, 0.8951737102111, 0.4476141708606, 0.2238105003685, 0.1119056770662,
0.0559528918938, 0.027976452617, 0.01398822714227, 0.006994113675353, 0.003497056850704
};

int i = 0;
double x_new, y_new;
double angleSum = 0.0;

for(i = 0; i < 15; i++)
{
if(y > 0)
{
x_new = x + y * tangent[i];
y_new = y - x * tangent[i];
x = x_new;
y = y_new;
angleSum += angle[i];
}
else
{
x_new = x - y * tangent[i];
y_new = y + x * tangent[i];
x = x_new;
y = y_new;
angleSum -= angle[i];
}
printf("Debug: i = %d angleSum = %f, angle = %f, ρ = %f\n", i, angleSum, angle[i], hypot(x, y));
}
return angleSum;
}

程序运行的输出结果如下：
Debug: i = 0 angleSum = 45.000000, angle = 45.000000, ρ = 316.227766
Debug: i = 1 angleSum = 71.565051, angle = 26.565051, ρ = 353.553391
Debug: i = 2 angleSum = 57.528808, angle = 14.036243, ρ = 364.434493
Debug: i = 3 angleSum = 64.653824, angle = 7.125016, ρ = 367.270602
Debug: i = 4 angleSum = 61.077490, angle = 3.576334, ρ = 367.987229
Debug: i = 5 angleSum = 62.867400, angle = 1.789911, ρ = 368.166866
Debug: i = 6 angleSum = 63.762574, angle = 0.895174, ρ = 368.211805
Debug: i = 7 angleSum = 63.314960, angle = 0.447614, ρ = 368.223042
Debug: i = 8 angleSum = 63.538770, angle = 0.223811, ρ = 368.225852
Debug: i = 9 angleSum = 63.426865, angle = 0.111906, ρ = 368.226554
Debug: i = 10 angleSum = 63.482818, angle = 0.055953, ρ = 368.226729
Debug: i = 11 angleSum = 63.454841, angle = 0.027976, ρ = 368.226773
Debug: i = 12 angleSum = 63.440853, angle = 0.013988, ρ = 368.226784
Debug: i = 13 angleSum = 63.433859, angle = 0.006994, ρ = 368.226787
Debug: i = 14 angleSum = 63.437356, angle = 0.003497, ρ = 368.226788

z = 63.437356