无需查找即可快速但不准确的 sin 函数
对于海洋着色器，我需要一个快速函数来计算 sin(x) 的非常近似的值。唯一的要求是它是周期性的，并且大致类似于正弦波。
sin 的泰勒级数太慢，因为我需要计算 x 的 9 次方才能得到完整的周期。
有什么建议吗？
编辑：抱歉我没有提到，我无法使用查找表，因为它位于顶点着色器上。查找表将涉及纹理样本，该样本在顶点着色器上比内置的 sin 函数慢。 它不必以任何方式准确，它只需看起来不错即可。
For an ocean shader, I need a fast function that computes a very approximate value for sin(x). The only requirements are that it is periodic, and roughly resembles a sine wave.
The taylor series of sin is too slow, since I'd need to compute up to the 9th power of x just to get a full period.
Any suggestions?
EDIT: Sorry I didn't mention, I can't use a lookup table since this is on the vertex shader. A lookup table would involve a texture sample, which on the vertex shader is slower than the built in sin function.
It doesn't have to be in any way accurate, it just has to look nice.
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根据您需要的数量使用切比雪夫近似值。如果您的输入角度被限制为良好行为（-π .. +π 或 0 .. 2π），那么这会特别容易，因此您不必首先将参数减少到合理的值。您可以使用 2 或 3 个术语而不是 9 个。
Use a Chebyshev approximation for as many terms as you need. This is particularly easy if your input angles are constrained to be well behaved (-π .. +π or 0 .. 2π) so you do not have to reduce the argument to a sensible value first. You might use 2 or 3 terms instead of 9.
您可以为某些值创建一个包含 sin 值的查找表，并在这些值之间使用线性插值。
You can make a look-up table with sin values for some values and use linear interpolation between that values.
sin(x) 的有理代数函数近似值（从零到 π/2 有效）为：
具有常数：
这些常数是通过最小二乘曲线拟合找到的。 （使用子程序 DHFTI，作者：Lawson & Hanson）。
如果输入在 [0, 2π] 之外，则需要采用 x mod 2 π。
要处理负数，您需要编写如下内容：
然后，要将范围扩展到 0 到 2π，请使用以下内容减少输入：
然后计算：
结果应在真实正弦值的 5% 左右。
A rational algebraic function approximation to sin(x), valid from zero to π/2 is:
with the constants:
These constants were found by least-squares curve fitting. (Using subroutine DHFTI, by Lawson & Hanson).
If the input is outside [0, 2π], you'll need to take x mod 2 π.
To handle negative numbers, you'll need to write something like:
Then, to extend the range to 0 to 2π, reduce the input with something like:
Then calculate:
The results should be within about 5% of the real sine.
好吧，你没有说你需要它有多准确。正弦可以通过区间 [0, pi/2]、[pi/2, 3*pi/2]、[3*pi/2, 2*pi 上的斜率 2/pi 和 -2/pi 的直线来近似]。在减少角度 mod 2*pi 后，可以通过乘法和加法的成本获得该近似值。
Well, you don't say how accurate you need it to be. The sine can be approximated by straight lines of slopes 2/pi and -2/pi on intervals [0, pi/2], [pi/2, 3*pi/2], [3*pi/2, 2*pi]. This approximation can be had for the cost of a multiplication and an addition after reducing the angle mod 2*pi.
使用查找表可能是控制速度和准确性之间权衡的最佳方法。
Using a lookup table is probably the best way to control the tradeoff between speed and accuracy.