近似e ^ x [英] Approximate e^x

问题描述

我想近似使用 e ^x 函数.

I'd like to approximate the e^x function.

是否可以使用基于多个样条线类型的方法来做到这一点?即在 x ₁ 和 x ₂ 之间，然后

Is it possible to do so using multiple splines type based approach? i.e between x₁ and x₂, then

y ₁ = a ₁ x + b ₁ ，在x _{2 和x 3 ，}

y₁ = a₁x + b₁, between x₂ and x₃,

然后

y ₂ = a ₂ x + b ₂

y₂ = a₂x + b₂

等

这是专用于fpga硬件，而不是通用CPU.因此，我需要自己创建函数.精度问题不那么重要了.此外，我真的负担不起一个以上的乘法电路和/或多个移位/加法器.我还想要比CORDIC函数小的东西，但实际上尺寸至关重要.

This is for dedicated fpga hardware and not a general purpose CPU. As such I need to create the function myself. Accuracy is much less of a concern. Furthermore I can't really afford more than one multiplication circuit and/or multiple shifts/adders. Also I want something much smaller than a CORDIC function, in fact size is critical.

推荐答案

这样的使用公式的策略怎么样

How about a strategy like this that uses the formula

e ^x = 2 ^x/ln(2)

1. 预先计算1/ln(2)
2. 将此常数乘以您的参数(1乘)
3. 使用二进制移位将2提高到幂的整数部分(假设exp +尾数格式)
4. 根据2的幂的余数(可能是第二个乘法)进行调整

1. Precalculate 1/ln(2)
2. Multiply this constant by your argument (1 multiplication)
3. Use binary shifts to raise 2 to the integer portion of the power (assumes exp+mantissa format)
4. Adjust based on the fractional power-of-2 remainder (likely a second multiplication)

我意识到这不是一个完整的解决方案，但是它只需要一个乘法就可以解决，剩下的问题可以简化为2的小数次幂，应该在硬件中更容易实现.

I realize this is not a complete solution, but it does only require a single multiplication and reduces the remaining problem to approximating a fractional power of 2, which should be easier to implement in hardware.

此外，如果您的应用程序具有足够的专业性，则可以尝试重新推导将在硬件上运行的所有数字代码，以将它们存储在基数 e 系统中，并实现浮动点硬件也可以在基本 e 中工作.那么根本就不需要转换.

Also, if your application is specialized enough, you could try to re-derive all of the numerical code that will run on your hardware to be in a base-e number system and implement your floating point hardware to work in base e as well. Then no conversion is needed at all.

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