math - Approximate e^x

Question

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

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

y₁ = a₁x + b₁, between x₂ and x₃,

then

y₂ = a₂x + b₂

etc

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.

Answer 1

How about a strategy like this that uses the formula

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

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)

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.

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.