What are some numerically stable forms of common mathematical expressions?

[Kalimaa23]

Hi.

I have an assignment lying around, in which I have to find numerically stable forms of some expressions. A few still elude me, so I was wondering if someone might have a suggestion.

$$e^{x}-e$$

This has large rounding errors if x is close to 1

$$sinh (x) - tanh (x)$$

Large errors for x close to 0

$$log(x+\sqrt{x^2+1})$$

No idea...

 

[pervect]

Hi.

I have an assignment lying around, in which I have to find numerically stable forms of some expressions. A few still elude me, so I was wondering if someone might have a suggestion.

$$e^{x}-e$$

This has large rounding errors if x is close to 1

did you try a Taylor series expansion, or is that not what's being asked for?

 

[Kalimaa23]

Yes, a Taylor expansion does seem obvious; but alas, the'yre asking analytical forms...

 

[arildno]

2.:
$$sinh(x)-tanh(x)=sinh(x)(\frac{cosh(x)-1}{cosh(x)})=2tanh(x)sinh^{2}(\frac{x}{2})$$

Use a similar trick for 1, by noting $$sinh(y)=\frac{e^{y}-e^{-y}}{2}$$

 

[Kalimaa23]

Thanks! This is just what I needed!