Improving Quality of Exponential and Logarithm Functions

[kphillisjr]

I was looking over the various math functions and noticed that there is a lot of error in the implementation of the Logarithm and Exponential functions. This is after applying the Intrinsic-free version of the square root function.

// The Square-Root Function.
sqrtf(x);
// Fast and Simple alternative... 
expf(0.5f*logf(x));

// the Power function
powf(a, b);
// alternative method:
expf(b*logf(a));

Now as far as improving the Accuracy of the Exponential and Logarithm Functions. I suggest looking into the work completed by Ping Tak Peter Tang in the following series of articles from the early 1990s and late 1980s.

• Table-driven implementation of the Expm1 function in IEEE floating-point arithmetic. ACM Transactions on Mathematical Software 18(2): 211-222 (1992)
• Table-lookup algorithms for elementary functions and their error analysis. IEEE Symposium on Computer Arithmetic 1991: 232-236
• Accurate and efficient testing of the exponential and logarithm functions. ACM Transactions on Mathematical Software 16(3): 185-200 (1990)
• Table-driven implementation of the exponential function in IEEE floating-point arithmetic. ACM Transactions on Mathematical Software ( 15(2): 144-157 (1989)

Edit: Added note about alternative method for calling the pow function.

[gingerBill]

I was looking over the various math functions and noticed that there is a lot of error in the implementation of the Logarithm and Exponential functions. This is after applying the Intrinsic-free version of the square root function. ...

@kphillisjr these functions are meant to be fast operations rather than precise ones. They are to be used when x is small.

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[kphillisjr]

Thanks for the response, and I found a good alternative ( see: https://github.com/yui0/slibs/blob/master/fmath.h ). I just have not tested the overall speed just yet.