Problem

When describing the exponential histograms and how to do the mapping in the DataModel all of the formulas have the property of monotonicity, e.g. for two points $a \lt b$ then $index_a \le index_b$. This is true for the first MapToIndex, but not for the second MapToIndex with an exact power of two cases added. When you make the correction, if any of the values around the power of two are in error because of floating point rounding, then they will violate the monotonicity property. For example, see the MapToIndex function around 2^8 image1. This happens similarly for values around 2^-8 and is only made worse for larger exponents.

Solutions

Use a different formula

The Main introduction of error in the current formula is that Log() of large numbers is not accurate. We can eliminate this error by bounding the log to a smaller range. If we split the value into exponent and frac, using Frexp, we can then only take the log of the fraction. Because the fraction is bounded between 0.5 and 1, the log will have a bounded error.

The change would look like this:

// MapToIndex for any scale, exact for powers of two.
func MapToIndex(value float64, scale int) int {
    // Special case for power-of-two values.
    frac, exp := math.Frexp(value)
    scaleFactor := math.Ldexp(math.Log2E, scale)
    return exp<<scale + int(math.Log(frac)*scaleFactor) - 1
}

Remove the "exact" optimization

By only recommending the first MapToIndex function, we can remove the error. This is the simplest solution but the least optimized for any scale above 0.

Appendix

Proof of New Formula