Please help me understand the mathematics involved in Wikipedia page of Floating point, section of Piecewise Linear approximation to exponential and logarithm. Following is the link Piecewise linear approximation to exponential and logarithm
This interpretation is useful for visualizing how the values of floating point numbers vary with the representation, and allow for certain efficient approximations of floating point operations by integer operations and bit shifts. For example, reinterpreting a float as an integer, taking the negative (or rather subtracting from a fixed number, due to bias and implicit 1), then reinterpreting as a float yields the reciprocal. Explicitly, ignoring significand, taking the reciprocal is just taking the additive inverse of the (unbiased) exponent, since the exponent of the reciprocal is the negative of the original exponent. (Hence actually subtracting the exponent from twice the bias, which corresponds to unbiasing, taking negative, and then biasing.) For the significand, near 1 the reciprocal is approximately linear: $\frac{1}{x} \approx 1 - x$ (since the derivative is -1; this is the first term of the Taylor series), and thus for the significand as well, taking the negative (or rather subtracting from a fixed number to handle the implicit 1) is approximately taking the reciprocal.
Also, the following paragraphs from the same section are unclear to me.
Concretely, each time the exponent increments, the value doubles (hence grows exponentially), while each time the significand increments (for a given exponent), the value increases by $2^{e-b}$ (hence grows linearly, with slope equal to the actual (unbiased) value of the exponent). This holds even for the last step from a given exponent, where the significand overflows into the exponent: with the implicit 1, the number after 1.11...1 is 2.0 (regardless of the exponent), i.e., an increment of the exponent:
(0...001)0...0 through (0...001)1...1, (0...010)0...0 are equal steps (linear)
Thus as a graph it is linear pieces (as the significand grows for a given exponent) connecting the evenly spaced powers of two (when the significand is 0), with each linear piece having twice the slope of the previous: it is approximately a scaled and shifted exponential $2^x$. Each piece takes the same horizontal space, but twice the vertical space of the last. Because the exponent is convex up, the value is always greater than or equal to the actual (shifted and scaled) exponential curve through the points with significand 0; by a slightly different shift one can more closely approximate an exponential, sometimes overestimating, sometimes underestimating. Conversely, interpreting a floating point number as an integer gives an approximate shifted and scaled logarithm, with each piece having half the slope of the last, taking the same vertical space but twice the horizontal space. Since the logarithm is convex down, the approximation is always less than the corresponding logarithmic curve; again, a different choice of scale and shift (as at above right) yields a closer approximation.
Is there a better way to learn the mathematics involved? Thank you.