Hi.
As per an advice in a well-known book (that one should not cite): In order
to compute a numerical approximation to the derivative:
f'(x) ~ [ f(x + h) - f(x) ] / h
the "h" should be a number representable in the set of floating point
numbers being used (e.g. "double").
It seems that a code like
---CUT---
final double delta = x + h - x;
final double deriv = (f.value(x + delta) - f.value(x)) / delta;
---CUT---
would be the simplest thing to do.
Besides
x + delta - x
being strange-looking and running the risk of being manually (and
incorrectly) changed to just
delta
the reference (not) cited above suggests to use a two- or three-steps
procedure to ensure that some optimizing compiler will not do the same:
---CUT---
double sum = x + delta;
dummy(sum); // function "dummy" does nothing.
delta = sum - x;
---CUT---
Thus, I initially intended to add a utility method in "MathPrecision":
---CUT---
public static double representableDelta(double x,
double originalDelta) {
final long sum = Double.doubleToLongBits(x + originalDelta);
final double delta = Double.longBitsToDouble(sum) - x;
return delta;
}
---CUT---
assuming that it is sufficiently "complicated" to evade the "smartest" JIT
compiler.
Then, I also implemented the simple thing:
---CUT---
public static double representableDelta2(double x,
double originalDelta) {
return x + originalDelta - x;
}
---CUT---
Remarkably (or not), a trivial unit test (10^8 random delta values) did not
find an occurrence where the returned values from one or the other method
would differ.
I think that the utility would be a useful addition to CM, if just to draw
attention to the issue of accuracy in numerical derivatives[1], but I wonder
whether the simple implementation is ensured to always work: I.e. does a rule
exist that states that no optimization may ever be perfomed to simplify
"x + delta - x" into "delta"?
Any thoughts? (even if only for a better name...)
Gilles
[1] A better way would have been to have the numerical derivatives framework
that someone had promised to contribute a few months ago...
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