This makes the choice of result scale of numeric power() for integer
exponents consistent with the choice for non-integer exponents, and
with the result scale of other numeric functions. Specifically, the
result scale will be at least as large as the scale of either input,
and sufficient to ensure that the result has at least 16 significant
digits.
Formerly, the result scale was based only on the scale of the first
input, without taking into account the weight of the result. For
results with negative weight, that could lead to results with very few
or even no non-zero significant digits (e.g., 10.0 ^ (-18) produced
0.0000000000000000).
Fix this by moving responsibility for the choice of result scale into
power_var_int(), which already has code to estimate the result weight.
Per report by Adrian Klaver and suggested fix by Tom Lane.
No back-patch -- arguably this is a bug fix, but one which is easy to
work around, so it doesn't seem worth the risk of changing query
results in stable branches.
Discussion: https://postgr.es/m/12a40226-70ac-3a3b-3d3a-fdaf9e32d312%40aklaver.com
When deciding on the local rscale to use for the Taylor series
expansion, ln_var() neglected to account for the fact that the result
is subsequently multiplied by a factor of 2^(nsqrt+1), where nsqrt is
the number of square root operations performed in the range reduction
step, which can be as high as 22 for very large inputs. This could
result in a loss of precision, particularly when combined with large
rscale values, for which a large number of Taylor series terms is
required (up to around 400).
Fix by computing a few extra digits in the Taylor series, based on the
weight of the multiplicative factor log10(2^(nsqrt+1)). It remains to
be proven whether or not the other 8 extra digits used for the Taylor
series is appropriate, but this at least deals with the obvious
oversight of failing to account for the effects of the final
multiplication.
Per report from Justin AnyhowStep. Reviewed by Tom Lane.
Discussion: https://postgr.es/m/16280-279f299d9c06e56f@postgresql.org
Commit 7d9a4737c2 greatly improved the
accuracy of the numeric transcendental functions, however it failed to
consider the case where the result from pow() is close to the overflow
threshold, for example 0.12 ^ -2345.6. For such inputs, where the
result has more than 2000 digits before the decimal point, the decimal
result weight estimate was being clamped to 2000, leading to a loss of
precision in the final calculation.
Fix this by replacing the clamping code with an overflow test that
aborts the calculation early if the final result is sure to overflow,
based on the overflow limit in exp_var(). This provides the same
protection against integer overflow in the subsequent result scale
computation as the original clamping code, but it also ensures that
precision is never lost and saves compute cycles in cases that are
sure to overflow.
The new early overflow test works with the initial low-precision
result (expected to be accurate to around 8 significant digits) and
includes a small fuzz factor to ensure that it doesn't kick in for
values that would not overflow exp_var(), so the overall overflow
threshold of pow() is unchanged and consistent for all inputs with
non-integer exponents.
Author: Dean Rasheed
Reviewed-by: Tom Lane
Discussion: http://www.postgresql.org/message-id/CAEZATCUj3U-cQj0jjoia=qgs0SjE3auroxh8swvNKvZWUqegrg@mail.gmail.com
See-also: http://www.postgresql.org/message-id/CAEZATCV7w+8iB=07dJ8Q0zihXQT1semcQuTeK+4_rogC_zq5Hw@mail.gmail.com
Set the "rscales" for intermediate-result calculations to ensure that
suitable numbers of significant digits are maintained throughout. The
previous coding hadn't thought this through in any detail, and as a result
could deliver results with many inaccurate digits, or in the worst cases
even fail with divide-by-zero errors as a result of losing all nonzero
digits of intermediate results.
In exp_var(), get rid entirely of the logic that separated the calculation
into integer and fractional parts: that was neither accurate nor
particularly fast. The existing range-reduction method of dividing by 2^n
can be applied across the full input range instead of only 0..1, as long as
we are careful to set an appropriate rscale for each step.
Also fix the logic in mul_var() for shortening the calculation when the
caller asks for fewer output digits than an exact calculation would
require. This bug doesn't affect simple multiplications since that code
path asks for an exact result, but it does contribute to accuracy issues
in the transcendental math functions.
In passing, improve performance of mul_var() a bit by forcing the shorter
input to be on the left, thus reducing the number of iterations of the
outer loop and probably also reducing the number of carry-propagation
steps needed.
This is arguably a bug fix, but in view of the lack of field complaints,
it does not seem worth the risk of back-patching.
Dean Rasheed
1. Using 100 digits after decimal point on the default
make runtest.
2. Using 1000 digits after decimal point in a new target
make bigtest.
At the end of 'make runtest', a hint about the new bigtest is
printed.
Jan
Dean Rasheed
Tom Lane
Jan Wieck