Improve type numeric's calculations for ln(), log(), exp(), pow().

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

src/test/regress/sql/numeric.sql

@@ -860,6 +860,57 @@ select 10.0 ^ -2147483647 as rounds_to_zero;
 select 10.0 ^ 2147483647 as overflows;
 select 117743296169.0 ^ 1000000000 as overflows;
 
+-- cases that used to return inaccurate results
+select 3.789 ^ 21;
+select 3.789 ^ 35;
+select 1.2 ^ 345;
+select 0.12 ^ (-20);
+
+-- cases that used to error out
+select 0.12 ^ (-25);
+select 0.5678 ^ (-85);
+
+--
+-- Tests for raising to non-integer powers
+--
+
+-- special cases
+select 0.0 ^ 0.0;
+select (-12.34) ^ 0.0;
+select 12.34 ^ 0.0;
+select 0.0 ^ 12.34;
+
+-- invalid inputs
+select 0.0 ^ (-12.34);
+select (-12.34) ^ 1.2;
+
+-- cases that used to generate inaccurate results
+select 32.1 ^ 9.8;
+select 32.1 ^ (-9.8);
+select 12.3 ^ 45.6;
+select 12.3 ^ (-45.6);
+
+-- big test
+select 1.234 ^ 5678;
+
+--
+-- Tests for EXP()
+--
+
+-- special cases
+select exp(0.0);
+select exp(1.0);
+select exp(1.0::numeric(71,70));
+
+-- cases that used to generate inaccurate results
+select exp(32.999);
+select exp(-32.999);
+select exp(123.456);
+select exp(-123.456);
+
+-- big test
+select exp(1234.5678);
+
 --
 -- Tests for generate_series
 --
@@ -880,3 +931,55 @@ select (i / (10::numeric ^ 131071))::numeric(1,0)
 select * from generate_series(1::numeric, 3::numeric) i, generate_series(i,3) j;
 select * from generate_series(1::numeric, 3::numeric) i, generate_series(1,i) j;
 select * from generate_series(1::numeric, 3::numeric) i, generate_series(1,5,i) j;
+
+--
+-- Tests for LN()
+--
+
+-- Invalid inputs
+select ln(-12.34);
+select ln(0.0);
+
+-- Some random tests
+select ln(1.2345678e-28);
+select ln(0.0456789);
+select ln(0.349873948359354029493948309745709580730482050975);
+select ln(0.99949452);
+select ln(1.00049687395);
+select ln(1234.567890123456789);
+select ln(5.80397490724e5);
+select ln(9.342536355e34);
+
+--
+-- Tests for LOG() (base 10)
+--
+
+-- invalid inputs
+select log(-12.34);
+select log(0.0);
+
+-- some random tests
+select log(1.234567e-89);
+select log(3.4634998359873254962349856073435545);
+select log(9.999999999999999999);
+select log(10.00000000000000000);
+select log(10.00000000000000001);
+select log(590489.45235237);
+
+--
+-- Tests for LOG() (arbitrary base)
+--
+
+-- invalid inputs
+select log(-12.34, 56.78);
+select log(-12.34, -56.78);
+select log(12.34, -56.78);
+select log(0.0, 12.34);
+select log(12.34, 0.0);
+select log(1.0, 12.34);
+
+-- some random tests
+select log(1.23e-89, 6.4689e45);
+select log(0.99923, 4.58934e34);
+select log(1.000016, 8.452010e18);
+select log(3.1954752e47, 9.4792021e-73);