Fix corner-case loss of precision in numeric ln().

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

src/backend/utils/adt/numeric.c

@@ -8500,6 +8500,7 @@ ln_var(const NumericVar *arg, NumericVar *result, int rscale)
 	NumericVar	ni;
 	NumericVar	elem;
 	NumericVar	fact;
+	int			nsqrt;
 	int			local_rscale;
 	int			cmp;
 
@@ -8530,12 +8531,14 @@ ln_var(const NumericVar *arg, NumericVar *result, int rscale)
 	 * rscale as we work so that we keep this many significant digits at each
 	 * step (plus a few more for good measure).
 	 */
+	nsqrt = 0;
 	while (cmp_var(&x, &const_zero_point_nine) <= 0)
 	{
 		local_rscale = rscale - x.weight * DEC_DIGITS / 2 + 8;
 		local_rscale = Max(local_rscale, NUMERIC_MIN_DISPLAY_SCALE);
 		sqrt_var(&x, &x, local_rscale);
 		mul_var(&fact, &const_two, &fact, 0);
+		nsqrt++;
 	}
 	while (cmp_var(&x, &const_one_point_one) >= 0)
 	{
@@ -8543,6 +8546,7 @@ ln_var(const NumericVar *arg, NumericVar *result, int rscale)
 		local_rscale = Max(local_rscale, NUMERIC_MIN_DISPLAY_SCALE);
 		sqrt_var(&x, &x, local_rscale);
 		mul_var(&fact, &const_two, &fact, 0);
+		nsqrt++;
 	}
 
 	/*
@@ -8555,8 +8559,12 @@ ln_var(const NumericVar *arg, NumericVar *result, int rscale)
 	 *
 	 * The convergence of this is not as fast as one would like, but is
 	 * tolerable given that z is small.
+	 *
+	 * The Taylor series result will be multiplied by 2^(nsqrt+1), which has a
+	 * decimal weight of (nsqrt+1) * log10(2), so work with this many extra
+	 * digits of precision (plus a few more for good measure).
 	 */
-	local_rscale = rscale + 8;
+	local_rscale = rscale + (int) ((nsqrt + 1) * 0.301029995663981) + 8;
 
 	sub_var(&x, &const_one, result);
 	add_var(&x, &const_one, &elem);

src/test/regress/expected/numeric_big.out

@@ -1612,6 +1612,25 @@ SELECT x, bc_result, ln(x::numeric), ln(x::numeric)-bc_result AS diff FROM t;
  1.0e1000 | 2302.5850929940457 | 2302.5850929940457 |  0.0000000000000
 (10 rows)
 
+-- inputs with 1000 decimal places
+--
+-- bc(1) results computed with a scale of 2000 and rounded to 1000 decimal
+-- places
+WITH t(x, bc_result) AS (VALUES
+(484990182159328900690402236933516249572671683638747490717351807610531884491845416923860371219625151551889257298200816555016472471293780254009492949585031653913930735918829139712249577547959394351523545901788627247613322896296041868431769047433229466634098452564756860190085118463828382895145244362033728480588969626012192733802377468089120757046364393407262957242230928854711898925295251902007136232994524624903257456111389508582206404271734668422903183500589303866613158037169610592539145461637447957948521714058034772237111009429638870236361143304703683377693378577075353794118557951847394763531830696578809001981568860219578880229402696449243344235099860421846016326538272155937175661905904288335499593232232926636205909086901191153907183842087577811871344870731324067822883041265129394268082883745408414994.8967939438561591657171240282983703914075472645212002662497023142663831371447287624846942598424990784971781730103682951722370983277124599054059027055336437808366784501932987082321905202623642371063626378290734289114618092750984153422293450048717769065428713836637664433167768445609659527458911187829232316677137895259433038764404970599325009178297626038331436654541552998098529141205301472138026818453893127265938030066392881979113522757891639646670670272542401773230506961559808927249585675430838495658225557294666522469887436551840596777627408780618586500922973500018513068499587683746133637919751545157547095670767246977244726331271787622126889459658539988980096764323712767863722912919120929339399753431689512753214200090670880647731689804555417871258907716687575767185444541243606329768784843125926070743277339790277626515824924290352180761378846035233155198504033292692893297993698953705472933411199778880561376633444249703838589180474329586470353212010427945060694794274109764269805332803290229,
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+ 1510.4332713542154696529645934345554302578243896764921637693542962119938599884313210100957753316832762996428481801312323020427109678979117469716796746760060470871840325255146954580681101106876674367471955788143763250819168311353856748872452260808797135108102729064040463343792765872545182299889360257515315869180266759715933989413256377582681707188367254513700731642913479683031478361835565783219287780434673712341147656477670848734998849030451414278832848680301511646182446524915091598080243532068451726548537866633622180283865668708517173065893429240665300584705585310049892047293928733753369421499719516009692095913169665213597158441636480707309244604139865130782756488091268094213446272360006907802989573582755585110277620911226015342778471352130366770729972784317323917141031824334355639769512749560550167491709646539950725523461943580211843652293561678342656010571108219244870234329176123205423872844099992204896411752620881541000940129833754169391528449211839693800724450201835161044717173715867437))
+SELECT trim_scale(ln(x::numeric)-bc_result) AS diff FROM t;
+ diff 
+------
+    0
+    0
+    0
+(3 rows)
+
 --
 -- Tests for LOG() (base 10)
 --

src/test/regress/sql/numeric_big.sql

@@ -1193,6 +1193,20 @@ WITH t(x, bc_result) AS (VALUES
 ('1.0e1000', 2302.5850929940457))
 SELECT x, bc_result, ln(x::numeric), ln(x::numeric)-bc_result AS diff FROM t;
 
+-- inputs with 1000 decimal places
+--
+-- bc(1) results computed with a scale of 2000 and rounded to 1000 decimal
+-- places
+
+WITH t(x, bc_result) AS (VALUES
+(484990182159328900690402236933516249572671683638747490717351807610531884491845416923860371219625151551889257298200816555016472471293780254009492949585031653913930735918829139712249577547959394351523545901788627247613322896296041868431769047433229466634098452564756860190085118463828382895145244362033728480588969626012192733802377468089120757046364393407262957242230928854711898925295251902007136232994524624903257456111389508582206404271734668422903183500589303866613158037169610592539145461637447957948521714058034772237111009429638870236361143304703683377693378577075353794118557951847394763531830696578809001981568860219578880229402696449243344235099860421846016326538272155937175661905904288335499593232232926636205909086901191153907183842087577811871344870731324067822883041265129394268082883745408414994.8967939438561591657171240282983703914075472645212002662497023142663831371447287624846942598424990784971781730103682951722370983277124599054059027055336437808366784501932987082321905202623642371063626378290734289114618092750984153422293450048717769065428713836637664433167768445609659527458911187829232316677137895259433038764404970599325009178297626038331436654541552998098529141205301472138026818453893127265938030066392881979113522757891639646670670272542401773230506961559808927249585675430838495658225557294666522469887436551840596777627408780618586500922973500018513068499587683746133637919751545157547095670767246977244726331271787622126889459658539988980096764323712767863722912919120929339399753431689512753214200090670880647731689804555417871258907716687575767185444541243606329768784843125926070743277339790277626515824924290352180761378846035233155198504033292692893297993698953705472933411199778880561376633444249703838589180474329586470353212010427945060694794274109764269805332803290229,
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+ 1510.4332713542154696529645934345554302578243896764921637693542962119938599884313210100957753316832762996428481801312323020427109678979117469716796746760060470871840325255146954580681101106876674367471955788143763250819168311353856748872452260808797135108102729064040463343792765872545182299889360257515315869180266759715933989413256377582681707188367254513700731642913479683031478361835565783219287780434673712341147656477670848734998849030451414278832848680301511646182446524915091598080243532068451726548537866633622180283865668708517173065893429240665300584705585310049892047293928733753369421499719516009692095913169665213597158441636480707309244604139865130782756488091268094213446272360006907802989573582755585110277620911226015342778471352130366770729972784317323917141031824334355639769512749560550167491709646539950725523461943580211843652293561678342656010571108219244870234329176123205423872844099992204896411752620881541000940129833754169391528449211839693800724450201835161044717173715867437))
+SELECT trim_scale(ln(x::numeric)-bc_result) AS diff FROM t;
+
 --
 -- Tests for LOG() (base 10)
 --