Fix corner-case loss of precision in numeric pow() calculation

Commit 7d9a4737 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

src/backend/utils/adt/numeric.c

@@ -7591,6 +7591,7 @@ exp_var(NumericVar *arg, NumericVar *result, int rscale)
 	val = numericvar_to_double_no_overflow(&x);
 
 	/* Guard against overflow */
+	/* If you change this limit, see also power_var()'s limit */
 	if (Abs(val) >= NUMERIC_MAX_RESULT_SCALE * 3)
 		ereport(ERROR,
 				(errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
@@ -7992,6 +7993,15 @@ power_var(NumericVar *base, NumericVar *exp, NumericVar *result)
 	 *
 	 * We want result = e ^ (exp * ln(base))
 	 * so result dweight = log10(result) = exp * ln(base) * log10(e)
+	 *
+	 * We also perform a crude overflow test here so that we can exit early if
+	 * the full-precision result is sure to overflow, and to guard against
+	 * integer overflow when determining the scale for the real calculation.
+	 * exp_var() supports inputs up to NUMERIC_MAX_RESULT_SCALE * 3, so the
+	 * result will overflow if exp * ln(base) >= NUMERIC_MAX_RESULT_SCALE * 3.
+	 * Since the values here are only approximations, we apply a small fuzz
+	 * factor to this overflow test and let exp_var() determine the exact
+	 * overflow threshold so that it is consistent for all inputs.
 	 *----------
 	 */
 	ln_dweight = estimate_ln_dweight(base);
@@ -8006,11 +8016,13 @@ power_var(NumericVar *base, NumericVar *exp, NumericVar *result)
 
 	val = numericvar_to_double_no_overflow(&ln_num);
 
-	val *= 0.434294481903252;	/* approximate decimal result weight */
+	/* initial overflow test with fuzz factor */
+	if (Abs(val) > NUMERIC_MAX_RESULT_SCALE * 3.01)
+		ereport(ERROR,
+				(errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
+				 errmsg("value overflows numeric format")));
 
-	/* limit to something that won't cause integer overflow */
-	val = Max(val, -NUMERIC_MAX_RESULT_SCALE);
-	val = Min(val, NUMERIC_MAX_RESULT_SCALE);
+	val *= 0.434294481903252;	/* approximate decimal result weight */
 
 	/* choose the result scale */
 	rscale = NUMERIC_MIN_SIG_DIGITS - (int) val;

src/test/regress/sql/numeric_big.sql

@@ -881,6 +881,16 @@ WITH t(b, p, bc_result) AS (VALUES
 (27.234987, 20.230957, 108142427112079606637962972621.121293))
 SELECT b, p, bc_result, b^p AS power, b^p - bc_result AS diff FROM t;
 
+-- Inputs close to overflow
+--
+-- bc(1) results computed with a scale of 2700 and truncated to 4 decimal
+-- places.
+
+WITH t(b, p, bc_result) AS (VALUES
+(0.12, -2829.8369, 58463948950011752465280493160293790845494328939320966633018493248607815580903065923369555885857984675501574162389726507612128133630191173383130639968378879506624785786843501848666498440326970769604109017960864573408272864266102690849952650095786874354625921641729880352858506454246180842452983243549491658464046163869265572232996388827878976066830374513768599285647145439771472435206769249126377164951470622827631950210853282324510655982757098065657709137845327135766013147354253426364240746381620690117663724329288646510198895137275207992825719846135857839292915100523542874885080351683587865157015032404901182924720371819942957083390475846809517968191151435281268695782594904484795360890092607679215675240583291240729468370895035823777914792823688291214492607109455017754453939895630226174304357121900605689015734289765672740769194115142607443713769825894380064727556869268488695795705030158832909348803019429370973064732712469794182891757241046263341655894972953512257981661670321890336672832647028099324621932563236459127918144141230217523147304565594514812518826936144181257723061181656522095236928347413997136815409159361412494284201481609684892562646522086577634100783077813105675590737823924220663206479031113753135119759722725207724879578900186075841393115040465401462266086907464970054073340036852442184414587772177753008511913377364966775792477387717262694468450099866775550614257191941835797445874557362115814601886902749237439492398087966544817154173072811937702110580330775581851211123491341435883319798273456296794954514173820352334127081705706502510709179711510240917772628308487366740741280043704807717608366220401933596364641284631036907635403895053036499618723044314773148779735006542501244942039455169872946018271985844759209768927953340447524637670938413827595013338859796135512187473850161303598087634723542727044978083220970836296653305188470017342167913572166172051819741354902582606590658382067039498769674611071582171914886494269818475850690414812481252963932223686078322390396586222238852602472958831686564971334200490182175112490433364675164900946902818404704835106260174052265784055642968397240262737313737007322288203637798365320295080314524864099419556398713380156353062937736280885716820226469419928595465390700629307079710611273715705695938635644841913194091407807776191951797748706106000922803167645881087385311847268311361092838264814899353459146959869764278464187826798546290981492648723002412475976344071283321798061003719251864595518596639432393032991023409676558943539937377229130132816883146259468718344018277257037013406135980469482324577407154032999045733141275895.3432),
+(1.2, 32908.8896, 58463467728170833376633133695001863276259293590926929026251227859007891876739460057725441400966420577009060860805883032969522911803372870882799865787473726926215148161529632590083389287080925059682489116446754279752928005457087175157581627230586554364417068189211136840990661174760199073702207450133797324318403866058202372178813998850887986769280847189341565507156189065295823921162851958925352114220880236114784962150135485415106748467247897246441194126125699204912883449386043559785865023459356275014504597646990160571664166410683323036984805434677654413174177920726210827006973855410386789516533036723888687725436216478665958434776205940192130053647653715221076841771578099896259902368829351569726536927952661429685419815305418450230567773264738536471211804481206474781470237730069753206249915908804615495060673071058534441654604668770343616386612119048579369195201590008082689834456232255266932976831478404670192731621439902738547169253818323045451045749609624500171633897705543164388470746657118050314064066768449450440405619135824055131398727045420324382226572368236570500391463795989258779677208133531636928003546809249007993065200108076924439703799231711400266122025052209803513232429907231051873161206025860851056337427740362763618748092029386371493898291580557004812947013231371383576580415676519066503391905962989205397824064923920045371823949776899815750413244195402085917098964452866825666226141169411712884994564949174271056284898570445214367063763956186792886147126466387576513166370247576466566827375268334148320298849218878848928271566491769458471357076035396330179659440244425914213309776100351793665960978678576150833311810944729586040624059867137538839913141142139636023129691775489034134511666020819676247950267220131499463010350308195762769192775344260909521732256844149916046793599150786757764962585268686580124987490115873389726527572428003433405659445349155536369077209682951123806333170190998931670309088422483075609203671527331975811507450670132060984691061148836994322505371265263690017938762760088575875666254883673433331627055180154954694693433502522592907190906966067656027637884202418119121728966267936832338377284832958974299187166554160783467156478554899314000348357280306042140481751668215838656488457943830180819301102535170705017482946779698265096226184239631924271857062033454725540956591929965181603262502135610768915716020374362368495244256420143645126927013882334008435586481691725030031204304273292938132599127402133470745819213047706793887965197191137237066440328777206799072470374264316425913530947082957300047105685634407092811630672103242089966046839626911122.7149))
+SELECT b, p, bc_result, b^p AS power, b^p - bc_result AS diff FROM t;
+
 --
 -- Tests for EXP()
 --