# [HACKERS] More inaccurate results from numeric pow()

```Doing some more testing of the numeric code patched in [1] I noticed
another case where the result is inaccurate -- computing 0.12 ^
-2345.6 gives a very large number containing 2162 digits, but only the
first 2006 correct, while the last 156 digits are wrong.```
```
The reason is this code in power_var():

/* limit to something that won't cause integer overflow */
val = Max(val, -NUMERIC_MAX_RESULT_SCALE);
val = Min(val, NUMERIC_MAX_RESULT_SCALE);

where "val" is the approximate decimal result weight. Here
NUMERIC_MAX_RESULT_SCALE is 2000, so it's clamping the estimated
result weight to 2000, and therefore reducing the rscale in the
subsequent calculations, causing the loss of precision at around 2000
digits.

In fact it's possible to predict exactly how large we need to allow
"val" to become, since the final result is computed using exp_var(),
which accepts inputs up to 6000, so the result weight "val" can be up
to around log10(exp(6000)) ~= 2606 before the final result causes an
overflow.

The obvious fix would be to modify the clamping limits. I think a
better answer though is to replace the clamping code with an overflow
test, immediately throwing an error if "val" is outside the allowed
range, per the attached patch.

This has the advantage that it avoids some expensive computations in
the case where the result will end up overflowing, but more
importantly it means that power_var() isn't so critically dependent on
the limits of exp_var() -- if someone in the future increased the
limits of exp_var() without touching power_var(), and power_var()
clamped to the old range, the problem would resurface. But doing an
overflow test in power_var() instead of clamping "val", it would
either compute an accurate result, or throw an overflow error early
on. There should be no possibility of it returning an inaccurate
result.

Regards,
Dean

[1]
http://www.postgresql.org/message-id/CAEZATCV7w+8iB=07dj8q0zihxqt1semcqutek+4_rogc_zq...@mail.gmail.com
```
```diff --git a/src/backend/utils/adt/numeric.c b/src/backend/utils/adt/numeric.c
new file mode 100644
index 3ba373a..b4a2829
@@ -8008,9 +8008,16 @@ power_var(NumericVar *base, NumericVar *

val *= 0.434294481903252;	/* approximate decimal result weight */

-	/* limit to something that won't cause integer overflow */
-	val = Max(val, -NUMERIC_MAX_RESULT_SCALE);
-	val = Min(val, NUMERIC_MAX_RESULT_SCALE);
+	/*
+	 * Apply a crude overflow test so we can exit early if the result is sure
+	 * to overflow.  exp_var() supports inputs up to 6000, so we know that the
+	 * result will overflow if the approximate decimal result weight exceeds
+	 * log10(exp(6000)) ~= 2610.
+	 */
+	if (Abs(val) > 2610)
+		ereport(ERROR,
+				(errcode(ERRCODE_NUMERIC_VALUE_OUT_OF_RANGE),
+				 errmsg("value overflows numeric format")));

/* choose the result scale */
rscale = NUMERIC_MIN_SIG_DIGITS - (int) val;
diff --git a/src/test/regress/expected/numeric_big.out b/src/test/regress/expected/numeric_big.out
new file mode 100644
index 1b3608a..8bb09c9
--- a/src/test/regress/expected/numeric_big.out
+++ b/src/test/regress/expected/numeric_big.out
@@ -1075,6 +1075,20 @@ SELECT b, p, bc_result, b^p AS power, b^
27.234987 |  20.230957 |           108142427112079606637962972621.121293 |           108142427112079606637962972621.121293 |                                        0.000000
(41 rows)

+-- 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;
+  b   |     p      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    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                                                                                                                                                                                                                                                   |  diff
+------+------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+--------
+ 0.12 | -2829.8369 | 58463948950011752465280493160293790845494328939320966633018493248607815580903065923369555885857984675501574162389726507612128133630191173383130639968378879506624785786843501848666498440326970769604109017960864573408272864266102690849952650095786874354625921641729880352858506454246180842452983243549491658464046163869265572232996388827878976066830374513768599285647145439771472435206769249126377164951470622827631950210853282324510655982757098065657709137845327135766013147354253426364240746381620690117663724329288646510198895137275207992825719846135857839292915100523542874885080351683587865157015032404901182924720371819942957083390475846809517968191151435281268695782594904484795360890092607679215675240583291240729468370895035823777914792823688291214492607109455017754453939895630226174304357121900605689015734289765672740769194115142607443713769825894380064727556869268488695795705030158832909348803019429370973064732712469794182891757241046263341655894972953512257981661670321890336672832647028099324621932563236459127918144141230217523147304565594514812518826936144181257723061181656522095236928347413997136815409159361412494284201481609684892562646522086577634100783077813105675590737823924220663206479031113753135119759722725207724879578900186075841393115040465401462266086907464970054073340036852442184414587772177753008511913377364966775792477387717262694468450099866775550614257191941835797445874557362115814601886902749237439492398087966544817154173072811937702110580330775581851211123491341435883319798273456296794954514173820352334127081705706502510709179711510240917772628308487366740741280043704807717608366220401933596364641284631036907635403895053036499618723044314773148779735006542501244942039455169872946018271985844759209768927953340447524637670938413827595013338859796135512187473850161303598087634723542727044978083220970836296653305188470017342167913572166172051819741354902582606590658382067039498769674611071582171914886494269818475850690414812481252963932223686078322390396586222238852602472958831686564971334200490182175112490433364675164900946902818404704835106260174052265784055642968397240262737313737007322288203637798365320295080314524864099419556398713380156353062937736280885716820226469419928595465390700629307079710611273715705695938635644841913194091407807776191951797748706106000922803167645881087385311847268311361092838264814899353459146959869764278464187826798546290981492648723002412475976344071283321798061003719251864595518596639432393032991023409676558943539937377229130132816883146259468718344018277257037013406135980469482324577407154032999045733141275895.3432 | 58463948950011752465280493160293790845494328939320966633018493248607815580903065923369555885857984675501574162389726507612128133630191173383130639968378879506624785786843501848666498440326970769604109017960864573408272864266102690849952650095786874354625921641729880352858506454246180842452983243549491658464046163869265572232996388827878976066830374513768599285647145439771472435206769249126377164951470622827631950210853282324510655982757098065657709137845327135766013147354253426364240746381620690117663724329288646510198895137275207992825719846135857839292915100523542874885080351683587865157015032404901182924720371819942957083390475846809517968191151435281268695782594904484795360890092607679215675240583291240729468370895035823777914792823688291214492607109455017754453939895630226174304357121900605689015734289765672740769194115142607443713769825894380064727556869268488695795705030158832909348803019429370973064732712469794182891757241046263341655894972953512257981661670321890336672832647028099324621932563236459127918144141230217523147304565594514812518826936144181257723061181656522095236928347413997136815409159361412494284201481609684892562646522086577634100783077813105675590737823924220663206479031113753135119759722725207724879578900186075841393115040465401462266086907464970054073340036852442184414587772177753008511913377364966775792477387717262694468450099866775550614257191941835797445874557362115814601886902749237439492398087966544817154173072811937702110580330775581851211123491341435883319798273456296794954514173820352334127081705706502510709179711510240917772628308487366740741280043704807717608366220401933596364641284631036907635403895053036499618723044314773148779735006542501244942039455169872946018271985844759209768927953340447524637670938413827595013338859796135512187473850161303598087634723542727044978083220970836296653305188470017342167913572166172051819741354902582606590658382067039498769674611071582171914886494269818475850690414812481252963932223686078322390396586222238852602472958831686564971334200490182175112490433364675164900946902818404704835106260174052265784055642968397240262737313737007322288203637798365320295080314524864099419556398713380156353062937736280885716820226469419928595465390700629307079710611273715705695938635644841913194091407807776191951797748706106000922803167645881087385311847268311361092838264814899353459146959869764278464187826798546290981492648723002412475976344071283321798061003719251864595518596639432393032991023409676558943539937377229130132816883146259468718344018277257037013406135980469482324577407154032999045733141275895.3432 | 0.0000
+  1.2 | 32908.8896 | 58463467728170833376633133695001863276259293590926929026251227859007891876739460057725441400966420577009060860805883032969522911803372870882799865787473726926215148161529632590083389287080925059682489116446754279752928005457087175157581627230586554364417068189211136840990661174760199073702207450133797324318403866058202372178813998850887986769280847189341565507156189065295823921162851958925352114220880236114784962150135485415106748467247897246441194126125699204912883449386043559785865023459356275014504597646990160571664166410683323036984805434677654413174177920726210827006973855410386789516533036723888687725436216478665958434776205940192130053647653715221076841771578099896259902368829351569726536927952661429685419815305418450230567773264738536471211804481206474781470237730069753206249915908804615495060673071058534441654604668770343616386612119048579369195201590008082689834456232255266932976831478404670192731621439902738547169253818323045451045749609624500171633897705543164388470746657118050314064066768449450440405619135824055131398727045420324382226572368236570500391463795989258779677208133531636928003546809249007993065200108076924439703799231711400266122025052209803513232429907231051873161206025860851056337427740362763618748092029386371493898291580557004812947013231371383576580415676519066503391905962989205397824064923920045371823949776899815750413244195402085917098964452866825666226141169411712884994564949174271056284898570445214367063763956186792886147126466387576513166370247576466566827375268334148320298849218878848928271566491769458471357076035396330179659440244425914213309776100351793665960978678576150833311810944729586040624059867137538839913141142139636023129691775489034134511666020819676247950267220131499463010350308195762769192775344260909521732256844149916046793599150786757764962585268686580124987490115873389726527572428003433405659445349155536369077209682951123806333170190998931670309088422483075609203671527331975811507450670132060984691061148836994322505371265263690017938762760088575875666254883673433331627055180154954694693433502522592907190906966067656027637884202418119121728966267936832338377284832958974299187166554160783467156478554899314000348357280306042140481751668215838656488457943830180819301102535170705017482946779698265096226184239631924271857062033454725540956591929965181603262502135610768915716020374362368495244256420143645126927013882334008435586481691725030031204304273292938132599127402133470745819213047706793887965197191137237066440328777206799072470374264316425913530947082957300047105685634407092811630672103242089966046839626911122.7149 | 58463467728170833376633133695001863276259293590926929026251227859007891876739460057725441400966420577009060860805883032969522911803372870882799865787473726926215148161529632590083389287080925059682489116446754279752928005457087175157581627230586554364417068189211136840990661174760199073702207450133797324318403866058202372178813998850887986769280847189341565507156189065295823921162851958925352114220880236114784962150135485415106748467247897246441194126125699204912883449386043559785865023459356275014504597646990160571664166410683323036984805434677654413174177920726210827006973855410386789516533036723888687725436216478665958434776205940192130053647653715221076841771578099896259902368829351569726536927952661429685419815305418450230567773264738536471211804481206474781470237730069753206249915908804615495060673071058534441654604668770343616386612119048579369195201590008082689834456232255266932976831478404670192731621439902738547169253818323045451045749609624500171633897705543164388470746657118050314064066768449450440405619135824055131398727045420324382226572368236570500391463795989258779677208133531636928003546809249007993065200108076924439703799231711400266122025052209803513232429907231051873161206025860851056337427740362763618748092029386371493898291580557004812947013231371383576580415676519066503391905962989205397824064923920045371823949776899815750413244195402085917098964452866825666226141169411712884994564949174271056284898570445214367063763956186792886147126466387576513166370247576466566827375268334148320298849218878848928271566491769458471357076035396330179659440244425914213309776100351793665960978678576150833311810944729586040624059867137538839913141142139636023129691775489034134511666020819676247950267220131499463010350308195762769192775344260909521732256844149916046793599150786757764962585268686580124987490115873389726527572428003433405659445349155536369077209682951123806333170190998931670309088422483075609203671527331975811507450670132060984691061148836994322505371265263690017938762760088575875666254883673433331627055180154954694693433502522592907190906966067656027637884202418119121728966267936832338377284832958974299187166554160783467156478554899314000348357280306042140481751668215838656488457943830180819301102535170705017482946779698265096226184239631924271857062033454725540956591929965181603262502135610768915716020374362368495244256420143645126927013882334008435586481691725030031204304273292938132599127402133470745819213047706793887965197191137237066440328777206799072470374264316425913530947082957300047105685634407092811630672103242089966046839626911122.7149 | 0.0000
+(2 rows)
+
--
-- Tests for EXP()
--
diff --git a/src/test/regress/sql/numeric_big.sql b/src/test/regress/sql/numeric_big.sql
new file mode 100644
index 60080e8..6dbcf96
--- a/src/test/regress/sql/numeric_big.sql
+++ b/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()
--
```
```--
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```
• [HACKERS] More inaccurate results from numeric pow() Dean Rasheed