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Accuracy information for the MASS libraries for AIX
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Abstract
This document provides accuracy data for the MASS scalar and vector libraries for AIX, in comparison with the standard system math libraries.
 
 
Content
Accuracy information for the MASS libraries for AIX
In some cases MASS is not as accurate as the system math library, and it may handle edge cases differently (sqrt(Inf), for example). The following tables provide sample accuracy data for the system library (libm), libmass, libmassv, libmassvp4, libmassvp5, and libmassvp6 libraries. The numbers are based on the results for 1000 random arguments for each floating-point base-two exponent in the specified ranges. (Denormalized arguments were omitted.) The AIX system quadruple-precision functions were used to compute the errors. There may be portions of the valid input argument range for which accuracy is not as good as illustrated in the table.

If the function (or the corresponding quadruple-precision function) does not exist, or if the measurement was not done, the entry is left blank.

The entries in the percent correctly rounded (PCR) column are obtained by counting the number of correctly rounded results out of the tested random arguments. A result is correctly rounded when the function returns the floating-point number that is closest to the exact (infinite-precision) result. The column labelled libm lists the results obtained with the system math library, while the columns labeled mass, vp4, vp5, vp6 list the results obtained with the libmass.a, libmassvp4.a, libmassvp5.a, and libmassvp6.a libraries, respectively.

Although the vector names (e.g. vacos) are used in the Function column, the libm and mass columns refers to the corresponding scalar function (e.g. acos).

The Range Key at the end of this section applies to both tables. The accuracy tests were run on a POWER5 processor.

Maximum Observed Error (ulps) -- double-precision functions

Function  Range libm  mass   v    vp4   vp5   vp6
==================================================
vacos        B   .59  1.43  2.13  2.13  2.13  2.13
vacosh       G  2.35  1.80  1.80  1.80  1.80  1.80
vasin        B   .61  1.89  2.18  2.18  2.18  2.18
vasinh       D   .96  2.09  2.24  2.24  2.24  2.24
vatan        B        1.04
vatan        D        1.48
vatan2       D  1.59  1.67  1.95  1.95  1.95  1.95
vatanh       B  1.93  2.58  2.76  2.76  2.76  2.76
vcbrt        D   .66  1.16  1.16  1.16  1.16  1.16
vcopysign    D        0.00
vcos         B  1.02  1.18   .98  1.11  1.11  1.11
vcos         D   .93  1.47  1.47  1.47  1.47  1.47
vcosh        D   .97  1.10  2.70  2.70  2.70  2.70
vcosisin     B  1.02  1.18   .98   .98   .98   .98
vcosisin     D   .94  1.47  1.47  1.47  1.47  1.47
vdint        D              0.00  0.00  0.00  0.00
vdiv         D   .50        1.42  1.42  1.42  1.42
vdnint       D        0.00        0.00  0.00  0.00
verf         C        1.24
verfc        C        2.29
vexp         D   .50   .67   .67   .67   .67   .67
vexpm1       D   .98  2.11  2.11  2.11  2.11  2.11
vhypot       D        0.00
vlgamma      H        1.95
vlog         C   .50  1.14  1.75  1.75  1.75  1.75
vlog10       C  1.56  2.59  2.59  2.59  2.59  2.59
vlog1p       H   .97  1.58  1.58  1.58  1.58  1.58
vpow         C   .50   .65   .60   .60   .60   .60
vqdrt        C   .84        1.42  1.42  1.42  1.42
vrcbrt       D  1.69        1.02  1.02  1.02  1.02
vrec         D   .50         .50   .52   .52   .52
vrqdrt       C   .89        1.04  1.04  1.04  1.04
vrsqrt       C   .95   .56   .63   .56   .99   .99
vsin         B   .91   .82   .79  1.39  1.39  1.39
vsin         D   .94  1.44  1.44  1.44  1.44  1.44
vsincos      B  1.02  1.18   .98  1.15  1.15  1.15
vsincos      D   .94  1.47  1.47  1.47  1.47  1.47
vsinh        D  1.47  3.13  1.94  1.94  1.94  1.94
vsqrt        C   .50   .60   .63  1.02   .75   .75
vtan         D   .53  2.67  3.23  3.23  3.23  3.23
vtanh        F  2.53  2.52  3.00  3.00  3.00  3.00

Maximum Observed Error (ulps) -- single-precision functions

Function  Range libm  mass    v    vp4   vp5   vp6
==================================================
vsacos       B   .50   .54   .56   .56   .56   .56
vsacosh      G   .50   .51   .51   .51   .51   .51
vsasin       B   .50   .54   .56   .56   .56   .56
vsasinh      D   .50   .52   .50   .50   .50   .50
vsatan       B         .54
vsatan       D         .53
vsatan2      D   .50   .50   .50   .70   .70   .70
vsatanh      B         .51   .51   .51   .51   .51
vscbrt       D   .50   .53   .53   .53   .53   .53
vscopysign   D        0.00
vscos        B   .50   .51   .51   .51   .51   .51
vscos        D   .50   .51   .51   .51   .51   .51
vscosh       D   .50   .53   .53   .53   .53   .53
vscosisin    B   .50         .51   .51   .51   .51
vscosisin    D   .50         .51   .51   .51   .51
vsdiv        D   .50         .50  1.54  1.46  1.44
vserf        C         .58
vserfc       C         .55
vsexp        D   .50   .53   .53   .53   .53   .53
vsexpm1      D   .50   .50   .50   .50   .50   .50
vshypot      D         .50
vslgamma     H         .50
vslog        C   .50   .51   .51   .51   .51   .51
vslog10      C   .50   .51   .51   .51   .51   .51
vslog1p      H   .50   .51   .51   .51   .51   .51
vspow        C   .50   .53   .53   .53   .53   .53
vsqdrt       C   .85         .56   .56   .56   .56
vsrcbrt      D   .50         .51   .51   .51   .51
vsrec        D   .50         .66   .58   .50   .50
vsrint       D        0.00
vsrqdrt      C   .50         .52   .52   .52   .52
vsrsqrt      C   .50         .83   .88   .83   .83
vssin        B   .50   .50   .50   .51   .51   .51
vssin        D   .50   .51   .51   .51   .51   .51
vssincos     B   .50         .51   .51   .51   .51
vssincos     D   .50         .51   .51   .51   .51
vssinh       D   .50   .53   .53   .53   .53   .53
vssqrt       C   .50         .80  1.08   .80   .80
vstan        D   .50   .51   .68   .68   .68   .68
vstanh       F   .50   .57   .74   .74   .74   .74

PCR -- double-precision functions

Function  Range libm   mass     v     vp4    vp5    vp6
========================================================
vacos        B  99.44  88.00  85.00  85.00  85.00  85.00
vacosh       G  86.78  92.00  92.00  92.00  92.00  92.00
vasin        B  98.82  87.00  69.00  69.00  69.00  69.00
vasinh       D  90.92  96.00  96.00  96.00  96.00  96.00
vatan        B         94.00
vatan        D         99.00
vatan2       D  74.66  86.00  84.00  84.00  84.00  84.00
vatanh       B  83.59  83.00  78.00  78.00  78.00  78.00
vcbrt        D  91.58  85.00  85.00  85.00  85.00  85.00
vcopysign    D        100.00
vcos         B  92.95  92.00  93.00  86.00  86.00  86.00
vcos         D  86.86  84.00  84.00  83.00  83.00  83.00
vcosh        D  95.64  92.00  58.00  58.00  58.00  58.00
vcosisin     B  81.31  95.00  95.00  95.00  95.00  95.00
vcosisin     D  86.03  84.00  84.00  84.00  84.00  84.00
vdint        D               100.00 100.00 100.00 100.00
vdiv         D 100.00         74.00  74.00  74.00  74.00
vdnint       D        100.00        100.00 100.00 100.00
verf         C        100.00
verfc        C         91.00
vexp         D  99.95  97.00  97.00  97.00  97.00  97.00
vexpm1       D  95.98  99.00  99.00  99.00  99.00  99.00
vhypot       D        100.00
vlgamma      H         66.00
vlog         C  99.99 100.00  99.00  99.00  99.00  99.00
vlog10       C  64.63  99.00  99.00  99.00  99.00  99.00
vlog1p       H  99.91  99.00  99.00  99.00  99.00  99.00
vpow         C  99.96  97.00  97.00  97.00  97.00  97.00
vqdrt        C  87.26         76.00  76.00  76.00  76.00
vrcbrt       D  70.51         87.00  87.00  87.00  87.00
vrec         D 100.00        100.00 100.00 100.00 100.00
vrqdrt       C  85.18         86.00  86.00  86.00  86.00
vrsqrt       C  85.36  99.00  97.00  97.00  85.00  85.00
vsin         B  81.31  97.00  97.00  92.00  92.00  92.00
vsin         D  86.03  84.00  84.00  83.00  83.00  83.00
vsincos      B  81.31  95.00  95.00  91.00  91.00  91.00
vsincos      D  86.03  84.00  84.00  84.00  84.00  84.00
vsinh        D  94.78  57.00  83.00  83.00  83.00  83.00
vsqrt        C 100.00  97.00  97.00  87.00  87.00  97.00
vtan         D  99.58  64.00  54.00  54.00  54.00  54.00
vtanh        F  96.97  91.00  57.00  57.00  57.00  57.00

PCR -- single-precision functions

Function  Range libm   mass     v     vp4    vp5    vp6
========================================================
vsacos       B 100.00  99.00  99.00  99.00  99.00  99.00
vsacosh      G 100.00 100.00 100.00 100.00 100.00 100.00
vsasin       B 100.00  99.00  98.00  98.00  98.00  98.00
vsasinh      D 100.00 100.00 100.00 100.00 100.00 100.00
vsatan       B         98.00
vsatan       D        100.00
vsatan2      D 100.00 100.00 100.00  99.00  99.00  99.00
vsatanh      B        100.00 100.00 100.00 100.00 100.00
vscbrt       D 100.00  99.00  99.00  99.00  99.00  99.00
vscopysign   D        100.00
vscos        B 100.00  99.00  99.00  99.00  99.00  99.00
vscos        D 100.00 100.00 100.00 100.00 100.00 100.00
vscosh       D 100.00  99.00  99.00  99.00  99.00  99.00
vscosisin    B 100.00        100.00 100.00 100.00 100.00
vscosisin    D 100.00        100.00 100.00 100.00 100.00
vsdiv        D 100.00        100.00  74.00  74.00  74.00
vserf        C        100.00
vserfc       C        100.00
vsexp        D 100.00  99.00  99.00  99.00  99.00  99.00
vsexpm1      D 100.00 100.00 100.00 100.00 100.00 100.00
vshypot      D        100.00
vslgamma     H        100.00
vslog        C 100.00 100.00 100.00 100.00 100.00 100.00
vslog10      C 100.00 100.00 100.00 100.00 100.00 100.00
vslog1p      H 100.00 100.00 100.00 100.00 100.00 100.00
vspow        C 100.00  99.00  99.00  99.00  99.00  99.00
vsqdrt       C  87.40         97.00  97.00  97.00  97.00
vsrcbrt      D 100.00        100.00 100.00 100.00 100.00
vsrec        D 100.00         92.00  99.00 100.00 100.00
vsrint       D        100.00
vsrqdrt      C 100.00         99.00  99.00  99.00  99.00
vsrsqrt      C 100.00         87.00  88.00  86.00  86.00
vssin        B 100.00 100.00 100.00 100.00 100.00 100.00
vssin        D 100.00 100.00 100.00 100.00 100.00 100.00
vssincos     B 100.00        100.00 100.00 100.00 100.00
vssincos     D 100.00        100.00 100.00 100.00 100.00
vssinh       D 100.00  99.00  99.00  99.00  99.00  99.00
vssqrt       C 100.00         86.00  84.00  86.00  86.00
vstan        D 100.00 100.00  98.00  98.00  98.00  98.00
vstanh       F 100.00 100.00  95.00  95.00  95.00  95.00

Range Key                   

A      0,1                          
B     -1,1
C      0,100
D   -100,100
E    -10,10
F    -20,20
G      1,100
H     -1,100
I      0,10

 
 
Original publication date
2005/12/5
 
 

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Document information
 Product categories:
 Software
 Software Development
 Traditional Progamming Language & Compilers
 Mathematical Acceleration Subsystem
 Libraries
 Operating system(s):
  AIX
 Software version:
  4.4
 Reference #:
  7007004
 IBM Group:
 Software Group
 Modified date:
 2007-06-25

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