1 December 2023 10:02:09.971 AM
toms715_test():
FORTRAN77 version
Test toms715().
1Test of anorm(x) vs double series expansion
2000 Random arguments were tested from the interval ( -0.663, 0.663)
ANORM(X) was larger 253 times,
agreed 1595 times, and
was smaller 152 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.4237E-15 = 2 ** -51.07
occurred for X =-0.637105E+00
The estimated loss of base 2 significant digits is 1.93
The root mean square relative error was 0.8569E-16 = 2 ** -53.37
The estimated loss of base 2 significant digits is 0.00
Test of anorm(x) vs Taylor series about x-1/2
2000 Random arguments were tested from the interval ( -5.657, -0.663)
ANORM(X) was larger 680 times,
agreed 476 times, and
was smaller 844 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.1178E-14 = 2 ** -49.59
occurred for X =-0.506677E+01
The estimated loss of base 2 significant digits is 3.41
The root mean square relative error was 0.2661E-15 = 2 ** -51.74
The estimated loss of base 2 significant digits is 1.26
Test of anorm(x) vs Taylor series about x-1/2
2000 Random arguments were tested from the interval (-37.000, -5.657)
ANORM(X) was larger 652 times,
agreed 683 times, and
was smaller 665 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.6139E-15 = 2 ** -50.53
occurred for X =-0.318733E+02
The estimated loss of base 2 significant digits is 2.47
The root mean square relative error was 0.1769E-15 = 2 ** -52.33
The estimated loss of base 2 significant digits is 0.67
1Special Tests
Check of identity anorm(X) + anorm(-X) = 1.0
X ANORM(-x) ANORM(x)+ANORM(-x)-1
0.503099E+01 0.243975E-06 0.000000E+00
0.887334E+01 0.354915E-18 0.000000E+00
0.916662E+01 0.244045E-19 0.000000E+00
0.582615E+01 0.283604E-08 0.000000E+00
0.826795E+01 0.681417E-16 0.000000E+00
0.349256E+01 0.239203E-03 0.000000E+00
0.657015E+01 0.251320E-10 0.000000E+00
0.126816E+01 0.102370E+00 0.000000E+00
0.852033E+01 0.795493E-17 0.000000E+00
0.504026E+01 0.232453E-06 0.000000E+00
Test of special arguments
ANORM ( 0.179769+309) = 0.100000E+01
ANORM ( 0.000000E+00) = 0.500000E+00
ANORM (-0.179769+309) = 0.000000E+00
Test of Error Returns
ANORM will be called with the argument -0.281395E+02
The result should not underflow
ANORM returned the value 0.160919-173
ANORM will be called with the argument -0.375194E+02
The result may underflow
ANORM returned the value 0.000000E+00
This concludes the tests
1Test of Dawson's Integral vs Taylor expansion
2000 Random arguments were tested from the interval ( 0.06, 1.00)
F(X) was larger 532 times,
agreed 963 times, and
was smaller 505 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.4422E-15 = 2 ** -51.01
occurred for X = 0.126888E+00
The estimated loss of base 2 significant digits is 1.99
The root mean square relative error was 0.1339E-15 = 2 ** -52.73
The estimated loss of base 2 significant digits is 0.27
1Test of Dawson's Integral vs Taylor expansion
2000 Random arguments were tested from the interval ( 1.00, 2.50)
F(X) was larger 663 times,
agreed 794 times, and
was smaller 543 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.6119E-15 = 2 ** -50.54
occurred for X = 0.215456E+01
The estimated loss of base 2 significant digits is 2.46
The root mean square relative error was 0.1632E-15 = 2 ** -52.44
The estimated loss of base 2 significant digits is 0.56
1Test of Dawson's Integral vs Taylor expansion
2000 Random arguments were tested from the interval ( 2.50, 5.00)
F(X) was larger 513 times,
agreed 1090 times, and
was smaller 397 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.3846E-15 = 2 ** -51.21
occurred for X = 0.361607E+01
The estimated loss of base 2 significant digits is 1.79
The root mean square relative error was 0.1116E-15 = 2 ** -52.99
The estimated loss of base 2 significant digits is 0.01
1Test of Dawson's Integral vs Taylor expansion
2000 Random arguments were tested from the interval ( 5.00,10.00)
F(X) was larger 429 times,
agreed 1139 times, and
was smaller 432 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.3784E-15 = 2 ** -51.23
occurred for X = 0.689120E+01
The estimated loss of base 2 significant digits is 1.77
The root mean square relative error was 0.1053E-15 = 2 ** -53.08
The estimated loss of base 2 significant digits is 0.00
1Special Tests
Estimated loss of base 2 significant digits in
X F(x)+F(-x)
0.057 0.00
3.145 0.00
1.086 0.00
3.761 0.00
2.133 0.00
2.656 0.00
0.009 0.00
1.078 0.00
2.692 0.00
0.447 0.00
Test of special arguments
F(XMIN) = 0.22250738585072014-307
Test of Error Returns
DAW will be called with the argument 0.223834+308
This should not underflow
DAW returned the value 0.223380-307
DAW will be called with the argument 0.224712+308
This may underflow
DAW returned the value 0.000000E+00
DAW will be called with the argument 0.225589+308
This may underflow
DAW returned the value 0.000000E+00
This concludes the tests
1Test of LGAMA(X) vs LN(2*SQRT(PI))-2X*LN(2)+LGAMA(2X)-LGAMA(X+1/2)
2000 Random arguments were tested from the interval ( 0.0, 0.9)
LGAMA(X) was larger 674 times,
agreed 975 times, and
was smaller 351 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.4233E-15 = 2 ** -51.07
occurred for X = 0.115943E+00
The estimated loss of base 2 significant digits is 1.93
The root mean square relative error was 0.1274E-15 = 2 ** -52.80
The estimated loss of base 2 significant digits is 0.20
1Test of LGAMA(X) vs LN(2*SQRT(PI))-(2X-1)*LN(2)+LGAMA(X-1/2)-LGAMA(2X-1)
2000 Random arguments were tested from the interval ( 1.3, 1.6)
LGAMA(X) was larger 262 times,
agreed 835 times, and
was smaller 903 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.7597E-15 = 2 ** -50.23
occurred for X = 0.162210E+01
The estimated loss of base 2 significant digits is 2.77
The root mean square relative error was 0.1318E-15 = 2 ** -52.75
The estimated loss of base 2 significant digits is 0.25
1Test of LGAMA(X) vs -LN(2*SQRT(PI))+X*LN(2)+LGAMA(X/2)+LGAMA(X/2+1/2)
2000 Random arguments were tested from the interval ( 4.0, 20.0)
LGAMA(X) was larger 655 times,
agreed 979 times, and
was smaller 366 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.4114E-15 = 2 ** -51.11
occurred for X = 0.805451E+01
The estimated loss of base 2 significant digits is 1.89
The root mean square relative error was 0.1279E-15 = 2 ** -52.80
The estimated loss of base 2 significant digits is 0.20
1Special Tests
Test of special arguments
LGAMA ( 0.222045E-15) = 0.360437E+02
LGAMA ( 0.500000E+00) = 0.572365E+00
LGAMA ( 0.100000E+01) = 0.000000E+00
LGAMA ( 0.200000E+01) = 0.000000E+00
1Test of Error Returns
LGAMA will be called with the argument 0.222507-307
This should not trigger an error message
LGAMA returned the value 0.708396E+03
LGAMA will be called with the argument 0.253442+306
This should not trigger an error message
LGAMA returned the value 0.177972+309
LGAMA will be called with the argument-0.100000E+01
This should trigger an error message
LGAMA returned the value Infinity
LGAMA will be called with the argument 0.000000E+00
This should trigger an error message
LGAMA returned the value Infinity
LGAMA will be called with the argument 0.177972+309
This should trigger an error message
LGAMA returned the value Infinity
This concludes the tests
1Test of Ei(x) vs series expansion
2000 Random arguments were tested from the interval ( 0.188, 0.310)
EI(X) was larger 422 times,
agreed 639 times, and
was smaller 939 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.5505E-15 = 2 ** -50.69
occurred for X = 0.300029E+00
The estimated loss of base 2 significant digits is 2.31
The root mean square relative error was 0.1817E-15 = 2 ** -52.29
The estimated loss of base 2 significant digits is 0.71
1Test of Ei(x) vs series expansion
2000 Random arguments were tested from the interval ( 0.435, 6.000)
EI(X) was larger 619 times,
agreed 682 times, and
was smaller 699 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.1593E-14 = 2 ** -49.16
occurred for X = 0.591180E+01
The estimated loss of base 2 significant digits is 3.84
The root mean square relative error was 0.2482E-15 = 2 ** -51.84
The estimated loss of base 2 significant digits is 1.16
1Test of Ei(x) vs series expansion
2000 Random arguments were tested from the interval ( 6.000, 12.000)
EI(X) was larger 613 times,
agreed 813 times, and
was smaller 574 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.1439E-14 = 2 ** -49.30
occurred for X = 0.604245E+01
The estimated loss of base 2 significant digits is 3.70
The root mean square relative error was 0.1584E-15 = 2 ** -52.49
The estimated loss of base 2 significant digits is 0.51
1Test of Ei(x) vs series expansion
2000 Random arguments were tested from the interval ( 12.000, 24.000)
EI(X) was larger 567 times,
agreed 848 times, and
was smaller 585 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.4427E-15 = 2 ** -51.00
occurred for X = 0.136210E+02
The estimated loss of base 2 significant digits is 2.00
The root mean square relative error was 0.1479E-15 = 2 ** -52.59
The estimated loss of base 2 significant digits is 0.41
1Test of Ei(x) vs series expansion
2000 Random arguments were tested from the interval ( 24.000, 48.000)
EI(X) was larger 561 times,
agreed 895 times, and
was smaller 544 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.4467E-15 = 2 ** -50.99
occurred for X = 0.286719E+02
The estimated loss of base 2 significant digits is 2.01
The root mean square relative error was 0.1388E-15 = 2 ** -52.68
The estimated loss of base 2 significant digits is 0.32
1Test of Ei(x) vs series expansion
2000 Random arguments were tested from the interval ( -0.250, -1.000)
EI(X) was larger 601 times,
agreed 895 times, and
was smaller 504 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.8323E-15 = 2 ** -50.09
occurred for X =-0.963175E+00
The estimated loss of base 2 significant digits is 2.91
The root mean square relative error was 0.1556E-15 = 2 ** -52.51
The estimated loss of base 2 significant digits is 0.49
1Test of Ei(x) vs series expansion
2000 Random arguments were tested from the interval ( -1.000, -4.000)
EI(X) was larger 712 times,
agreed 578 times, and
was smaller 710 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.8403E-15 = 2 ** -50.08
occurred for X =-0.228801E+01
The estimated loss of base 2 significant digits is 2.92
The root mean square relative error was 0.2266E-15 = 2 ** -51.97
The estimated loss of base 2 significant digits is 1.03
1Test of Ei(x) vs series expansion
2000 Random arguments were tested from the interval ( -4.000,-10.000)
EI(X) was larger 559 times,
agreed 891 times, and
was smaller 550 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.5990E-15 = 2 ** -50.57
occurred for X =-0.624667E+01
The estimated loss of base 2 significant digits is 2.43
The root mean square relative error was 0.1431E-15 = 2 ** -52.63
The estimated loss of base 2 significant digits is 0.37
Test of special arguments
EI ( 0.375000E+00) = 0.969138E-02
The relative error is 0.1398E-16 = 2 ** -55.99
The estimated loss of base 2 significant digits is 0.00
Test of Error Returns
EONE will be called with the argument 0.701800E+03
This should not underflow
EONE returned the value 0.231901-307
EONE will be called with the argument 0.701844E+03
This should underflow
EONE returned the value-0.000000E+00
EI will be called with the argument 0.716300E+03
This should not overflow
EI returned the value 0.170079+309
EI will be called with the argument 0.716356E+03
This should overflow
EI returned the value 0.179000+309
EXPEI will be called with the argument 0.449423+308
This should not underflow
EXPEI returned the value 0.222507-307
EI will be called with the argument 0.000000E+00
This should overflow
EI returned the value-0.179000+309
This concludes the tests
1Test of erf(x) vs double series expansion
2000 Random arguments were tested from the interval ( 0.000, 0.469)
ERF(X) was larger 384 times,
agreed 1286 times, and
was smaller 330 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.2346E-15 = 2 ** -51.92
occurred for X = 0.447558E+00
The estimated loss of base 2 significant digits is 1.08
The root mean square relative error was 0.9124E-16 = 2 ** -53.28
The estimated loss of base 2 significant digits is 0.00
Test of erfc(x) vs exp(x+1/4) SUM i^n erfc(x+1/2)
2000 Random arguments were tested from the interval ( 0.469, 2.000)
ERFC(X) was larger 588 times,
agreed 846 times, and
was smaller 566 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.6084E-15 = 2 ** -50.55
occurred for X = 0.119422E+01
The estimated loss of base 2 significant digits is 2.45
The root mean square relative error was 0.1562E-15 = 2 ** -52.51
The estimated loss of base 2 significant digits is 0.49
1Test of exp(x*x) erfc(x) vs SUM i^n erfc(x+1/2)
2000 Random arguments were tested from the interval ( 0.469, 2.000)
ERFCX(X) was larger 817 times,
agreed 618 times, and
was smaller 565 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.6622E-15 = 2 ** -50.42
occurred for X = 0.141916E+01
The estimated loss of base 2 significant digits is 2.58
The root mean square relative error was 0.2179E-15 = 2 ** -52.03
The estimated loss of base 2 significant digits is 0.97
Test of erfc(x) vs exp(x+1/4) SUM i^n erfc(x+1/2)
2000 Random arguments were tested from the interval ( 2.000, 26.000)
ERFC(X) was larger 595 times,
agreed 845 times, and
was smaller 560 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.6225E-15 = 2 ** -50.51
occurred for X = 0.244585E+02
The estimated loss of base 2 significant digits is 2.49
The root mean square relative error was 0.1484E-15 = 2 ** -52.58
The estimated loss of base 2 significant digits is 0.42
1Test of exp(x*x) erfc(x) vs SUM i^n erfc(x+1/2)
2000 Random arguments were tested from the interval ( 2.000, 20.000)
ERFCX(X) was larger 474 times,
agreed 1014 times, and
was smaller 512 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.9356E-15 = 2 ** -49.93
occurred for X = 0.301848E+01
The estimated loss of base 2 significant digits is 3.07
The root mean square relative error was 0.1389E-15 = 2 ** -52.68
The estimated loss of base 2 significant digits is 0.32
1Special Tests
Estimated loss of base 2significant digits in
X Erf(x)+Erf(-x) Erf(x)+Erfc(x)-1 Erfcx(x)-exp(x*x)*erfc(x)
0.000 0.00 0.00 0.00
-0.500 0.00 0.00 0.03
-1.000 0.00 0.00 0.00
-1.500 0.00 0.00 0.00
-2.000 0.00 0.00 0.00
-2.500 0.00 0.00 0.00
-3.000 0.00 0.00 0.02
-3.500 0.00 0.00 0.33
-4.000 0.00 0.00 0.00
-4.500 0.00 0.00 0.00
Test of special arguments
ERF ( 0.179769+309) = 0.100000E+01
ERF ( 0.000000E+00) = 0.000000E+00
ERFC ( 0.000000E+00) = 0.100000E+01
ERFC (-0.179769+309) = 0.200000E+01
Test of Error Returns
ERFC will be called with the argument 0.199074E+02
This should not underflow
ERFC returned the value 0.217879-173
ERFC will be called with the argument 0.265433E+02
This may underflow
ERFC returned the value 0.222508-307
ERFCX will be called with the argument 0.237712+308
This should not underflow
ERFCX returned the value 0.237341-307
ERFCX will be called with the argument-0.239659E+02
This should not overflow
ERFCX returned the value 0.554007+250
ERFCX will be called with the argument-0.266287E+02
This may overflow
ERFCX returned the value 0.179000+309
This concludes the tests
1Test of GAMMA(X) vs Duplication Formula
2000 Random arguments were tested from the interval ( 0.000, 2.000)
GAMMA(X) was larger 639 times,
agreed 840 times, and
was smaller 521 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.4839E-15 = 2 ** -50.88
occurred for X = 0.120131E+01
The estimated loss of base 2 significant digits is 2.12
The root mean square relative error was 0.1351E-15 = 2 ** -52.72
The estimated loss of base 2 significant digits is 0.28
1Test of GAMMA(X) vs Duplication Formula
2000 Random arguments were tested from the interval ( 2.000, 10.000)
GAMMA(X) was larger 663 times,
agreed 721 times, and
was smaller 616 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.7028E-15 = 2 ** -50.34
occurred for X = 0.656492E+01
The estimated loss of base 2 significant digits is 2.66
The root mean square relative error was 0.1867E-15 = 2 ** -52.25
The estimated loss of base 2 significant digits is 0.75
1Test of GAMMA(X) vs Duplication Formula
2000 Random arguments were tested from the interval ( 10.000,171.124)
GAMMA(X) was larger 847 times,
agreed 483 times, and
was smaller 670 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.8467E-15 = 2 ** -50.07
occurred for X = 0.155165E+03
The estimated loss of base 2 significant digits is 2.93
The root mean square relative error was 0.2610E-15 = 2 ** -51.77
The estimated loss of base 2 significant digits is 1.23
1Test of GAMMA(X) vs Duplication Formula
2000 Random arguments were tested from the interval ( -4.750, -4.250)
GAMMA(X) was larger 880 times,
agreed 545 times, and
was smaller 575 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.9033E-15 = 2 ** -49.98
occurred for X =-0.438643E+01
The estimated loss of base 2 significant digits is 3.02
The root mean square relative error was 0.2240E-15 = 2 ** -51.99
The estimated loss of base 2 significant digits is 1.01
1Special Tests
Test of special arguments
GAMMA (-0.500000E+00) = -0.354491E+01
GAMMA ( 0.224755-307) = 0.444929+308
GAMMA ( 0.100000E+01) = 0.100000E+01
GAMMA ( 0.200000E+01) = 0.100000E+01
GAMMA ( 0.169908E+03) = 0.266542+305
1Test of Error Returns
GAMMA will be called with the argument-0.100000E+01
This should trigger an error message
GAMMA returned the value NaN
GAMMA will be called with the argument 0.000000E+00
This should trigger an error message
GAMMA returned the value Infinity
GAMMA will be called with the argument 0.222507-307
This should trigger an error message
GAMMA returned the value 0.449423+308
GAMMA will be called with the argument 0.171624E+03
This should trigger an error message
GAMMA returned the value Infinity
This concludes the tests
1Test of I0(X) vs Multiplication Theorem
2000 Random arguments were tested from the interval ( 0.00, 2.00)
I0(X) was larger 494 times,
agreed 973 times, and
was smaller 533 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.5078E-15 = 2 ** -50.81
occurred for X = 0.107724E+01
The estimated loss of base 2 significant digits is 2.19
The root mean square relative error was 0.1433E-15 = 2 ** -52.63
The estimated loss of base 2 significant digits is 0.37
1Test of I0(X) vs Taylor series
2000 Random arguments were tested from the interval ( 2.00, 7.50)
I0(X) was larger 697 times,
agreed 601 times, and
was smaller 702 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.8889E-15 = 2 ** -50.00
occurred for X = 0.618930E+01
The estimated loss of base 2 significant digits is 3.00
The root mean square relative error was 0.2206E-15 = 2 ** -52.01
The estimated loss of base 2 significant digits is 0.99
1Test of I0(X) vs Taylor series
2000 Random arguments were tested from the interval ( 7.50,15.00)
I0(X) was larger 832 times,
agreed 303 times, and
was smaller 865 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.1538E-14 = 2 ** -49.21
occurred for X = 0.127215E+02
The estimated loss of base 2 significant digits is 3.79
The root mean square relative error was 0.4151E-15 = 2 ** -51.10
The estimated loss of base 2 significant digits is 1.90
1Test of I0(X) vs Taylor series
2000 Random arguments were tested from the interval (15.00,30.00)
I0(X) was larger 583 times,
agreed 814 times, and
was smaller 603 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.6851E-15 = 2 ** -50.37
occurred for X = 0.150006E+02
The estimated loss of base 2 significant digits is 2.63
The root mean square relative error was 0.1576E-15 = 2 ** -52.49
The estimated loss of base 2 significant digits is 0.51
1Special Tests
Test with extreme arguments
I0(XMIN) = 0.10000000000000000E+01
I0(0) = 0.10000000000000000E+01
I0(-0.28449822688652987E+00 ) = 0.10203374025948893E+01
I0( 0.28449822688652987E+00 ) = 0.10203374025948893E+01
E**-X * I0(XMAX) = 0.29754474593158999-154
Tests near the largest argument for unscaled functions
I0( 0.69235094188622168E+03 ) = 0.73285657728857090+299
I0( 0.73629899972079636E+03 ) = 0.17900000000000000+309
This concludes the tests.
1Test of I1(X) vs Multiplication Theorem
2000 Random arguments were tested from the interval ( 0.00, 1.00)
I1(X) was larger 692 times,
agreed 675 times, and
was smaller 633 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.6379E-15 = 2 ** -50.48
occurred for X = 0.130257E+00
The estimated loss of base 2 significant digits is 2.52
The root mean square relative error was 0.1813E-15 = 2 ** -52.29
The estimated loss of base 2 significant digits is 0.71
1Test of I1(X) vs Taylor series
2000 Random arguments were tested from the interval ( 1.00, 7.50)
I1(X) was larger 718 times,
agreed 578 times, and
was smaller 704 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.1016E-14 = 2 ** -49.81
occurred for X = 0.688286E+01
The estimated loss of base 2 significant digits is 3.19
The root mean square relative error was 0.2172E-15 = 2 ** -52.03
The estimated loss of base 2 significant digits is 0.97
1Test of I1(X) vs Taylor series
2000 Random arguments were tested from the interval ( 7.50,15.00)
I1(X) was larger 806 times,
agreed 348 times, and
was smaller 846 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.1219E-14 = 2 ** -49.54
occurred for X = 0.125173E+02
The estimated loss of base 2 significant digits is 3.46
The root mean square relative error was 0.3900E-15 = 2 ** -51.19
The estimated loss of base 2 significant digits is 1.81
1Test of I1(X) vs Taylor series
2000 Random arguments were tested from the interval (15.00,30.00)
I1(X) was larger 633 times,
agreed 759 times, and
was smaller 608 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.6949E-15 = 2 ** -50.35
occurred for X = 0.150215E+02
The estimated loss of base 2 significant digits is 2.65
The root mean square relative error was 0.1605E-15 = 2 ** -52.47
The estimated loss of base 2 significant digits is 0.53
1Special Tests
Test with extreme arguments
I1(XMIN) = 0.11125369292536007-307
I1(0) = 0.00000000000000000E+00
I1(-0.74433904881837309E+00 ) = -0.39854607729907382E+00
I1( 0.74433904881837309E+00 ) = 0.39854607729907382E+00
E**-X * I1(XMAX) = 0.29754474593158999-154
Tests near the largest argument for unscaled functions
I1( 0.69235162141875753E+03 ) = 0.73282458365806542+299
I1( 0.73629972238772166E+03 ) = 0.17900000000000000+309
This concludes the tests.
1Test of J0(X) VS Taylor expansion
2000 random arguments were tested from the interval ( 0.0, 4.0)
ABS(J0(X)) was larger 501 times
agreed 1003 times, and
was smaller 496 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.1122E-14 = 2 ** -49.66
occurred for X = 0.244691E+01
The estimated loss of base 2 significant digits is 3.34
The root mean square relative error was 0.1579E-15 = 2 ** -52.49
The estimated loss of base 2 significant digits is 0.51
1Test of J0(X) VS Taylor expansion
2000 random arguments were tested from the interval ( 4.0, 8.0)
ABS(J0(X)) was larger 659 times
agreed 652 times, and
was smaller 689 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.6201E-15 = 2 ** -50.52
occurred for X = 0.431690E+01
The estimated loss of base 2 significant digits is 2.48
The root mean square relative error was 0.1933E-15 = 2 ** -52.20
The estimated loss of base 2 significant digits is 0.80
1Test of J0(X) VS Taylor expansion
2000 random arguments were tested from the interval ( 8.0, 20.0)
ABS(J0(X)) was larger 699 times
agreed 665 times, and
was smaller 636 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.6142E-15 = 2 ** -50.53
occurred for X = 0.973690E+01
The estimated loss of base 2 significant digits is 2.47
The root mean square relative error was 0.1863E-15 = 2 ** -52.25
The estimated loss of base 2 significant digits is 0.75
1Special Tests
Accuracy near zeros
X BESJ0(X) Loss of base 2 digits
0.2406250000E+01 -0.739276482217003E-03 2.72
0.5519531250E+01 -0.186086517975737E-03 6.12
Test with extreme arguments
J0 will be called with the argument 0.1797693135+309
This may stop execution.
J0 returned the value -0.41869868495853734-154
This concludes the tests.
1Test of J1(X) VS Maclaurin expansion
2000 random arguments were tested from the interval ( 0.0, 1.0)
ABS(J1(X)) was larger 219 times
agreed 1573 times, and
was smaller 208 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.2220E-15 = 2 ** -52.00
occurred for X = 0.517111E+00
The estimated loss of base 2 significant digits is 1.00
The root mean square relative error was 0.7076E-16 = 2 ** -53.65
The estimated loss of base 2 significant digits is 0.00
1Test of J1(X) VS local Taylor expansion
2000 random arguments were tested from the interval ( 1.0, 4.0)
ABS(J1(X)) was larger 630 times
agreed 765 times, and
was smaller 605 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.1785E-14 = 2 ** -48.99
occurred for X = 0.386321E+01
The estimated loss of base 2 significant digits is 4.01
The root mean square relative error was 0.2214E-15 = 2 ** -52.00
The estimated loss of base 2 significant digits is 1.00
1Test of J1(X) VS local Taylor expansion
2000 random arguments were tested from the interval ( 4.0, 8.0)
ABS(J1(X)) was larger 692 times
agreed 610 times, and
was smaller 698 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.8900E-15 = 2 ** -50.00
occurred for X = 0.697672E+01
The estimated loss of base 2 significant digits is 3.00
The root mean square relative error was 0.2112E-15 = 2 ** -52.07
The estimated loss of base 2 significant digits is 0.93
1Test of J1(X) VS local Taylor expansion
2000 random arguments were tested from the interval ( 8.0, 20.0)
ABS(J1(X)) was larger 696 times
agreed 582 times, and
was smaller 722 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.8008E-15 = 2 ** -50.15
occurred for X = 0.101043E+02
The estimated loss of base 2 significant digits is 2.85
The root mean square relative error was 0.2043E-15 = 2 ** -52.12
The estimated loss of base 2 significant digits is 0.88
1Special Tests
Accuracy near zeros
X BESJ1(X) Loss of base 2 digits
0.3832031250E+01 -0.131003930013275E-03 8.37
0.7015625000E+01 0.115034607023044E-04 11.02
Test with extreme arguments
J1 will be called with the argument 0.1797693135+309
This may stop execution.
J1 returned the value 0.42287458488299958-154
This concludes the tests.
1Test of K0(X) vs Multiplication Theorem
2000 random arguments were tested from the interval ( 0.0, 1.0)
ABS(K0(X)) was larger 535 times,
agreed 1071 times, and
was smaller 394 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.5184E-15 = 2 ** -50.78
occurred for X = 0.987953E+00
The estimated loss of base 2 significant digits is 2.22
The root mean square relative error was 0.1209E-15 = 2 ** -52.88
The estimated loss of base 2 significant digits is 0.12
1Test of K0(X) vs Multiplication Theorem
2000 random arguments were tested from the interval ( 1.0, 8.0)
ABS(K0(X)) was larger 826 times,
agreed 514 times, and
was smaller 660 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.8382E-15 = 2 ** -50.08
occurred for X = 0.187771E+01
The estimated loss of base 2 significant digits is 2.92
The root mean square relative error was 0.2465E-15 = 2 ** -51.85
The estimated loss of base 2 significant digits is 1.15
1Test of K0(X) vs Multiplication Theorem
2000 random arguments were tested from the interval ( 8.0, 20.0)
ABS(K0(X)) was larger 646 times,
agreed 609 times, and
was smaller 745 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum absolute error of 0.7285E-15 = 2 ** -50.29
occurred for X = 0.164197E+02
The estimated loss of base 2 significant digits is 2.71
The root mean square absolute error was 0.2115E-15 = 2 ** -52.07
The estimated loss of base 2 significant digits is 0.93
1Special Tests
Test with extreme arguments
K0(XMIN) = 0.70851235004792250E+03
K0(0) = 0.17900000000000000+309
K0(-0.61041516186224920E+00 ) = 0.17900000000000000+309
E**X * K0(XMAX) = 0.93476438793292451-154
K0( 0.66125877272454943E+03 ) = 0.32118560786711748-288
K0( 0.79351052726945932E+03 ) = 0.00000000000000000E+00
1Test of K1(X) vs Multiplication Theorem
2000 random arguments were tested from the interval ( 0.0, 1.0)
ABS(K1(X)) was larger 626 times,
agreed 729 times, and
was smaller 645 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.8679E-15 = 2 ** -50.03
occurred for X = 0.964556E+00
The estimated loss of base 2 significant digits is 2.97
The root mean square relative error was 0.1831E-15 = 2 ** -52.28
The estimated loss of base 2 significant digits is 0.72
1Test of K1(X) vs Multiplication Theorem
2000 random arguments were tested from the interval ( 1.0, 8.0)
ABS(K1(X)) was larger 701 times,
agreed 546 times, and
was smaller 753 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.9703E-15 = 2 ** -49.87
occurred for X = 0.198302E+01
The estimated loss of base 2 significant digits is 3.13
The root mean square relative error was 0.2487E-15 = 2 ** -51.84
The estimated loss of base 2 significant digits is 1.16
1Test of K1(X) vs Multiplication Theorem
2000 random arguments were tested from the interval ( 8.0, 20.0)
ABS(K1(X)) was larger 760 times,
agreed 592 times, and
was smaller 648 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum absolute error of 0.7424E-15 = 2 ** -50.26
occurred for X = 0.166829E+02
The estimated loss of base 2 significant digits is 2.74
The root mean square absolute error was 0.2160E-15 = 2 ** -52.04
The estimated loss of base 2 significant digits is 0.96
1Special Tests
Test with extreme arguments
K1(XLEAST) = 0.44843049327354256+308
K1(XMIN) = 0.17900000000000000+309
K1(0) = 0.17900000000000000+309
K1(-0.88999635885180317E+00 ) = 0.17900000000000000+309
E**X * K1(XMAX) = 0.93476438793292451-154
K1( 0.66125943635324325E+03 ) = 0.32121497573487907-288
K1( 0.79351132362389194E+03 ) = 0.00000000000000000E+00
1
Test of PSI(X) vs (PSI(X/2)+PSI(X/2+1/2))/2 + ln(2)
2000 random arguments were tested from the interval ( 0.0, 1.0)
ABS(PSI(X)) was larger 577 times
agreed 690 times, and
was smaller 733 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.7373E-15 = 2 ** -50.27
occurred for X = 0.984891E+00
The estimated loss of base 2 significant digits is 2.73
The root mean square relative error was 0.2174E-15 = 2 ** -52.03
The estimated loss of base 2 significant digits is 0.97
Test of PSI(X) vs (PSI(X/2)+PSI(X/2+1/2))/2 + ln(2)
2000 random arguments were tested from the interval ( 2.0, 8.0)
ABS(PSI(X)) was larger 492 times
agreed 983 times, and
was smaller 525 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.7847E-15 = 2 ** -50.18
occurred for X = 0.250865E+01
The estimated loss of base 2 significant digits is 2.82
The root mean square relative error was 0.1427E-15 = 2 ** -52.64
The estimated loss of base 2 significant digits is 0.36
1
Test of PSI(X) vs (PSI(X/2)+PSI(X/2+1/2))/2 + ln(2)
2000 random arguments were tested from the interval ( 8.0, 20.0)
ABS(PSI(X)) was larger 433 times
agreed 1390 times, and
was smaller 177 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.3279E-15 = 2 ** -51.44
occurred for X = 0.155047E+02
The estimated loss of base 2 significant digits is 1.56
The root mean square relative error was 0.9829E-16 = 2 ** -53.18
The estimated loss of base 2 significant digits is 0.00
Test of PSI(X) vs (PSI(X/2)+PSI(X/2+1/2))/2 + ln(2)
500 random arguments were tested from the interval (-17.6,-16.9)
ABS(PSI(X)) was larger 158 times
agreed 211 times, and
was smaller 131 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.5814E-15 = 2 ** -50.61
occurred for X =-0.175603E+02
The estimated loss of base 2 significant digits is 2.39
The root mean square relative error was 0.1732E-15 = 2 ** -52.36
The estimated loss of base 2 significant digits is 0.64
1Special Tests
Accuracy near positive zero
PSI( 0.1460938E+01) = -0.67240239024288055E-03
Loss of base 2 digits = 0.54
Test with extreme arguments
PSI will be called with the argument 0.2225073859-307
This should not stop execution.
PSI returned the value -0.17900000000000000+309
PSI will be called with the argument 0.1797693135+309
This should not stop execution.
PSI returned the value 0.70978271289338397E+03
Test of error returns
PSI will be called with the argument 0.0000000000E+00
This may stop execution.
PSI returned the value 0.17900000000000000+309
PSI will be called with the argument -0.1351079888E+17
This may stop execution.
PSI returned the value 0.17900000000000000+309
This concludes the tests.
1Test of I(X,ALPHA) vs Multiplication Theorem
2000 Random arguments were tested from the interval ( 0.00, 2.00)
I(X,ALPHA) was larger 768 times,
agreed 453 times, and
was smaller 779 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.2156E-14 = 2 ** -48.72
occurred for X = 0.121769E+01 and NU = 0.160411E+00
The estimated loss of base 2 significant digits is 4.28
The root mean square relative error was 0.4911E-15 = 2 ** -50.85
The estimated loss of base 2 significant digits is 2.15
1Test of I(X,ALPHA) vs Taylor series
2000 Random arguments were tested from the interval ( 2.00, 4.00)
I(X,ALPHA) was larger 789 times,
agreed 448 times, and
was smaller 763 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.1936E-14 = 2 ** -48.88
occurred for X = 0.264314E+01 and NU = 0.501470E-01
The estimated loss of base 2 significant digits is 4.12
The root mean square relative error was 0.5642E-15 = 2 ** -50.65
The estimated loss of base 2 significant digits is 2.35
1Test of I(X,ALPHA) vs Taylor series
2000 Random arguments were tested from the interval ( 4.00,10.00)
I(X,ALPHA) was larger 764 times,
agreed 408 times, and
was smaller 828 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.3525E-14 = 2 ** -48.01
occurred for X = 0.995942E+01 and NU = 0.476622E-01
The estimated loss of base 2 significant digits is 4.99
The root mean square relative error was 0.4694E-15 = 2 ** -50.92
The estimated loss of base 2 significant digits is 2.08
1Test of I(X,ALPHA) vs Taylor series
2000 Random arguments were tested from the interval (10.00,20.00)
I(X,ALPHA) was larger 843 times,
agreed 383 times, and
was smaller 774 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.3203E-14 = 2 ** -48.15
occurred for X = 0.100148E+02 and NU = 0.322212E+00
The estimated loss of base 2 significant digits is 4.85
The root mean square relative error was 0.4381E-15 = 2 ** -51.02
The estimated loss of base 2 significant digits is 1.98
1Check of Error Returns
The following summarizes calls with indicated parameters
NCALC different from MB indicates some form of error
See documentation for RIBESL for details
ARG ALPHA MB IZ RES NCALC
0.1000000E+01 0.1500000E+01 5 2 0.0000000E+00 -1
0.1000000E+01 0.5000000E+00 -5 2 0.0000000E+00 -6
0.1000000E+01 0.5000000E+00 5 5 0.0000000E+00 -1
0.0000000E+00 0.1722874E-01 2 1 0.0000000E+00 2
0.0000000E+00 0.0000000E+00 2 1 0.1000000E+01 2
0.0000000E+00 0.1000000E+01 2 1 0.0000000E+00 -1
RIBESL will be called with the argument-0.100000E+01
This should trigger an error message.
NCALC returned the value -1
and RIBESL returned the value 0.000000E+00
Tests near the largest argument for scaled functions
RIBESL will be called with the argument 0.999878E+04
NCALC returned the value 2
and RIBESL returned the value NaN
RIBESL will be called with the argument 0.100012E+05
This should trigger an error message.
NCALC returned the value -1
and RIBESL returned the value 0.000000E+00
Tests near the largest argument for unscaled functions
RIBESL will be called with the argument 0.708913E+03
NCALC returned the value 2
and RIBESL returned the value 0.112931+307
RIBESL will be called with the argument 0.709087E+03
This should trigger an error message.
NCALC returned the value -1
and RIBESL returned the value 0.000000E+00
This concludes the tests.
1Test of J(X,ALPHA) vs Multiplication Theorem
2000 Random arguments were tested from the interval ( 0.00, 2.00)
J(X,ALPHA) was larger 670 times,
agreed 643 times, and
was smaller 687 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.1046E-14 = 2 ** -49.76
occurred for X = 0.198702E+01 and NU = 0.115905E+00
with J(X,ALPHA) = 0.318499E+00
The estimated loss of base 2 significant digits is 3.24
The root mean square relative error was 0.2050E-15 = 2 ** -52.12
The estimated loss of base 2 significant digits is 0.88
1Test of J(X,ALPHA) vs Taylor series
2000 Random arguments were tested from the interval ( 2.00,10.00)
J(X,ALPHA) was larger 822 times,
agreed 316 times, and
was smaller 862 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.1009E-13 = 2 ** -46.49
occurred for X = 0.702428E+01 and NU = 0.672654E+00
with J(X,ALPHA) = 0.140270E+00
The estimated loss of base 2 significant digits is 6.51
The root mean square relative error was 0.8908E-15 = 2 ** -50.00
The estimated loss of base 2 significant digits is 3.00
1Test of J(X,ALPHA) vs Taylor series
2000 Random arguments were tested from the interval (10.00,20.00)
J(X,ALPHA) was larger 964 times,
agreed 158 times, and
was smaller 878 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.1409E-13 = 2 ** -46.01
occurred for X = 0.181389E+02 and NU = 0.107148E+00
with J(X,ALPHA) = -0.187191E-01
The estimated loss of base 2 significant digits is 6.99
The root mean square relative error was 0.1334E-14 = 2 ** -49.41
The estimated loss of base 2 significant digits is 3.59
1Test of J(X,ALPHA) vs Taylor series
2000 Random arguments were tested from the interval (30.00,40.00)
J(X,ALPHA) was larger 608 times,
agreed 765 times, and
was smaller 627 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.3634E-14 = 2 ** -47.97
occurred for X = 0.345621E+02 and NU = 0.380305E+00
with J(X,ALPHA) = -0.257802E-01
The estimated loss of base 2 significant digits is 5.03
The root mean square relative error was 0.2201E-15 = 2 ** -52.01
The estimated loss of base 2 significant digits is 0.99
1Check of Error Returns
The following summarizes calls with indicated parameters
NCALC different from MB indicates some form of error
See documentation for RJBESL for details
ARG ALPHA MB B(1) NCALC
0.1000000E+01 0.1500000E+01 5 0.0000000E+00 -1
0.1000000E+01 0.5000000E+00 -5 0.0000000E+00 -6
0.0000000E+00 0.1000000E+01 2 0.0000000E+00 -1
-0.1000000E+01 0.5000000E+00 5 0.0000000E+00 -1
Tests near the largest acceptable argument for RJBESL
RJBESL will be called with the argument 0.999878E+04
NCALC returned the value 2
and RJBESL returned U(1) = 0.630030E-02
RJBESL will be called with the argument 0.100012E+05
This should trigger an error message.
NCALC returned the value -1
and RJBESL returned U(1) = 0.000000E+00
This concludes the tests.
1Test of K(X,ALPHA) vs Multiplication Theorem
2000 Random arguments were tested from the interval ( 0.00, 1.00)
K(X,ALPHA) was larger 640 times,
agreed 694 times, and
was smaller 666 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.1248E-14 = 2 ** -49.51
occurred for X = 0.922674E+00, NU = 0.570961E+00 and IZE = 1
The estimated loss of base 2 significant digits is 3.49
The root mean square relative error was 0.2111E-15 = 2 ** -52.07
The estimated loss of base 2 significant digits is 0.93
1Test of K(X,ALPHA) vs Multiplication Theorem
2000 Random arguments were tested from the interval ( 1.00,10.00)
K(X,ALPHA) was larger 686 times,
agreed 676 times, and
was smaller 638 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.6447E-15 = 2 ** -50.46
occurred for X = 0.370920E+01, NU = 0.595446E+00 and IZE = 1
The estimated loss of base 2 significant digits is 2.54
The root mean square relative error was 0.1949E-15 = 2 ** -52.19
The estimated loss of base 2 significant digits is 0.81
1Test of K(X,ALPHA) vs Multiplication Theorem
2000 Random arguments were tested from the interval (10.00,20.00)
K(X,ALPHA) was larger 696 times,
agreed 622 times, and
was smaller 682 times.
There are 53 base 2 significant digits in a floating-point number
The maximum absolute error of 0.7945E-15 = 2 ** -50.16
occurred for X = 0.168555E+02, NU = 0.431457E+00 and IZE = 1
The estimated loss of base 2 significant digits is 2.84
The root mean square absolute error was 0.2041E-15 = 2 ** -52.12
The estimated loss of base 2 significant digits is 0.88
1Check of Error Returns
The following summarizes calls with indicated parameters
NCALC different from MB indicates some form of error
See documentation for RKBESL for details
ARG ALPHA MB IZ RES NCALC
-0.1000000E+01 0.5000000E+00 5 2 0.0000000E+00 -2
0.1000000E+01 0.1500000E+01 5 2 0.0000000E+00 -2
0.1000000E+01 0.5000000E+00 -5 2 0.0000000E+00 -7
0.1000000E+01 0.5000000E+00 5 5 0.0000000E+00 -2
0.2225074-307 0.0000000E+00 2 2 0.7085124E+03 2
0.1000000E-09 0.0000000E+00 20 2 0.2314178E+02 20
0.1000000E-09 0.0000000E+00 20 2 0.2314178E+02 20
0.6612588E+03 0.0000000E+00 2 1 0.3211860-288 2
0.7053427E+03 0.0000000E+00 2 1 0.0000000E+00 -2
0.4503600E+17 0.0000000E+00 2 2 0.5905818E-08 2
0.1797693+309 0.0000000E+00 2 2 0.9347644-154 2
1Test of Y(X,ALPHA) vs Multiplication Theorem
1983 Random arguments were tested from the interval ( 0.00, 2.00)
Y(X,ALPHA) was larger 783 times,
agreed 403 times, and
was smaller 797 times.
NOTE: first 17 arguments in test interval skipped
because multiplication theorem did not converge
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.2075E-14 = 2 ** -48.78
occurred for X = 0.196853E+01 and NU = 0.604741E+00
with Y(X,ALPHA) = 0.147133E+00
The estimated loss of base 2 significant digits is 4.22
The root mean square relative error was 0.3763E-15 = 2 ** -51.24
The estimated loss of base 2 significant digits is 1.76
1Test of Y(X,ALPHA) vs Taylor series
2000 Random arguments were tested from the interval ( 2.00,10.00)
Y(X,ALPHA) was larger 710 times,
agreed 486 times, and
was smaller 804 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.5188E-14 = 2 ** -47.45
occurred for X = 0.831191E+01 and NU = 0.682244E+00
with Y(X,ALPHA) = 0.508201E-01
The estimated loss of base 2 significant digits is 5.55
The root mean square relative error was 0.3385E-15 = 2 ** -51.39
The estimated loss of base 2 significant digits is 1.61
1Test of Y(X,ALPHA) vs Taylor series
2000 Random arguments were tested from the interval (10.00,20.00)
Y(X,ALPHA) was larger 680 times,
agreed 561 times, and
was smaller 759 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.3682E-14 = 2 ** -47.95
occurred for X = 0.159974E+02 and NU = 0.156113E+00
with Y(X,ALPHA) = 0.135694E+00
The estimated loss of base 2 significant digits is 5.05
The root mean square relative error was 0.2436E-15 = 2 ** -51.87
The estimated loss of base 2 significant digits is 1.13
1Test of Y(X,ALPHA) vs Taylor series
2000 Random arguments were tested from the interval (30.00,40.00)
Y(X,ALPHA) was larger 613 times,
agreed 768 times, and
was smaller 619 times.
There are 53 base 2 significant digits in a floating-point number
The maximum relative error of 0.1342E-14 = 2 ** -49.40
occurred for X = 0.354141E+02 and NU = 0.269416E+00
with Y(X,ALPHA) = 0.465376E-01
The estimated loss of base 2 significant digits is 3.60
The root mean square relative error was 0.1808E-15 = 2 ** -52.30
The estimated loss of base 2 significant digits is 0.70
1Check of Error Returns
The following summarizes calls with indicated parameters
NCALC different from MB indicates some form of error
See documentation for RYBESL for details
ARG ALPHA MB B(1) NCALC
0.1000000E+01 0.1500000E+01 5 0.0000000E+00 -1
0.1000000E+01 0.5000000E+00 -5 0.0000000E+00 -6
0.2225074-307 0.0000000E+00 2 0.0000000E+00 -1
0.6675222-307 0.0000000E+00 2 -0.4503536E+03 2
0.6675222-307 0.1000000E+01 2 -0.9537058+307 1
Tests near the largest acceptable argument for RYBESL
RYBESL will be called with the arguments 0.335544E+08 0.500000E+00
NCALC returned the value 2
and RYBESL returned U(1) = 0.296749E-04
RYBESL will be called with the arguments 0.536871E+09 0.500000E+00
This should trigger an error message.
NCALC returned the value -1
and RYBESL returned U(1) = 0.000000E+00
This concludes the tests.
1Test of Y0(X) VS Multiplication Theorem
2000 random arguments were tested from the interval ( 0.0, 3.0)
ABS(Y0(X)) was larger 662 times
agreed 669 times, and
was smaller 669 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.1758E-14 = 2 ** -49.01
occurred for X = 0.867707E+00
The estimated loss of base 2 significant digits is 3.99
The root mean square relative error was 0.2001E-15 = 2 ** -52.15
The estimated loss of base 2 significant digits is 0.85
1Test of Y0(X) VS Multiplication Theorem
2000 random arguments were tested from the interval ( 3.0, 5.5)
ABS(Y0(X)) was larger 691 times
agreed 599 times, and
was smaller 710 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.9720E-15 = 2 ** -49.87
occurred for X = 0.390803E+01
The estimated loss of base 2 significant digits is 3.13
The root mean square relative error was 0.1966E-15 = 2 ** -52.18
The estimated loss of base 2 significant digits is 0.82
1Test of Y0(X) VS Multiplication Theorem
2000 random arguments were tested from the interval ( 5.5, 8.0)
ABS(Y0(X)) was larger 740 times
agreed 525 times, and
was smaller 735 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.7884E-15 = 2 ** -50.17
occurred for X = 0.686902E+01
The estimated loss of base 2 significant digits is 2.83
The root mean square relative error was 0.1914E-15 = 2 ** -52.21
The estimated loss of base 2 significant digits is 0.79
1Test of Y0(X) VS Multiplication Theorem
500 random arguments were tested from the interval ( 8.0, 20.0)
ABS(Y0(X)) was larger 191 times
agreed 128 times, and
was smaller 181 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum absolute error of 0.8042E-15 = 2 ** -50.14
occurred for X = 0.129987E+02
The estimated loss of base 2 significant digits is 2.86
The root mean square absolute error was 0.1884E-15 = 2 ** -52.24
The estimated loss of base 2 significant digits is 0.76
1Special Tests
Accuracy near zeros
X BESY0(X) Loss of base 2 digits
0.8906250000E+00 -0.260031427229336E-02 5.39
0.3957031250E+01 0.260534549114568E-03 1.91
0.7085937500E+01 -0.340794487147973E-04 8.81
Test with extreme arguments
Y0 will be called with the argument 0.2225073859-307
This should not stop execution.
Y0 returned the value -0.45105297100712858E+03
Y0 will be called with the argument 0.0000000000E+00
This may stop execution.
Y0 returned the value -Infinity
Y0 will be called with the argument 0.1797693135+309
This may stop execution.
Y0 returned the value 0.42287458488299958-154
This concludes the tests.
1Test of Y1(X) VS Multiplication Theorem
2000 random arguments were tested from the interval ( 0.0, 4.0)
ABS(Y1(X)) was larger 710 times
agreed 622 times, and
was smaller 668 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.1394E-14 = 2 ** -49.35
occurred for X = 0.212194E+01
The estimated loss of base 2 significant digits is 3.65
The root mean square relative error was 0.2109E-15 = 2 ** -52.07
The estimated loss of base 2 significant digits is 0.93
1Test of Y1(X) VS Multiplication Theorem
2000 random arguments were tested from the interval ( 4.0, 8.0)
ABS(Y1(X)) was larger 758 times
agreed 531 times, and
was smaller 711 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.1072E-14 = 2 ** -49.73
occurred for X = 0.524169E+01
The estimated loss of base 2 significant digits is 3.27
The root mean square relative error was 0.2165E-15 = 2 ** -52.04
The estimated loss of base 2 significant digits is 0.96
1Test of Y1(X) VS Multiplication Theorem
500 random arguments were tested from the interval ( 8.0, 20.0)
ABS(Y1(X)) was larger 190 times
agreed 125 times, and
was smaller 185 times.
There are 53 base 2 significant digits in a floating-point number.
The maximum relative error of 0.7203E-15 = 2 ** -50.30
occurred for X = 0.171171E+02
The estimated loss of base 2 significant digits is 2.70
The root mean square relative error was 0.1947E-15 = 2 ** -52.19
The estimated loss of base 2 significant digits is 0.81
1Special Tests
Accuracy near zeros
X BESY1(X) Loss of base 2 digits
0.2195312500E+01 -0.952823930977230E-03 4.68
0.5429687500E+01 -0.219818300806240E-05 13.38
Test with extreme arguments
Y1 will be called with the argument 0.2225073859-307
This should not stop execution.
Y1 returned the value -0.28611174857570283+308
Y1 will be called with the argument -0.1000000000E+01
This may stop execution.
Y1 returned the value NaN
Y1 will be called with the argument 0.1797693135+309
This may stop execution.
Y1 returned the value 0.41869868495853734-154
This concludes the tests.
toms715_test():
Normal end of execution.
1 December 2023 10:02:10.122 AM