Mail Archives: djgpp-workers/1998/10/08/20:56:21
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Dear Eric Rudd,
Attached are the complete Elefunt-style reports for 14 libm.a
elementary math functions that correspond to yours. I have
sent these same reports on your functions separately.
K.B. Williams
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L I B M . A
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Test of acosh(x) vs. xacosh(x):
There are 53 base 2 significant digits
Test 1: acosh(x) for 1000 values in (+1, +1.5)
There were 1000 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 3.2368517907E-16 =3D 2 ^ -51.46
=09occurred for x =3D 1.0610233129427708576
Estimated loss of base 2 significant digits is 1.54
The root-mean-square relative error was 3.019862925E-18 =3D 2 ^ -58.20
Estimated loss of base 2 significant digits is 0.00
Test 2: acosh(x) for 1000 values in (+1.5, +2)
There were 87 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.9621563894E-16 =3D 2 ^ -52.18
=09occurred for x =3D 1.5482657370560612975
Estimated loss of base 2 significant digits is 0.82
The root-mean-square relative error was 2.262676156E-18 =3D 2 ^ -58.62
Estimated loss of base 2 significant digits is 0.00
Test 3: acosh(x) for 1000 values in (+2, +10)
There were 0 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.3756046301E-16 =3D 2 ^ -52.69
=09occurred for x =3D 4.3926985753321030614
Estimated loss of base 2 significant digits is 0.31
The root-mean-square relative error was 1.671895855E-18 =3D 2 ^ -59.05
Estimated loss of base 2 significant digits is 0.00
Test 4: acosh(x) for 1000 values in (+35, +55)
There were 0 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.2090929508E-16 =3D 2 ^ -52.88
=09occurred for x =3D 35.419737122262844764
Estimated loss of base 2 significant digits is 0.12
The root-mean-square relative error was 1.812411134E-18 =3D 2 ^ -58.94
Estimated loss of base 2 significant digits is 0.00
Test 5: acosh(x) for 1000 values in (2^27, 2^108)
There were 0 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 9.0354849357E-17 =3D 2 ^ -53.30
=09occurred for x =3D 134461047.87544175982
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.604293275E-18 =3D 2 ^ -59.11
Estimated loss of base 2 significant digits is 0.00
SPECIAL VALUES:
Testing acosh(x) for x very close to 1:
=09x =3D 1 + 2.22044604925E-16 (EPSILON), acosh(x) =3D 2.10734242554E-08
Abs error =3D 3.308722450212110699E-24, or 2 ^ -78.00
This translates to a loss of 0.00 base 2 digits.
The following call should not trigger an error message:
=09x =3D +1, acosh(x) =3D +0 (Expect +0)
BOUNDARY VALUES:
Testing acosh(x) for x =3D maximum floating point value
=09x =3D 1.797693135E+308, acosh(x) =3D +710.4758601
The following calls might trigger error messages:
Testing acosh(x) for x =3D minimum floating point value
=09x =3D 2.225073859E-308, acosh(x) =3D +NaN
Testing acosh(x) for x =3D 0
=09x =3D 0, acosh(x) =3D +NaN
Elapsed time: 48.352 seconds
Test of asin(x) vs. xasin(x) & acos(x) vs. xacos(x):
There are 53 base 2 significant digits
Test 1: asin(x) for 1000 values in (-0.125, 0.125)
There were 1000 function values less than 1
Result was smaller 500 times, equal 0 times, and larger 500 times
The maximum relative error of 1.0251104979E-16 =3D 2 ^ -53.12
=09occurred for x =3D -0.03142305120255911577
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.440394126E-18 =3D 2 ^ -59.27
Estimated loss of base 2 significant digits is 0.00
Test 2: acos(x) for 1000 values in (-0.125, 0.125)
There were 0 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 7.5964020636E-17 =3D 2 ^ -53.55
=09occurred for x =3D 0.11734471429872479764
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.282201485E-18 =3D 2 ^ -59.44
Estimated loss of base 2 significant digits is 0.00
Test 3: asin(x) for 1000 values in (0.5, 1)
There were 683 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.1083824505E-16 =3D 2 ^ -53.00
=09occurred for x =3D 0.84228994318635708183
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.554495051E-18 =3D 2 ^ -59.16
Estimated loss of base 2 significant digits is 0.00
Test 4: acos(x) for 1000 values in (0.5, 1)
There were 919 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.0802920352E-16 =3D 2 ^ -53.04
=09occurred for x =3D 0.87562128789477500224
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.454716514E-18 =3D 2 ^ -59.25
Estimated loss of base 2 significant digits is 0.00
Test 5: acos(x) for 1000 values in (-1, -0.5)
There were 0 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.2334476616E-16 =3D 2 ^ -52.85
=09occurred for x =3D -0.5471588891409965294
Estimated loss of base 2 significant digits is 0.15
The root-mean-square relative error was 1.742384394E-18 =3D 2 ^ -58.99
Estimated loss of base 2 significant digits is 0.00
SPECIAL VALUES:
asin(x) + asin(-x) for random values in [0, 1)
=09x =3D -0.5771310922, asin(x) + asin(-x) =3D +0
=09x =3D -0.8380738201, asin(x) + asin(-x) =3D +0
=09x =3D -0.6321389080, asin(x) + asin(-x) =3D +0
=09x =3D -0.8597400239, asin(x) + asin(-x) =3D +0
=09x =3D -0.7003973234, asin(x) + asin(-x) =3D +0
asin(x) - x for Small x
=09x =3D 1.0837106757E-16, asin(x) - x =3D +0
=09x =3D 5.4185533787E-17, asin(x) - x =3D +0
=09x =3D 2.7092766894E-17, asin(x) - x =3D +0
=09x =3D 1.3546383447E-17, asin(x) - x =3D +0
=09x =3D 6.7731917234E-18, asin(x) - x =3D +0
Testing underflow for very small argument
=09x =3D 5.152919016E-231, asin(x) =3D +5.152919016E-231
BOUNDARY VALUES:
The following calls might trigger error messages:
=09x =3D 1.25, asin(x) =3D NaN
=09x =3D -1.25, acos(x) =3D NaN
Elapsed time: 118.516 seconds
Test of asinh(x) vs. xasinh(x):
There are 53 base 2 significant digits
Test 1: asinh(x) for 1000 values in (+0, +0.5)
There were 1000 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 3.1961726799E-16 =3D 2 ^ -51.47
=09occurred for x =3D 0.44287572408013914371
Estimated loss of base 2 significant digits is 1.53
The root-mean-square relative error was 2.815994188E-18 =3D 2 ^ -58.30
Estimated loss of base 2 significant digits is 0.00
Test 2: asinh(x) for 1000 values in (+0.5, +1)
There were 1000 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 3.0149608857E-16 =3D 2 ^ -51.56
=09occurred for x =3D 0.78735623677740540494
Estimated loss of base 2 significant digits is 1.44
The root-mean-square relative error was 3.249403416E-18 =3D 2 ^ -58.09
Estimated loss of base 2 significant digits is 0.00
Test 3: asinh(x) for 1000 values in (+10, +1E+10)
There were 0 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 9.2380316196E-17 =3D 2 ^ -53.27
=09occurred for x =3D 40952562.187855124474
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.440594616E-18 =3D 2 ^ -59.27
Estimated loss of base 2 significant digits is 0.00
Test 4: asinh(x) for 1000 values in (+1E+10, +1E+11)
There were 0 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 7.3187596544E-17 =3D 2 ^ -53.60
=09occurred for x =3D 10677340152.406639099
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.289237846E-18 =3D 2 ^ -59.43
Estimated loss of base 2 significant digits is 0.00
SPECIAL VALUES:
The following calls should not trigger error messages:
Testing asinh(x) vs. asinh(-x) for random Values in [0, 1)
=09x =3D 0.36583767906, asinh(x) + asinh(-x) =3D -6.0715321659E-18
=09x =3D 0.51691089516, asinh(x) + asinh(-x) =3D -1.8241701552E-17
=09x =3D 0.79613069565, asinh(x) + asinh(-x) =3D +4.6132802439E-17
=09x =3D 0.01756874969, asinh(x) + asinh(-x) =3D +4.4723339615E-19
=09x =3D 0.21730847028, asinh(x) + asinh(-x) =3D +1.3552527156E-17
Testing asinh(x) - x for small values of x
=09x =3D 1.34273157558E-08, asinh(x) - x =3D -6.2200104655E-26
=09x =3D 6.71365787792E-09, asinh(x) - x =3D +3.5139020162E-26
=09x =3D 3.35682893896E-09, asinh(x) - x =3D +0 (Exactly)
=09x =3D 1.67841446948E-09, asinh(x) - x =3D +0 (Exactly)
=09x =3D 8.39207234740E-10, asinh(x) - x =3D +0 (Exactly)
Testing underflow in asinh(x) for very small x:
=09x =3D 5.1529190157E-231, asinh(x) =3D 5.152919016E-231
BOUNDARY VALUES:
Testing asinh(x) for x =3D maximum floating point value
=09x =3D 1.797693135E+308, asinh(x) =3D 710.4758601
Testing asinh(x) for x =3D minimum floating point value
=09x =3D 2.225073859E-308, asinh(x) =3D 2.225073859E-308
Testing asinh(x) for x =3D 0
=09x =3D 0, asinh(x) =3D 0
Elapsed time: 38.242 seconds
Test of atan(x) vs. xatan(x) & atan2(x,y) vs. xatan2(x,y):
There are 53 base 2 significant digits
Test 1: atan(x) for 1000 values in (-0.0625, 0.0625)
There were 1000 function values less than 1
Result was smaller 500 times, equal 0 times, and larger 500 times
The maximum relative error of 1.0421426307E-16 =3D 2 ^ -53.09
=09occurred for x =3D 0.03403460958828241623
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.452690816E-18 =3D 2 ^ -59.26
Estimated loss of base 2 significant digits is 0.00
Test 2: atan(x) for 1000 values in (0.0625, 1)
There were 1000 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.0702279779E-16 =3D 2 ^ -53.05
=09occurred for x =3D 0.55989791057534077368
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.497816495E-18 =3D 2 ^ -59.21
Estimated loss of base 2 significant digits is 0.00
Test 3: atan2(x,y) for 1000 values of
=09x in (-0.0625, +0.0625) and
=09y in (-0.0625, +0.0625)
There were 336 function values less than 1
Result was smaller 507 times, equal 0 times, and larger 493 times
The maximum relative error of 1.7657411940E-16 =3D 2 ^ -52.33
=09occurred for x =3D +0.0530634786128044017
=09 and y =3D -0.03150928925582063833
Estimated loss of base 2 significant digits is 0.67
The root-mean-square relative error was 1.678737984E-18 =3D 2 ^ -59.05
Estimated loss of base 2 significant digits is 0.00
Test 4: atan2(x,y) for 1000 values of
=09x in (+0.0625, +1.5) and
=09y in (-1.5, -0.0625)
There were 691 function values less than 1
Result was smaller 1000 times, equal 0 times, and larger 0 times
The maximum relative error of 1.7865341237E-16 =3D 2 ^ -52.31
=09occurred for x =3D +0.48157306521369103169
=09 and y =3D -0.26816512017396942902
Estimated loss of base 2 significant digits is 0.69
The root-mean-square relative error was 1.813595286E-18 =3D 2 ^ -58.94
Estimated loss of base 2 significant digits is 0.00
SPECIAL VALUES:
atan(x) + atan(-x) for random values in [0, 5)
=09x =3D 2.8856554609, atan(x) + atan(-x) =3D 8.153200337E-17
=09x =3D 4.1903691004, atan(x) + atan(-x) =3D -1.051676107E-16
=09x =3D 3.1606945399, atan(x) + atan(-x) =3D -9.627715292E-17
=09x =3D 4.2987001193, atan(x) + atan(-x) =3D -2.341876693E-17
=09x =3D 3.5019866169, atan(x) + atan(-x) =3D 1.301042607E-18
atan(x) - x for small x
=09x =3D 1.0837106757E-16, atan(x) - x =3D 0
=09x =3D 5.4185533787E-17, atan(x) - x =3D 0
=09x =3D 2.7092766894E-17, atan(x) - x =3D 0
=09x =3D 1.3546383447E-17, atan(x) - x =3D 0
=09x =3D 6.7731917234E-18, atan(x) - x =3D 0
Testing atan(y/x) vs. atan2(y, x)
=09x =3D +1.0000000000, y =3D +1.0000000000
=09atan(y/x) - atan2(y,x) =3D +0*pi (Expect +0*pi)
=09atan(-y/x) - atan2(y,-x) =3D -1*pi (Expect -1*pi)
=09x =3D -1.0000000000, y =3D +1.0000000000
=09atan(y/x) - atan2(y,x) =3D -1*pi (Expect -1*pi)
=09atan(-y/x) - atan2(y,-x) =3D +0*pi (Expect +0*pi)
=09x =3D +1.0000000000, y =3D -1.0000000000
=09atan(y/x) - atan2(y,x) =3D +0*pi (Expect +0*pi)
=09atan(-y/x) - atan2(y,-x) =3D +1*pi (Expect +1*pi)
=09x =3D -1.0000000000, y =3D -1.0000000000
=09atan(y/x) - atan2(y,x) =3D +1*pi (Expect +1*pi)
=09atan(-y/x) - atan2(y,-x) =3D +0*pi (Expect +0*pi)
Testing underflow for very small argument
=09x =3D 5.152919016E-231, atan(x) =3D 5.152919016E-231
BOUNDARY VALUES:
The following calls should not trigger error messages:
=09x =3D 1.797693135E+308, atan(x) =3D 1.570796327
=09x =3D 2.225073859E-308, atan(x) =3D 2.225073859E-308
=09x =3D 1, y =3D 0, atan2(y, x) =3D 0
=09x =3D 2.225073859E-308, y =3D 1.797693135E+308
=09atan2(y, x) =3D 1.570796326794896558
=09x =3D 1.797693135E+308, y =3D 2.225073859E-308
=09atan2(y, x) =3D 0
The following call might trigger an error message:
=09x =3D 0, y =3D 0, atan2(y, x) =3D 0
Elapsed time: 75.604 seconds
Test of atanh(x) vs. xatanh(x):
There are 53 base 2 significant digits
Test 1: atanh(x) for 1000 values in (+0, +0.5)
There were 1000 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 3.2515530270E-16 =3D 2 ^ -51.45
=09occurred for x =3D 0.17308396554553232005
Estimated loss of base 2 significant digits is 1.55
The root-mean-square relative error was 2.913050972E-18 =3D 2 ^ -58.25
Estimated loss of base 2 significant digits is 0.00
Test 2: atanh(x) for 1000 values in (+0.5, +1)
There were 523 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.7507418012E-16 =3D 2 ^ -52.34
=09occurred for x =3D 0.6190820900169302643
Estimated loss of base 2 significant digits is 0.66
The root-mean-square relative error was 1.780596748E-18 =3D 2 ^ -58.96
Estimated loss of base 2 significant digits is 0.00
Test 3: atanh(x) for 1000 values in (+0.9375, +1)
There were 0 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.1949355554E-16 =3D 2 ^ -52.89
=09occurred for x =3D 0.96481376601427204509
Estimated loss of base 2 significant digits is 0.11
The root-mean-square relative error was 1.529888946E-18 =3D 2 ^ -59.18
Estimated loss of base 2 significant digits is 0.00
SPECIAL VALUES:
The following calls should not trigger error messages:
Testing atanh(x) vs. atanh(-x) for random Values in [0, 1)
=09x =3D 0.0137849244, atanh(x) + atanh(-x) =3D +0
=09x =3D 0.5640895348, atanh(x) + atanh(-x) =3D +0
=09x =3D 0.4798730761, atanh(x) + atanh(-x) =3D +0
=09x =3D 0.3521589870, atanh(x) + atanh(-x) =3D +0
=09x =3D 0.7828831063, atanh(x) + atanh(-x) =3D +0
Testing atanh(x) - x for small values of x
=09x =3D +3.88728617569E-09, atanh(x) - x =3D +0
=09x =3D -1.94364308785E-09, atanh(x) - x =3D +0
=09x =3D +9.71821543923E-10, atanh(x) - x =3D +0
=09x =3D -4.85910771961E-10, atanh(x) - x =3D +0
=09x =3D +2.42955385981E-10, atanh(x) - x =3D +0
Testing underflow in atanh(x) for very small x:
=09x =3D 5.152919016E-231, atanh(x) =3D 5.152919016E-231
BOUNDARY VALUES:
Testing atanh(x) for x =3D minimum floating point value
=09x =3D 2.225073859E-308, atanh(x) =3D 2.225073859E-308
Testing atanh(x) for x =3D 0
=09x =3D 0, atanh(x) =3D 0
Testing atanh(x) for x =3D 1
=09x =3D 1, atanh(x) =3D Inf
Testing atanh(x) for x =3D -1
=09x =3D -1, atanh(x) =3D Inf
The following call might trigger an error message:
Testing atanh(x) for x =3D maximum floating point value
=09x =3D 1.797693135E+308, atanh(x) =3D NaN
Elapsed time: 20.110 seconds
Test of exp(x) vs. xexp(x):
There are 53 base 2 significant digits
Test 1: exp(x) for values in (-0.5*log(2), +0.5*log(2))
There were 500 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.0893159428E-16 =3D 2 ^ -53.03
=09occurred for x =3D 0.00499950396965237465
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.478194474E-18 =3D 2 ^ -59.23
Estimated loss of base 2 significant digits is 0.00
Test 2: exp(x) for values in (-670, -0.5*log(2))
There were 1000 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.0889853782E-16 =3D 2 ^ -53.03
=09occurred for x =3D -123.37073719402391703
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.483123352E-18 =3D 2 ^ -59.23
Estimated loss of base 2 significant digits is 0.00
Test 3: exp(x) for values in (10*log(2), +709)
There were 486 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.0606691617E-16 =3D 2 ^ -53.07
=09occurred for x =3D -581.52212120706690257
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.50317233E-18 =3D 2 ^ -59.21
Estimated loss of base 2 significant digits is 0.00
SPECIAL VALUES:
The following calls should not trigger error messages:
Testing exp(x) vs. 1/exp(-x) for random Values in [2, 3)
x =3D 2.2282373610, exp(x) * exp(-x) - 1.0 =3D +3.2634485392E-17
x =3D 2.8648585006, exp(x) * exp(-x) - 1.0 =3D +2.2876665839E-17
x =3D 2.7771097699, exp(x) * exp(-x) - 1.0 =3D +1.0744443529E-16
x =3D 2.4352933619, exp(x) * exp(-x) - 1.0 =3D +1.0408340856E-17
x =3D 2.4758829272, exp(x) * exp(-x) - 1.0 =3D +7.3508907295E-17
exp(0) - 1 =3D 0
exp(log(1.4932217896051502065E-300)) =3D 1.4932217896050938411E-300
exp(log(1.7976931080746005285E+308)) =3D 1.7976931080745582167E+308
exp(354.89135644669198655) =3D 1.3407807929942437823E+154
=09=09 should equal 1.3407807929942439312E+154
BOUNDARY VALUES:
The following calls might trigger error messages
=09exp(-6.703903965E+153) =3D 0
=09exp(+6.703903965E+153) =3D Inf
Elapsed time: 12.857 seconds
Test of log1p(x) vs xlog(1+x):
There are 53 base 2 significant digits
Test 1: log1p(x) for 1000 values in (-0.3125, 0.3125)
There were 1000 function values less than 1
Result was smaller 500 times, equal 0 times, and larger 500 times
The maximum relative error of 2.9361051270E-16 =3D 2 ^ -51.60
=09occurred for x =3D -0.30078537659858722852
Estimated loss of base 2 significant digits is 1.40
The root-mean-square relative error was 1.776431658E-18 =3D 2 ^ -58.97
Estimated loss of base 2 significant digits is 0.00
Test 2: log1p(x) for 1000 values in (-1.49E-08, 1.49E-08)
There were 1000 function values less than 1
Result was smaller 500 times, equal 0 times, and larger 500 times
The maximum relative error of 1.0629367556E-16 =3D 2 ^ -53.06
=09occurred for x =3D -7.7356111280752981553E-09
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.44920799E-18 =3D 2 ^ -59.26
Estimated loss of base 2 significant digits is 0.00
Test 3: log1p(x) for 1000 values in (16, 240)
There were 0 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.0362311867E-16 =3D 2 ^ -53.10
=09occurred for x =3D 68.404795572969462114
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.537764643E-18 =3D 2 ^ -59.17
Estimated loss of base 2 significant digits is 0.00
SPECIAL VALUES:
BOUNDARY VALUES:
The following call should not trigger an error message:
=09log1p(0) =3D 0
=09log1p(1.797693135E+308) =3D +709.7827129
The following calls might trigger error messages
=09log1p(-2) =3D -Inf
=09log1p(Inf) =3D Inf
=09log1p(NaN) =3D NaN
Elapsed time: 6.923 seconds
Test of log(x) vs xlog(x) and log10(x) vs xlog10(x)
There are 53 base 2 significant digits
Test 1: log(x) for 1000 values in (1-7.6E-06, 1+7.6E-06)
There were 1000 function values less than 1
Result was smaller 500 times, equal 0 times, and larger 500 times
The maximum relative error of 1.0721056860E-16 =3D 2 ^ -53.05
=09occurred for x =3D 1.0000038928985701681
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.464338293E-18 =3D 2 ^ -59.24
Estimated loss of base 2 significant digits is 0.00
Test 2: log(x) for 1000 values in (sqrt(0.5), 15/16)
There were 1000 function values less than 1
Result was smaller 1000 times, equal 0 times, and larger 0 times
The maximum relative error of 1.1028787318E-16 =3D 2 ^ -53.01
=09occurred for x =3D 0.77687386147623893606
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.527689657E-18 =3D 2 ^ -59.18
Estimated loss of base 2 significant digits is 0.00
Test 3: log10(x) for 1000 values in (sqrt(0.1), 0.9)
There were 1000 function values less than 1
Result was smaller 1000 times, equal 0 times, and larger 0 times
The maximum relative error of 1.3041314065E-16 =3D 2 ^ -52.77
=09occurred for x =3D 0.5493380372236760234
Estimated loss of base 2 significant digits is 0.23
The root-mean-square relative error was 1.524164499E-18 =3D 2 ^ -59.19
Estimated loss of base 2 significant digits is 0.00
Test 4: log(x) for 1000 values in (16, 240)
There were 0 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.0466833896E-16 =3D 2 ^ -53.09
=09occurred for x =3D 55.666506884655021281
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.613411428E-18 =3D 2 ^ -59.10
Estimated loss of base 2 significant digits is 0.00
SPECIAL VALUES:
Testing identity log(x) =3D -log(1/x) for x in [15, 16)
=09x =3D 15.0296392966, log(x) + log(1/x) =3D +4.44089209850E-16
=09x =3D 15.9356973198, log(x) + log(1/x) =3D +0.00000000000E+00
=09x =3D 15.4694980257, log(x) + log(1/x) =3D +0.00000000000E+00
=09x =3D 15.6709178670, log(x) + log(1/x) =3D +0.00000000000E+00
=09x =3D 15.6511964467, log(x) + log(1/x) =3D +0.00000000000E+00
=09log(1) =3D +0
BOUNDARY VALUES:
The following calls should not trigger error messages:
=09log(2.225073859E-308) =3D -708.3964185
=09log10(2.225073859E-308) =3D -307.6526556
=09log(1.797693135E+308) =3D +709.7827129
=09log10(1.797693135E+308) =3D +308.2547156
The following calls might trigger error messages
=09log(-2) =3D -Inf
=09log10(-2) =3D -Inf
=09log(0) =3D -Inf
=09log10(0) =3D -Inf
=09log(Inf) =3D Inf
=09log10(Inf) =3D Inf
=09log(NaN) =3D NaN
=09log10(NaN) =3D NaN
Elapsed time: 39.121 seconds
Test of pow(x,p) vs. xpow(x,p):
There are 53 base 2 significant digits
Test 1: pow(x,1) for 1000 values in (+0.5, +1)
There were 1000 function values less than 1
Result was smaller 0 times, equal 1000 times, and larger 0 times
The maximum relative error =3D 2 ^ -(INF)
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative is 2 ^ -(INF)
Estimated loss of base 2 significant digits is 0.00
Test 2: pow(x*x,1.5) for 1000 values in (+0.5, +1)
There were 1000 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.0476217290E-16 =3D 2 ^ -53.08
=09occurred for x =3D 0.79636908624085223085
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.473215821E-18 =3D 2 ^ -59.24
Estimated loss of base 2 significant digits is 0.00
Test 3: pow(x*x,1.5) for 1000 values in (+1.0, +5.6438E+102)
There were 0 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.0814404692E-16 =3D 2 ^ -53.04
=09occurred for x =3D 3.571946118822868591E+102
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.471661693E-18 =3D 2 ^ -59.24
Estimated loss of base 2 significant digits is 0.00
Test 4: pow(x,y) for 1000 values of
=09x in (+0.01, +10) and
=09y in (-154.127, +154.127)
There were 502 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.0368674662E-16 =3D 2 ^ -53.10
=09occurred for x =3D +0.98497431317877415236
=09 and y =3D -46.469707366900621537
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.442938876E-18 =3D 2 ^ -59.27
Estimated loss of base 2 significant digits is 0.00
SPECIAL VALUES:
These calls should not trigger error messages:
=09Comparing pow(x, y) vs. pow(1/x, -y)
=09for Random Values of x & y in [1, 11):
=09 x y=09 Relative Error
=09 - -=09 --------------
=09 6.7713109218 9.3807382007 -2.40378963634E-16
=09 7.3213890798 9.5974002387 +4.50320402026E-16
=09 8.0039732337 10.7611979909 +3.62893656918E-16
=09 1.1678296328 7.8897789482 -1.30575269811E-16
=09 7.8642676150 2.4286762138 +1.89839344468E-16
=09 2.7636657928 9.6017786381 -6.29412197311E-16
=09 8.4606401035 1.2589809415 +0.00000000000E+00
=09 9.5148453466 5.5189793857 -3.47770233757E-16
=09 4.7846462152 9.1443398525 -1.41339072470E-16
=09 8.1849413176 5.3750309180 -1.80066630694E-16
=09pow(2, -1022) =3D 2.225073859E-308
=09pow(2, +1023) =3D 8.988465674E+307
=09pow(0, 2) =3D 0
=09pow(-2, 0) =3D 1
=09pow(-2, 2) =3D 4
BOUNDARY VALUES:
=09These calls might trigger error messages:
=09pow(-1.25, -2.375) =3D NaN
=09pow(0, 0) =3D 1
Elapsed time: 97.253 seconds
Test of sin(x) vs. xsin(x) & cos(x) vs. xcos(x) (1000 arguments per test)=
:
There are 53 base 2 significant digits
Test 1: sin(x) for 1000 values in (0.0, pi/2)
There were 1000 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.1835466336E-16 =3D 2 ^ -52.91
=09occurred for x =3D 0.79496100541356973235
Estimated loss of base 2 significant digits is 0.09
The root-mean-square relative error was 1.437555555E-18 =3D 2 ^ -59.27
Estimated loss of base 2 significant digits is 0.00
Test 2: cos(x) for 1000 values in (0.0, pi/2)
There were 1000 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.7089545251E-16 =3D 2 ^ -52.38
=09occurred for x =3D 1.0051782379222931052
Estimated loss of base 2 significant digits is 0.62
The root-mean-square relative error was 1.742827944E-18 =3D 2 ^ -58.99
Estimated loss of base 2 significant digits is 0.00
Test 3: sin(x) for 1000 values in (6*pi, 6.5*pi)
There were 1000 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.0741688947E-16 =3D 2 ^ -53.05
=09occurred for x =3D 18.880857098354820067
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.345283366E-18 =3D 2 ^ -59.37
Estimated loss of base 2 significant digits is 0.00
Test 4: cos(x) for 1000 values in (7*pi, 7.5*pi)
There were 1000 function values less than 1
Result was smaller 1000 times, equal 0 times, and larger 0 times
The maximum relative error of 1.0895401077E-16 =3D 2 ^ -53.03
=09occurred for x =3D 23.022813364246385248
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.334640908E-18 =3D 2 ^ -59.38
Estimated loss of base 2 significant digits is 0.00
Test 5: sin(x) for 1000 values in (1e22, 1e31)
There were 1000 function values less than 1
Result was smaller 476 times, equal 0 times, and larger 524 times
The maximum relative error of 1.8492920380E-16 =3D 2 ^ -52.26
=09occurred for x =3D 9.003336786771030225E+30
Estimated loss of base 2 significant digits is 0.74
The root-mean-square relative error was 1.762518855E-18 =3D 2 ^ -58.98
Estimated loss of base 2 significant digits is 0.00
Test 6: cos(x) for 1000 values in (1e22, 1e31)
There were 1000 function values less than 1
Result was smaller 504 times, equal 0 times, and larger 496 times
The maximum relative error of 1.8918815032E-16 =3D 2 ^ -52.23
=09occurred for x =3D 6.096968451256923128E+30
Estimated loss of base 2 significant digits is 0.77
The root-mean-square relative error was 1.704421621E-18 =3D 2 ^ -59.03
Estimated loss of base 2 significant digits is 0.00
SPECIAL VALUES:
sin(x) has proper period if 1.000000000000000 is close to 1
Absolute Error =3D 0.00000000000E+00
This translates to a loss of 0.00 base 2 digits
sin(x) + sin(-x) for random values in [0, 6PI)
=09x =3D 10.8786647960, sin(x) + sin(-x) =3D 5.009014037E-17
=09x =3D 15.7973193378, sin(x) + sin(-x) =3D -1.639855786E-18
=09x =3D 11.9155376961, sin(x) + sin(-x) =3D 1.675092356E-17
=09x =3D 16.2057176579, sin(x) + sin(-x) =3D 2.539743589E-17
=09x =3D 13.2021785142, sin(x) + sin(-x) =3D 1.680513367E-17
sin(x) - x for Small x
=09x =3D 1.0837106757E-16, sin(x) - x =3D 0
=09x =3D 5.4185533787E-17, sin(x) - x =3D 0
=09x =3D 2.7092766894E-17, sin(x) - x =3D 0
=09x =3D 1.3546383447E-17, sin(x) - x =3D 0
=09x =3D 6.7731917234E-18, sin(x) - x =3D 0
cos(x) - cos(-x) for random values in [0, 7PI)
=09x =3D 0.3690766390, cos(x) - cos(-x) =3D 2.526191062E-17
=09x =3D 15.1514152500, cos(x) - cos(-x) =3D 1.837722682E-17
=09x =3D 15.0953128982, cos(x) - cos(-x) =3D 3.881443777E-17
=09x =3D 3.1418230884, cos(x) - cos(-x) =3D 2.520770051E-17
=09x =3D 3.8785036485, cos(x) - cos(-x) =3D 2.043721095E-17
=09x =3D 5.152919016E-231, sin(x) =3D 5.152919016E-231
=09sin(x) for consecutive x around 94906265.624251552887:
The following calls should not trigger any error messages:
=09sin(9.490626562425154E+07) =3D 0.94195705717885624964
=09sin(94906265.6242515529) =3D 0.94195705217603886705
=09sin(9.490626562425157E+07) =3D 0.94195704717322136209
BOUNDARY VALUE:
The following call might trigger an error message:
=09sin(9.00719925474099400000E+15) =3D -0.1272965509
Elapsed time: 82.912 seconds
Test of sinh(x) vs. xsinh(x) & cosh(x) vs. xcosh(x) (1000 arguments per t=
est):
There are 53 base 2 significant digits
Test 1: sinh(x) for 1000 values in (0, 0.5)
There were 1000 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.0983727745E-16 =3D 2 ^ -53.02
=09occurred for x =3D 0.48194285691752902778
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.439955063E-18 =3D 2 ^ -59.27
Estimated loss of base 2 significant digits is 0.00
Test 2: cosh(x) for 1000 values in (0, 0.5)
There were 0 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.2608207665E-16 =3D 2 ^ -52.82
=09occurred for x =3D 0.44273878372048675045
Estimated loss of base 2 significant digits is 0.18
The root-mean-square relative error was 1.988461183E-18 =3D 2 ^ -58.80
Estimated loss of base 2 significant digits is 0.00
Test 3: sinh(x) for 1000 values in (3, 709.78)
There were 0 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.1800305413E-16 =3D 2 ^ -52.91
=09occurred for x =3D 12.151441322013930346
Estimated loss of base 2 significant digits is 0.09
The root-mean-square relative error was 1.491125388E-18 =3D 2 ^ -59.22
Estimated loss of base 2 significant digits is 0.00
Test 4: cosh(x) for 1000 values in (3, 709.78)
There were 0 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.6374448851E-16 =3D 2 ^ -52.44
=09occurred for x =3D 12.526859651711694355
Estimated loss of base 2 significant digits is 0.56
The root-mean-square relative error was 1.499407356E-18 =3D 2 ^ -59.21
Estimated loss of base 2 significant digits is 0.00
Test 5: sinh(x) for 1000 values in (1E-07, 2E-07)
There were 1000 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.0623460271E-16 =3D 2 ^ -53.06
=09occurred for x =3D 1.2220234628147488077E-07
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.471604769E-18 =3D 2 ^ -59.24
Estimated loss of base 2 significant digits is 0.00
SPECIAL VALUES
sinh(x) vs. sinh(-x) for random values
=09x =3D 1.7313932765, sinh(x) + sinh(-x) =3D +0
=09x =3D 2.5142214602, sinh(x) + sinh(-x) =3D +0
=09x =3D 1.8964167239, sinh(x) + sinh(-x) =3D +0
=09x =3D 2.5792200716, sinh(x) + sinh(-x) =3D +0
=09x =3D 2.1011919701, sinh(x) + sinh(-x) =3D +0
sinh(x) - x for selected values of x
=09x =3D 1E-07, sinh(x) - x =3D +1.720535674E-22
sinh(x) =3D 1.00000000000000168E-07, Expected: +1.00000000000000168E-07
=09Error =3D +0 (Exactly)
=09x =3D 1E-18, sinh(x) - x =3D +0 (Exactly)
sinh(x) =3D 1.00000000000000007E-18, Expected: +1.00000000000000007E-18
=09Error =3D +0 (Exactly)
sinh(x) - x for small values of x
=09x =3D 1.083710676E-16, sinh(x) - x =3D +0
=09x =3D 5.418553379E-17, sinh(x) - x =3D +0
=09x =3D 2.709276689E-17, sinh(x) - x =3D +0
=09x =3D 1.354638345E-17, sinh(x) - x =3D +0
=09x =3D 6.773191723E-18, sinh(x) - x =3D +0
cosh(x) vs. cosh(-x) for random values
=09x =3D 0.0503488898, cosh(x) - cosh(-x) =3D +0
=09x =3D 2.0669336845, cosh(x) - cosh(-x) =3D +0
=09x =3D 2.0592802845, cosh(x) - cosh(-x) =3D +0
=09x =3D 0.4286028642, cosh(x) - cosh(-x) =3D +0
=09x =3D 0.5290997378, cosh(x) - cosh(-x) =3D +0
Testing underflow in sinh for very small argument.
=09x =3D 5.152919016E-231, sinh(x) =3D +5.152919016E-231
BOUNDARY VALUES:
The following call should not trigger an error message:
=09x =3D 709.7824688, sinh(x) =3D +8.986271493E+307
The following calls might trigger error messages:
=09x =3D 9.007199255E+15, sinh(x) =3D +Inf
=09x =3D 9.007199255E+15, cosh(x) =3D +Inf
Elapsed time: 77.857 seconds
Test of sqrt(x) vs. xsqrt(x):
There are 53 Base 2 Significant Digits
Test 1: sqrt(x) for 1000 values in (0.5, 1)
There were 1000 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 6.4197456505E-17 =3D 2 ^ -53.79
=09occurred for x =3D 0.57068483077915321822
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.102093148E-18 =3D 2 ^ -59.65
Estimated loss of base 2 significant digits is 0.00
Test 2: sqrt(x) for 1000 values in (1, 2)
There were 0 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 7.8143136818E-17 =3D 2 ^ -53.51
=09occurred for x =3D 1.0086810605853808731
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.240065553E-18 =3D 2 ^ -59.48
Estimated loss of base 2 significant digits is 0.00
SPECIAL VALUES:
These calls should not trigger error messages
=09sqrt(0) =3D 0 (Exactly)
=09sqrt(1) =3D 1 (Exactly)
=09EpsNeg =3D 1.1102230246251565404E-16
=09sqrt(1 - EpsNeg) =3D 1 (Exactly)
=09Eps =3D 2.2204460492503130808E-16
=09sqrt(1 + Eps) =3D 1 (Exactly)
=09Min =3D 2.2250738585072013832E-308
=09sqrt(Min) =3D 1.4916681462400413485E-154
=09Max =3D 1.7976931348623157082E+308
=09sqrt(Max) =3D 1.3407807929942596355E+154
Testing x =3D sqrt(x*x) for integral x in [1, 1000]
=09x =3D sqrt(x*x) failed 0 out of 1000 times
BOUNDARY VALUE:
=09Finding Square Root of a Negative Number:
=09This call might trigger an error message
=09Result: sqrt(-1.0) =3D NaN
Elapsed time: 52.967 seconds
Test of tan(x) vs. xtan(x) & cot(x) vs. xcot(x):
There are 53 base 2 significant digits
Test 1: tan(x) for 1000 values in (0, pi/4)
There were 1000 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.0750342024E-16 =3D 2 ^ -53.05
=09occurred for x =3D 0.12752326864337668044
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.458087448E-18 =3D 2 ^ -59.25
Estimated loss of base 2 significant digits is 0.00
Test 2: tan(x) for 1000 values in (0.875*pi, 1.125*pi)
There were 1000 function values less than 1
Result was smaller 500 times, equal 0 times, and larger 500 times
The maximum relative error of 1.0669706984E-16 =3D 2 ^ -53.06
=09occurred for x =3D 3.3940761664991017099
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.495524765E-18 =3D 2 ^ -59.21
Estimated loss of base 2 significant digits is 0.00
Test 3: tan(x) for 1000 values in (6.000*pi, 6.250*pi)
There were 1000 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.1017651031E-16 =3D 2 ^ -53.01
=09occurred for x =3D 19.320797363434405014
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.477251929E-18 =3D 2 ^ -59.23
Estimated loss of base 2 significant digits is 0.00
SPECIAL VALUES:
tan(x) + tan(-x) for random values in [0, 6PI)
=09x =3D 10.8786647960, tan(x) + tan(-x) =3D 7.060324547E-16
=09x =3D 15.7973193378, tan(x) + tan(-x) =3D -1.795709848E-18
=09x =3D 11.9155376961, tan(x) + tan(-x) =3D -3.491130995E-17
=09x =3D 16.2057176579, tan(x) + tan(-x) =3D -5.551115123E-17
=09x =3D 13.2021785142, tan(x) + tan(-x) =3D -2.818925648E-18
tan(x) - x for Small x
=09x =3D 1.0837106757E-16, tan(x) - x =3D 0
=09x =3D 5.4185533787E-17, tan(x) - x =3D 0
=09x =3D 2.7092766894E-17, tan(x) - x =3D 0
=09x =3D 1.3546383447E-17, tan(x) - x =3D 0
=09x =3D 6.7731917234E-18, tan(x) - x =3D 0
=09x =3D 5.152919016E-231, tan(x) =3D 5.152919016E-231
=09tan(11) =3D -225.95084645419512981
=09relative error =3D -5.406928322E-17
=09Estimated Loss of Base 2 Significant Digits is 0.00
BOUNDARY VALUES:
The next call should not trigger an error message:
=09tan(67108864) =3D -0.4638148881
The next calls might trigger error messages:
=09tan(9.007199255E+15) =3D 1.6062574
Attempting to calculate tan(x) for x very near pi/2
PI/2 =3D 1.570796326794896558
TanArg =3D 1.5707963267948961139
Diff: =3D 4.4408920985006261617E-16
=09tan(1.5707963267948961139) =3D 1.978937966E+15
Attempting to calculate tan(x) for x at pi/2
PI/2 =3D 1.570796326794896558
TanArg =3D 1.570796326794896558
Diff: =3D 0
=09tan(1.570796326794896558) =3D 1.633123935E+16
Elapsed time: 117.143 seconds
Test of tanh(x) vs. xtanh(x):
There are 53 base 2 significant digits
Test 1: tanh(x) for 1000 values in (+0, +0.5493061443)
There were 1000 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.1300378520E-16 =3D 2 ^ -52.97
=09occurred for x =3D 0.52211873825830057427
Estimated loss of base 2 significant digits is 0.03
The root-mean-square relative error was 1.453454815E-18 =3D 2 ^ -59.26
Estimated loss of base 2 significant digits is 0.00
Test 2: tanh(x) for 1000 values in (+0.0625, 54+log(2))
There were 1000 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 9.0909745608E-17 =3D 2 ^ -53.29
=09occurred for x =3D 0.96377010819298736255
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 6.376711103E-19 =3D 2 ^ -60.44
Estimated loss of base 2 significant digits is 0.00
SPECIAL VALUES:
The following calls should not trigger error messages:
Testing tanh(x) vs. tanh(-x) for random Values in [0, 3)
=09x =3D 1.7313932765, tanh(x) + tanh(-x) =3D +4.9331198848E-18
=09x =3D 2.5142214602, tanh(x) + tanh(-x) =3D -3.3176586478E-17
=09x =3D 1.8964167239, tanh(x) + tanh(-x) =3D -1.3606737265E-17
=09x =3D 2.5792200716, tanh(x) + tanh(-x) =3D +1.1004652051E-17
=09x =3D 2.1011919701, tanh(x) + tanh(-x) =3D +2.9707139526E-17
Testing tanh(x) - x for small values of x
=09x =3D +1.81816480884E-09, tanh(x) - x =3D -1.9185097215E-27
=09x =3D -9.09082404421E-10, tanh(x) - x =3D +2.5243548967E-28
=09x =3D +4.54541202211E-10, tanh(x) - x =3D -2.5243548967E-29
=09x =3D -2.27270601105E-10, tanh(x) - x =3D +1.2621774484E-29
=09x =3D +1.13635300553E-10, tanh(x) - x =3D +0
=09x =3D -5.68176502763E-11, tanh(x) - x =3D +0
=09x =3D +2.84088251382E-11, tanh(x) - x =3D +0
=09x =3D -1.42044125691E-11, tanh(x) - x =3D +7.8886090522E-31
=09x =3D +7.10220628454E-12, tanh(x) - x =3D +0
=09x =3D -3.55110314227E-12, tanh(x) - x =3D -1.9721522631E-31
Testing tanh(x) - 1 for large x
=09x =3D 54.0116330637, tanh(x) - 1 =3D +0
=09x =3D 56.7675446430, tanh(x) - 1 =3D +0
=09x =3D 59.5132516890, tanh(x) - 1 =3D +0
=09x =3D 60.0847221745, tanh(x) - 1 =3D +0
=09x =3D 60.7901884916, tanh(x) - 1 =3D +0
Testing underflow in tanh(x) for very small x:
=09x =3D 5.1529190157E-231, tanh(x) =3D +5.1529190157E-231
BOUNDARY VALUES:
Testing tanh(x) for x =3D maximum floating point value
=09x =3D 1.7976931349E+308, tanh(x) =3D +1
Testing tanh(x) for x =3D minimum floating point value
=09x =3D 2.2250738585E-308, tanh(x) =3D +2.2250738585E-308
Testing tanh(x) for x =3D 0
=09x =3D 0, tanh(x) =3D +0
Elapsed time: 40.440 seconds
--part0_907893097_boundary--
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