Mail Archives: djgpp-workers/1998/10/08/21:13:54
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Dear Eric Rudd,
Attached are the complete Elefunt-style reports for your 14
elementary math functions. Separately, I will send results
for the same functions from libm.a.
K.B. Williams
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++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
E R I C R U D D
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
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 1.0258433649E-16 =3D 2 ^ -53.11
=09occurred for x =3D 1.1294475649854087695
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.417351111E-18 =3D 2 ^ -59.29
Estimated loss of base 2 significant digits is 0.00
Test 2: acosh(x) for 1000 values in (+1.5, +2)
There were 86 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.0918790744E-16 =3D 2 ^ -53.02
=09occurred for x =3D 1.5526026855623642486
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.710818274E-18 =3D 2 ^ -59.02
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 1.0516288465E-16 =3D 2 ^ -53.08
=09occurred for x =3D 4.0298723923854478102
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.555497906E-18 =3D 2 ^ -59.16
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 1.0376585372E-16 =3D 2 ^ -53.10
=09occurred for x =3D 35.158872579655003676
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.796270932E-18 =3D 2 ^ -58.95
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 9.1555022666E-17 =3D 2 ^ -53.28
=09occurred for x =3D 134369328.13235986233
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.655397329E-18 =3D 2 ^ -59.07
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 0.000000000000000000E+00
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: 48.022 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 1.0857972338E-16 =3D 2 ^ -53.03
=09occurred for x =3D -0.12491219690790450469
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.50617046E-18 =3D 2 ^ -59.20
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 7.6581568459E-17 =3D 2 ^ -53.54
=09occurred for x =3D 0.12052473614429452808
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.331354333E-18 =3D 2 ^ -59.38
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 1.0669897925E-16 =3D 2 ^ -53.06
=09occurred for x =3D 0.85718133091287229686
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.550192258E-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 1.0827462060E-16 =3D 2 ^ -53.04
=09occurred for x =3D 0.9920864099741708575
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.454348734E-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 1.0353697128E-16 =3D 2 ^ -53.10
=09occurred for x =3D -0.53321585540049354357
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.647600245E-18 =3D 2 ^ -59.07
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 -1.138412281E-17
=09x =3D -0.8380738201, asin(x) + asin(-x) =3D +3.854338723E-17
=09x =3D -0.6321389080, asin(x) + asin(-x) =3D -3.01408204E-17
=09x =3D -0.8597400239, asin(x) + asin(-x) =3D -8.554355141E-17
=09x =3D -0.7003973234, asin(x) + asin(-x) =3D +1.707618422E-17
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 +3.7615819226313200257E-37
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: 118.407 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 1.0714432652E-16 =3D 2 ^ -53.05
=09occurred for x =3D 0.1258854302688110216
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.423663762E-18 =3D 2 ^ -59.29
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 1.0619274996E-16 =3D 2 ^ -53.06
=09occurred for x =3D 0.53387192123691773471
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.493840108E-18 =3D 2 ^ -59.22
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 1.0248114983E-16 =3D 2 ^ -53.12
=09occurred for x =3D 13886045.620451968163
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.450627909E-18 =3D 2 ^ -59.26
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 7.3099384883E-17 =3D 2 ^ -53.60
=09occurred for x =3D 17294333489.316173553
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.241381602E-18 =3D 2 ^ -59.48
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 0.01058228995, asinh(x) + asinh(-x) =3D -2.8799120207E-20
=09x =3D 0.93402556185, asinh(x) + asinh(-x) =3D -1.2739375527E-17
=09x =3D 0.16316900591, asinh(x) + asinh(-x) =3D -1.1831356207E-17
=09x =3D 0.98555079159, asinh(x) + asinh(-x) =3D +1.8323016715E-17
=09x =3D 0.43746369532, asinh(x) + asinh(-x) =3D -1.1275702594E-17
Testing asinh(x) - x for small values of x
=09x =3D 1.47761084771E-10, asinh(x) - x =3D +0 (Exactly)
=09x =3D 7.38805423853E-11, asinh(x) - x =3D +0 (Exactly)
=09x =3D 3.69402711926E-11, asinh(x) - x =3D +0 (Exactly)
=09x =3D 1.84701355963E-11, asinh(x) - x =3D +0 (Exactly)
=09x =3D 9.23506779816E-12, 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: 38.132 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 1.0950564256E-16 =3D 2 ^ -53.02
=09occurred for x =3D -0.03154852077133271648
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.455683391E-18 =3D 2 ^ -59.25
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 1.0969720604E-16 =3D 2 ^ -53.02
=09occurred for x =3D 0.55019477716455711391
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.510828517E-18 =3D 2 ^ -59.20
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 343 function values less than 1
Result was smaller 494 times, equal 0 times, and larger 506 times
The maximum relative error of 1.0590048549E-16 =3D 2 ^ -53.07
=09occurred for x =3D +0.01287574575272250679
=09 and y =3D +0.00740817891357730546
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.460126163E-18 =3D 2 ^ -59.25
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 679 function values less than 1
Result was smaller 1000 times, equal 0 times, and larger 0 times
The maximum relative error of 1.0625170124E-16 =3D 2 ^ -53.06
=09occurred for x =3D +1.3887105332370135446
=09 and y =3D -0.36675920621244040287
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.530107939E-18 =3D 2 ^ -59.18
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 8.012254055E-17
=09x =3D 4.1903691004, atan(x) + atan(-x) =3D -1.050591905E-16
=09x =3D 3.1606945399, atan(x) + atan(-x) =3D -9.530137096E-17
=09x =3D 4.2987001193, atan(x) + atan(-x) =3D -2.298508606E-17
=09x =3D 3.5019866169, atan(x) + atan(-x) =3D 1.19262239E-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 NaN
Elapsed time: 75.275 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 1.0619315414E-16 =3D 2 ^ -53.06
=09occurred for x =3D 0.12901738229572382033
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.48226141E-18 =3D 2 ^ -59.23
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 1.0548334639E-16 =3D 2 ^ -53.07
=09occurred for x =3D 0.77695821567161105214
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.503171087E-18 =3D 2 ^ -59.21
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 1.0945444175E-16 =3D 2 ^ -53.02
=09occurred for x =3D 0.96519019789746529625
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.462110875E-18 =3D 2 ^ -59.25
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 0.6359492850, atanh(x) + atanh(-x) =3D -3.9519169187E-17
=09x =3D 0.1285980174, atanh(x) + atanh(-x) =3D +7.8875708048E-18
=09x =3D 0.9465689253, atanh(x) + atanh(-x) =3D +6.5160550566E-17
=09x =3D 0.8955649986, atanh(x) + atanh(-x) =3D +8.7386695102E-17
=09x =3D 0.6267650394, atanh(x) + atanh(-x) =3D -3.1983964088E-17
Testing atanh(x) - x for small values of x
=09x =3D +2.86247149993E-09, atanh(x) - x =3D +7.8759872777E-27
=09x =3D -1.43123574997E-09, atanh(x) - x =3D -1.0097419587E-27
=09x =3D +7.15617874983E-10, atanh(x) - x =3D +1.0097419587E-28
=09x =3D -3.57808937492E-10, atanh(x) - x =3D -2.5243548967E-29
=09x =3D +1.78904468746E-10, atanh(x) - x =3D +1.2621774484E-29
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: 20.165 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 1.0224554558E-16 =3D 2 ^ -53.12
=09occurred for x =3D 0.04108901996323588435
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.501070881E-18 =3D 2 ^ -59.21
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 1.2339975877E-16 =3D 2 ^ -52.85
=09occurred for x =3D -587.06208632840912287
Estimated loss of base 2 significant digits is 0.15
The root-mean-square relative error was 1.566138606E-18 =3D 2 ^ -59.15
Estimated loss of base 2 significant digits is 0.00
Test 3: exp(x) for values in (10*log(2), +709)
There were 485 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.2845459531E-16 =3D 2 ^ -52.79
=09occurred for x =3D 417.97821086348255903
Estimated loss of base 2 significant digits is 0.21
The root-mean-square relative error was 1.532459093E-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 exp(x) vs. 1/exp(-x) for random Values in [2, 3)
x =3D 2.8617004099, exp(x) * exp(-x) - 1.0 =3D -8.6627753582E-17
x =3D 2.2555257589, exp(x) * exp(-x) - 1.0 =3D +6.5052130349E-19
x =3D 2.4448363764, exp(x) * exp(-x) - 1.0 =3D -3.3339216804E-17
x =3D 2.1838928054, exp(x) * exp(-x) - 1.0 =3D -3.5128150389E-17
x =3D 2.7767009994, exp(x) * exp(-x) - 1.0 =3D +8.7928796189E-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 1.3407807929942437946E+154
BOUNDARY VALUES:
The following calls might trigger error messages
=09exp(-6.703903965E+153) =3D 0
=09exp(+6.703903965E+153) =3D Inf
Elapsed time: 12.967 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 1.0650956607E-16 =3D 2 ^ -53.06
=09occurred for x =3D -0.11849718672685342902
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.508402193E-18 =3D 2 ^ -59.20
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 1.0715172857E-16 =3D 2 ^ -53.05
=09occurred for x =3D 7.5406790020347796771E-09
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.430648083E-18 =3D 2 ^ -59.28
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 1.0820847826E-16 =3D 2 ^ -53.04
=09occurred for x =3D 54.187053196160064772
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.602079504E-18 =3D 2 ^ -59.11
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 NaN
=09log1p(Inf) =3D Inf
=09log1p(NaN) =3D NaN
Elapsed time: 6.868 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 1.0411266233E-16 =3D 2 ^ -53.09
=09occurred for x =3D 0.99999808609278451055
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.436708314E-18 =3D 2 ^ -59.27
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 1.0567865921E-16 =3D 2 ^ -53.07
=09occurred for x =3D 0.88031567536312071987
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.548032766E-18 =3D 2 ^ -59.16
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 1.0946909834E-16 =3D 2 ^ -53.02
=09occurred for x =3D 0.86591654498549097685
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.465023468E-18 =3D 2 ^ -59.24
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 1.0229846481E-16 =3D 2 ^ -53.12
=09occurred for x =3D 68.932035932723195515
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.614035289E-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 15.3696275962, log(x) + log(1/x) =3D -1.14925430283E-16
=09x =3D 15.4413296515, log(x) + log(1/x) =3D -1.70870262384E-16
=09x =3D 15.4396488513, log(x) + log(1/x) =3D -7.80625564190E-18
=09x =3D 15.8432722448, log(x) + log(1/x) =3D +4.55364912444E-18
=09x =3D 15.2108576698, log(x) + log(1/x) =3D +2.90566182226E-17
=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 NaN
=09log10(-2) =3D NaN
=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: 22.088 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 1.0346296654E-16 =3D 2 ^ -53.10
=09occurred for x =3D 0.80720338542507696911
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.460855037E-18 =3D 2 ^ -59.25
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 1.2227782396E-16 =3D 2 ^ -52.86
=09occurred for x =3D 3.6568268313913339637E+102
Estimated loss of base 2 significant digits is 0.14
The root-mean-square relative error was 1.508462855E-18 =3D 2 ^ -59.20
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 482 function values less than 1
Result was smaller 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 1.0791730584E-16 =3D 2 ^ -53.04
=09occurred for x =3D +6.8263375984826311083
=09 and y =3D -32.096227765425815903
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.523589938E-18 =3D 2 ^ -59.19
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 -1.60154831972E-16
=09 7.3213890798 9.5974002387 +3.78052186467E-16
=09 8.0039732337 10.7611979909 +3.01052892140E-16
=09 1.1678296328 7.8897789482 -8.44786291503E-17
=09 7.8642676150 2.4286762138 +1.74266585742E-16
=09 2.7636657928 9.6017786381 -7.19459938430E-16
=09 8.4606401035 1.2589809415 -3.59115848443E-17
=09 9.5148453466 5.5189793857 -3.20261064876E-16
=09 4.7846462152 9.1443398525 -1.78951276814E-16
=09 8.1849413176 5.3750309180 -2.13741200790E-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 NaN
Elapsed time: 48.516 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 1.0526282457E-16 =3D 2 ^ -53.08
=09occurred for x =3D 0.54511621658548070624
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.332141667E-18 =3D 2 ^ -59.38
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 1.0855515206E-16 =3D 2 ^ -53.03
=09occurred for x =3D 1.041137131745529798
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.355497134E-18 =3D 2 ^ -59.36
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 1.0524986309E-16 =3D 2 ^ -53.08
=09occurred for x =3D 19.386445705396567973
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.340301864E-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 1.0553942575E-16 =3D 2 ^ -53.07
=09occurred for x =3D 23.300499868982196716
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.32885462E-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 0 times, equal 0 times, and larger 1000 times
The maximum relative error of 9.9999886100E-01 =3D 2 ^ 0.00
=09occurred for x =3D 2.5158646203860639495E+30
Estimated loss of base 2 significant digits is 53.00
The root-mean-square relative error was 0.0223280432 =3D 2 ^ -5.48
Estimated loss of base 2 significant digits is 47.52
Test 6: cos(x) for 1000 values in (1e22, 1e31)
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 1.9999448000E+00 =3D 2 ^ 1.00
=09occurred for x =3D 9.809717521657076246E+30
Estimated loss of base 2 significant digits is 54.00
The root-mean-square relative error was 0.0386393328 =3D 2 ^ -4.69
Estimated loss of base 2 significant digits is 48.31
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 -1.802486112E-18
=09x =3D 11.9155376961, sin(x) + sin(-x) =3D 1.799775606E-17
=09x =3D 16.2057176579, sin(x) + sin(-x) =3D 2.377113263E-17
=09x =3D 13.2021785142, sin(x) + sin(-x) =3D 1.539567085E-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 2.531612073E-17
=09x =3D 15.1514152500, cos(x) - cos(-x) =3D 1.664250335E-17
=09x =3D 15.0953128982, cos(x) - cos(-x) =3D 3.6429193E-17
=09x =3D 3.1418230884, cos(x) - cos(-x) =3D 2.520770051E-17
=09x =3D 3.8785036485, cos(x) - cos(-x) =3D 1.897353802E-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 0.94195705717881517138
=09sin(94906265.6242515529) =3D 0.94195705217599789982
=09sin(9.490626562425157E+07) =3D 0.94195704717318034585
BOUNDARY VALUE:
The following call might trigger an error message:
=09sin(9.00719925474099400000E+15) =3D -0.1272850492
Elapsed time: 40.989 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 1.0776915449E-16 =3D 2 ^ -53.04
=09occurred for x =3D 0.49118904194295348598
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.448107192E-18 =3D 2 ^ -59.26
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 1.1002571313E-16 =3D 2 ^ -53.01
=09occurred for x =3D 0.09230620490505507536
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.959463537E-18 =3D 2 ^ -58.82
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 1.2084371037E-16 =3D 2 ^ -52.88
=09occurred for x =3D 698.75941880554739782
Estimated loss of base 2 significant digits is 0.12
The root-mean-square relative error was 1.505484181E-18 =3D 2 ^ -59.20
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 1.1966912678E-16 =3D 2 ^ -52.89
=09occurred for x =3D 649.61772647364887234
Estimated loss of base 2 significant digits is 0.11
The root-mean-square relative error was 1.53035235E-18 =3D 2 ^ -59.18
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 1.0481553608E-16 =3D 2 ^ -53.08
=09occurred for x =3D 1.2229195398547262994E-07
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.473982282E-18 =3D 2 ^ -59.23
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 +5.247538515E-17
=09x =3D 2.5142214602, sinh(x) + sinh(-x) =3D -5.074066167E-17
=09x =3D 1.8964167239, sinh(x) + sinh(-x) =3D +7.090682208E-17
=09x =3D 2.5792200716, sinh(x) + sinh(-x) =3D +3.955169525E-16
=09x =3D 2.1011919701, sinh(x) + sinh(-x) =3D +3.851086117E-16
sinh(x) - x for selected values of x
=09x =3D 1E-07, sinh(x) - x =3D +1.666704311E-22
sinh(x) =3D 1.00000000000000162E-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 +1.57209315E-17
=09x =3D 2.0669336845, cosh(x) - cosh(-x) =3D +2.406928823E-16
=09x =3D 2.0592802845, cosh(x) - cosh(-x) =3D -4.33680869E-17
=09x =3D 0.4286028642, cosh(x) - cosh(-x) =3D +3.1441863E-17
=09x =3D 0.5290997378, cosh(x) - cosh(-x) =3D -5.854691731E-17
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: 38.956 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 6.4453188962E-17 =3D 2 ^ -53.78
=09occurred for x =3D 0.54861755350939500619
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.131241903E-18 =3D 2 ^ -59.62
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 7.6786241501E-17 =3D 2 ^ -53.53
=09occurred for x =3D 1.0262172042625992141
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.221327575E-18 =3D 2 ^ -59.51
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: 26.538 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 1.0704281786E-16 =3D 2 ^ -53.05
=09occurred for x =3D 0.00798261765561560782
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.434050209E-18 =3D 2 ^ -59.27
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 1.0609457747E-16 =3D 2 ^ -53.07
=09occurred for x =3D 3.3933884377398548793
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.495539483E-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 1.0917280795E-16 =3D 2 ^ -53.02
=09occurred for x =3D 19.097575406236842355
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.46843322E-18 =3D 2 ^ -59.24
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 7.103692634E-16
=09x =3D 15.7973193378, tan(x) + tan(-x) =3D -1.734723476E-18
=09x =3D 11.9155376961, tan(x) + tan(-x) =3D -2.748452507E-17
=09x =3D 16.2057176579, tan(x) + tan(-x) =3D -5.518589058E-17
=09x =3D 13.2021785142, tan(x) + tan(-x) =3D 4.553649124E-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 1.606298915
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 1.978945886E+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 1.633177873E+16
Elapsed time: 58.516 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 1.0523478542E-16 =3D 2 ^ -53.08
=09occurred for x =3D 0.12980574531194788435
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 1.429128813E-18 =3D 2 ^ -59.28
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 7.0064282068E-17 =3D 2 ^ -53.66
=09occurred for x =3D 0.80282164248038045962
Estimated loss of base 2 significant digits is 0.00
The root-mean-square relative error was 6.13588643E-19 =3D 2 ^ -60.50
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 +5.5294310797E-18
=09x =3D 2.5142214602, tanh(x) + tanh(-x) =3D -3.2255014631E-17
=09x =3D 1.8964167239, tanh(x) + tanh(-x) =3D -1.3281476613E-17
=09x =3D 2.5792200716, tanh(x) + tanh(-x) =3D +1.0354130747E-17
=09x =3D 2.1011919701, tanh(x) + tanh(-x) =3D +3.26886955E-17
Testing tanh(x) - x for small values of x
=09x =3D +1.81816480884E-09, tanh(x) - x =3D -2.0194839174E-27
=09x =3D -9.09082404421E-10, tanh(x) - x =3D +2.5243548967E-28
=09x =3D +4.54541202211E-10, tanh(x) - x =3D +0
=09x =3D -2.27270601105E-10, tanh(x) - x =3D +1.2621774484E-29
=09x =3D +1.13635300553E-10, tanh(x) - x =3D -6.3108872418E-30
=09x =3D -5.68176502763E-11, tanh(x) - x =3D +0
=09x =3D +2.84088251382E-11, tanh(x) - x =3D +1.5777218104E-30
=09x =3D -1.42044125691E-11, tanh(x) - x =3D +0
=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: 20.165 seconds
--part0_907893057_boundary--
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