[worst cases for decimal64]

The decimal32 format from IEEE 754R has a 7-digit significand, and an exponent emin ≤ e ≤ emax, with emin = -95, emax = +96, the significand being between 1.000000 and 9.999999 for normal numbers, and between 0.000001 and 0.999999 for subnormal numbers.

The smallest positive normal number is thus 1.0E-95, the smallest positive subnormal is 1.0E-101, and the largest number is 9.999999E+96.

The exponential function

The column x represents the (exact) input. The column f(x) represents the approximate value of exp(x), rounded to the nearest 15-digit decimal value. The column rnd represents the rounding mode for which this x is a bad-case (N for nearest, Z for directed rounding). The column m is -log10(|f(x)-round(f(x))|/ulp(f(x))): it represents the number of consecutive 0 or 9 after the first 7 digits of f(x), or of f(x) - 1/2ulp(f(x)) for rounding to nearest. This table gives all |x| ≥ 1E-7 in the decimal32 format (values |x| < 1E-7 give trivial worst cases) such that f(x) rounded to the nearest 14-digit decimal value ends with 7 zeroes (or 5000000 for rounding to nearest). This corresponds roughly to m ≥ 7.3. The worst case (apart from the tiny values) is x=0.2408597E-2, with m=8.3.
xf(x)rndm
0.4999999E-61.00000050000002N7.6
0.9999995E-61.00000100000000Z12.5
0.1999998E-51.00000200000000Z11.6
0.3999992E-51.00000400000000Z10.7
0.5999982E-51.00000600000000Z10.1
0.7999968E-51.00000800000000Z9.8
0.9999950E-51.00001000000000Z9.5
0.1999980E-41.00002000000000Z8.6
0.2999955E-41.00002999999999Z8.0
0.3999920E-41.00003999999998Z7.7
0.4999875E-41.00004999999996Z7.4
0.6449792E-41.00006450000004N7.4
0.7549715E-41.00007549999998N7.7
0.7949684E-41.00007949999996N7.4
0.8049676E-41.00008049999995N7.3
0.3349439E-31.00033499999997Z7.5
0.4533972E-31.00045350000005N7.3
0.7232384E-31.00072349999995N7.3
0.2408597E-21.00241150000001N8.3
0.2877855E-21.00288199999998Z7.8
0.6440714E-21.00646150000001N7.9
0.7750388E-21.00778050000003N7.5
0.2143316E-11.02166449999996N7.4
0.8258902E02.28391300000005Z7.3
0.9720898E02.64346300000002Z7.7
0.3185954E12.41903550000002e1N7.6
0.3725083E14.14746750000002e1N7.7
0.6739320E18.44985949999996e2N7.4
-0.5000000E-79.99999950000001e-1N7.9
-0.1000000E-69.99999900000005e-1Z7.3
-0.4500001E-69.99999550000001e-1N7.9
-0.2000002E-59.99998000000000e-1Z10.6
-0.2450003E-59.99997550000001e-1N7.9
-0.4000008E-59.99996000000000e-1Z9.7
-0.6000018E-59.99994000000000e-1Z9.1
-0.8000032E-59.99992000000000e-1Z8.8
-0.1000005E-49.99990000000000e-1Z8.5
-0.2000020E-49.99980000000003e-1Z7.6
-0.8660375E-49.99913399999997e-1Z7.5
-0.3046464E-39.99695400000003e-1Z7.6
-0.9743245E-39.99026149999997e-1N7.6
-0.8340082E-29.91694600000002e-1Z7.6
-0.1242082E-19.87656000000002e-1Z7.7
-0.1941098E-19.80776199999995e-1Z7.3
-0.6846253E-19.33828450000005e-1N7.3
-0.8838634E-19.15407150000003e-1N7.5
-0.6603752E11.35527349999998e-3N7.8
-0.7546489E21.68273100000003e-33Z7.6