[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.
 x f(x) rnd m 0.4999999E-6 1.00000050000002 N 7.6 0.9999995E-6 1.00000100000000 Z 12.5 0.1999998E-5 1.00000200000000 Z 11.6 0.3999992E-5 1.00000400000000 Z 10.7 0.5999982E-5 1.00000600000000 Z 10.1 0.7999968E-5 1.00000800000000 Z 9.8 0.9999950E-5 1.00001000000000 Z 9.5 0.1999980E-4 1.00002000000000 Z 8.6 0.2999955E-4 1.00002999999999 Z 8.0 0.3999920E-4 1.00003999999998 Z 7.7 0.4999875E-4 1.00004999999996 Z 7.4 0.6449792E-4 1.00006450000004 N 7.4 0.7549715E-4 1.00007549999998 N 7.7 0.7949684E-4 1.00007949999996 N 7.4 0.8049676E-4 1.00008049999995 N 7.3 0.3349439E-3 1.00033499999997 Z 7.5 0.4533972E-3 1.00045350000005 N 7.3 0.7232384E-3 1.00072349999995 N 7.3 0.2408597E-2 1.00241150000001 N 8.3 0.2877855E-2 1.00288199999998 Z 7.8 0.6440714E-2 1.00646150000001 N 7.9 0.7750388E-2 1.00778050000003 N 7.5 0.2143316E-1 1.02166449999996 N 7.4 0.8258902E0 2.28391300000005 Z 7.3 0.9720898E0 2.64346300000002 Z 7.7 0.3185954E1 2.41903550000002e1 N 7.6 0.3725083E1 4.14746750000002e1 N 7.7 0.6739320E1 8.44985949999996e2 N 7.4 -0.5000000E-7 9.99999950000001e-1 N 7.9 -0.1000000E-6 9.99999900000005e-1 Z 7.3 -0.4500001E-6 9.99999550000001e-1 N 7.9 -0.2000002E-5 9.99998000000000e-1 Z 10.6 -0.2450003E-5 9.99997550000001e-1 N 7.9 -0.4000008E-5 9.99996000000000e-1 Z 9.7 -0.6000018E-5 9.99994000000000e-1 Z 9.1 -0.8000032E-5 9.99992000000000e-1 Z 8.8 -0.1000005E-4 9.99990000000000e-1 Z 8.5 -0.2000020E-4 9.99980000000003e-1 Z 7.6 -0.8660375E-4 9.99913399999997e-1 Z 7.5 -0.3046464E-3 9.99695400000003e-1 Z 7.6 -0.9743245E-3 9.99026149999997e-1 N 7.6 -0.8340082E-2 9.91694600000002e-1 Z 7.6 -0.1242082E-1 9.87656000000002e-1 Z 7.7 -0.1941098E-1 9.80776199999995e-1 Z 7.3 -0.6846253E-1 9.33828450000005e-1 N 7.3 -0.8838634E-1 9.15407150000003e-1 N 7.5 -0.6603752E1 1.35527349999998e-3 N 7.8 -0.7546489E2 1.68273100000003e-33 Z 7.6