On Sun, May 1, 2016 at 1:02 AM, boB Stepp <robertvst...@gmail.com> wrote: > > py3: 1.9999999999999999 > 2.0 > py3: 1.999999999999999 > 1.999999999999999 ... > It has been many years since I did problems in converting decimal to > binary representation (Shades of two's-complement!), but I am under > the (apparently mistaken!) impression that in these 0.999...999 > situations that the floating point representation should not go "up" > in value to the next integer representation.
https://en.wikipedia.org/wiki/Double-precision_floating-point_format A binary64 float has 52 signifcand bits, with an implicit integer value of 1, so it's effectively 53 bits. That leaves 11 bits for the exponent, and 1 bit for the sign. The 11-bit exponent value is biased by 1023, i.e. 2**0 is stored as 1023. The minimum binary exponent is (1-1023) == -1022, and the maximum binary exponent is (2046-1023) == 1023. A biased exponent of 0 signifies either signed 0 (mantissa is zero) or a subnormal number (mantissa is nonzero). A biased exponent of 2047 signifies either signed infinity (mantissa is zero) or a non-number, i.e. NaN (mantissia is nonzero). The largest finite value has all 53 bits set: >>> sys.float_info.max 1.7976931348623157e+308 >>> sum(Decimal(2**-n) for n in range(53)) * 2**1023 Decimal('1.797693134862315708145274237E+308') The smallest finite value has the 52 fractional bits unset: >>> sys.float_info.min 2.2250738585072014e-308 >>> Decimal(2)**-1022 Decimal('2.225073858507201383090232717E-308') The machine epsilon value is 2**-52: >>> sys.float_info.epsilon 2.220446049250313e-16 >>> Decimal(2)**-52 Decimal('2.220446049250313080847263336E-16') Your number is just shy of 2, i.e. implicit 1 plus a 52-bit fractional value and a binary exponent of 0. >>> sum(Decimal(2**-n) for n in range(53)) Decimal('1.999999999999999777955395075') The next increment by epsilon jumps to 2.0. The 52-bit mantissa rolls over to all 0s, and the exponent increments by 1, i.e. (1 + 0.0) * 2**1. Python's float type has a hex() method to let you inspect this: >>> (1.9999999999999998).hex() '0x1.fffffffffffffp+0' >>> (2.0).hex() '0x1.0000000000000p+1' where the integer part is 0x1; the 52-bit mantissa is 13 hexadecimal digits; and the binary exponent comes after 'p'. You can also parse a float hex string using float.fromhex(): >>> float.fromhex('0x1.0000000000000p-1022') 2.2250738585072014e-308 >>> float.fromhex('0x1.fffffffffffffp+1023') 1.7976931348623157e+308 _______________________________________________ Tutor maillist - Tutor@python.org To unsubscribe or change subscription options: https://mail.python.org/mailman/listinfo/tutor
