Maximum number of decimal digits that can affect a double

by

When you have a subnormal number with odd significand, that is, an odd multiple of 2^(-1074), you have a number whose last nonzero digit in the decimal representation is the 1074th after the decimal point. You then have around 300 zeros directly following the decimal point, so the number has around 750-770 significant decimal digits. The smallest positive subnormal, 2^(-1074) has 751 significant digits, and the largest positive subnormal, (2^52-1)*2^(-1074) has 767 significant digits (I think that is the maximum).

So there is at least one sequence d1, ..., d766 of decimal digits such that there is an IEEE754 double in the open interval

(d1.d2...d766E-308, d1.d2...(d766 + 1)E-308)

The answer does not change much if we consider “contains the midpoint of two consecutive IEEE754 doubles”, since subnormal doubles have all roughly the same amount of significant decimal digits, and the midpoint of two consecutive such too.

In the worst case, the entire digit sequence must be consumed (consider "0.5000000...0001" with arbitrarily many zeros before the final 1 that determines that the result shall be 0.5 + 0.5^53 and not 0.5 when rounding away from zero or up).

However, there are only

floor(DBL_MANT_DIG * log 2 / log 10) + 2 = 17

significant decimal digits necessary to distinguish between different double values, so a relatively easy, albeit probably not very efficient, method of parsing would be to parse the first (at least 17) digits (and the exponent) to the closest double, and compare the input string with the exact representation of that double value (and its neighbour).

More Related Contents:

  1. What is the rationale for all comparisons returning false for IEEE754 NaN values?
  2. What range of numbers can be represented in a 16-, 32- and 64-bit IEEE-754 systems?
  3. Why is NaN not equal to NaN? [duplicate]
  4. What is the difference between quiet NaN and signaling NaN?
  5. Why does IEEE 754 reserve so many NaN values?
  6. How to perform round to even with floating point numbers
  7. Next higher/lower IEEE double precision number
  8. What uses do floating point NaN payloads have?
  9. Why not use Double or Float to represent currency?
  10. Large numbers erroneously rounded in JavaScript
  11. biggest integer that can be stored in a double
  12. Why do I see a double variable initialized to some value like 21.4 as 21.399999618530273?
  13. Which is the first integer that an IEEE 754 float is incapable of representing exactly?
  14. How is floating point stored? When does it matter?
  15. Float and double datatype in Java
  16. Coercing floating-point to be deterministic in .NET?
  17. How to calculate float type precision and does it make sense?
  18. Formatting doubles for output in C#
  19. Python float – str – float weirdness
  20. How to check if C++ compiler uses IEEE 754 floating point standard
  21. Floating point comparison revisited
  22. How do I use floating point number literals when using generic types?
  23. Turn float into string
  24. How do I get the minimum or maximum value of an iterator containing floating point numbers?
  25. Ranges of floating point datatype in C?
  26. How computer does floating point arithmetic?
  27. Why does the floating-point value of 4*0.1 look nice in Python 3 but 3*0.1 doesn’t?
  28. What’s the relative speed of floating point add vs. floating point multiply
  29. on what systems does Python not use IEEE-754 double precision floats
  30. Does the C++ standard specify anything on the representation of floating point numbers?

Leave a Comment