C中数据类型的最大值

Question

I am trying to understand the maximum value that I can store in C. I tried doing printf("%f", pow(2, x)) . The answer holds good until x = 1023. It says Inf when x = 1024.

I am sorry that it is a basic question but I am trying to understand how C assigns datatypes' sizes based on my machine.

I have a Mac (64-bit processor). A clear understanding that I have is that my processor being a 64-bit one, it will be able to do calculations up to the value (2 ⁶⁴ ). Clearly pow(2, 1023) is greater than that. But my program is working fine till x = 1023. How is this possible? Is GNU compiler has something to do with this?

If this is a duplicate of other question kindly give the link.

Answer 1

In C the pow() functions returns a double , and the double type is typically a 64-bit IEEE format representation of a floating point number.

The basic idea of floating point is to express a number in the same general way as eg 1.234×10 ⁵⁶ . Here you have a mantissa 1.234 and an exponent 56. C++, and probably also C, allows decimal representation for floating point numbers (but not for integer types), but in practice the internal representation will be binary, with a power of 2 rather than a power of 10.

The limit you ran up against was the supported range for the exponent in your compiler's representation of double numbers; probably 64-bit IEEE 754 .

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The limits of the various built-in integral numerical types are available as symbolic constants from <limits.h> . The limits of the built-in floating point types are available as symbolic constants from <float.h> . See the table over at cppreference.com for more details.

In C++ these limits are also available via the numeric_limits class template from <limits> .

Answer 2

"64-bit processor" typically means that it can deal with integers that contain at most 64 bits at a time (ie in a single instruction), not that it can only process numbers with 64 binary digits or less. Using arbitrary precision arithmetic you can do calculations on numbers that are arbitrarily large, provided that you have enough memory (and time), just like how us humans can do operations on big values with only 10 fingers. Read more here: What is the biggest number you can generate using a 64-bit processor?

However pow(2, 1023) is a little bit different. It's not an integer but a floating-point number (of type double in C) represented by a sign, a mantissa and an exponent like this (-1) ^sign × 1 × 2 ¹⁰²³ . Not all the digits are stored so it's only accurate to the first few digits. However most systems use binary floating-point types so they can store the precise value of a power of 2 up to a large exponent depending on the exponent range. Most modern systems' floating-point types conform to IEEE-754 standard with double maps to binary64/double precision , therefore the maximum value will be

2 ¹⁰²³ × (1 + (1 − 2 ⁻⁵² )) ≈ 1.7976931348623157 × 10 ³⁰⁸

Answer 3

The maximum value for a double is DBL_MAX . This is defined by <float.h> in C, or <cfloat> in C++. The numeric value may vary across systems, but you can always refer to it by the macro DBL_MAX .

You can print this:

printf("%f\n", DBL_MAX);

The integer data types all have similar macros defined in <limits.h> : eg ULLONG_MAX is the biggest value for unsigned long long . If printing with printf make sure to use the correct format specifier.