C语言中float的幅度范围？

Question

In The C Programming Language book by Dennis Ritchie, it is mentioned that "A float number is typically a 32-bit quantity, with at least six significant digits and magnitude generally between about 10^-38 and 10^+38."

How is that possible since we have only 32-bits? Shouldn't the top limit be 2^32? I tried printing out a float by looping and multiplying a float by 10, 38 times and this was the output 100000006944061730000000000000000000000.000000 . It also has to keep track of the sign , so how is it storing all this in just 32-bits?

Answer 1

See Single-precision floating-point format for a details about a typical C float .

Range of magnitude for float in c programming language?

#include <float.h>
printf("float magnitude range %e to %e\n", FLT_MIN, FLT_MAX);
// typical result
// float magnitude range 1.175494e-38 to 3.402823e+38

How is that possible since we have only 32-bits?

Typical float does indeed only store about 2 ³² different values. Yet they are not distributed linearly but logarithmically in linear groups.

2 ²³ different values in the range [2 ^-126 to 2 ^-127 )
...
2 ²³ different values in the range [0.5 to 1.0)
2 ²³ different values in the range [1.0 to 2.0)
2 ²³ different values in the range [2.0 to 4.0)
...
2 ²³ different values in the range [2 ¹²⁷ to 2 ¹²⁸ )

And their negative counter parts.
Also +/- zeros, small sub-normal numbers, +/- infinity and Not-a-Number

It also has to keep track of the sign , so how is it storing all this in just 32-bits?

 1 bit for sign
 8 bits for the binary exponent
23 bits for the significant (with a MSBit usually implied as 1)
--
32 bits

---

When printing a float , only about 6 or so ( FLT_DIG, FLT_DECIMAL_DIG ) significant digits usually are important.

printf("%.*e\n", FLT_DIG-1, FLT_TRUE_MIN);
printf("%.*e\n", FLT_DIG-1, FLT_MIN);
printf("%.*e\n", FLT_DIG-1, acos(-1));
printf("%.*e\n", FLT_DECIMAL_DIG - 1, nextafterf(FLT_MAX, 0.0));
printf("%.*e\n", FLT_DECIMAL_DIG - 1, FLT_MAX);

Output

1.40130e-45    // min sub-normal
1.17549e-38    // min normal
3.14159e+00    // pi
3.40282326e+38 // number before max
3.40282347e+38 // max