Ranges of floating point datatype in C?

Question

I am reading a C book, talking about ranges of floating point, the author gave the table:

Type     Smallest Positive Value  Largest value      Precision
====         


        

Answer 1

Infinity, NaN and subnormals

These are important caveats that no other answer has mentioned so far.

First read this introduction to IEEE 754 and subnormal numbers: What is a subnormal floating point number?

Then, for single precision floats (32-bit):

• IEEE 754 says that if the exponent is all ones (0xFF == 255), then it represents either NaN or Infinity.

  This is why the largest non-infinite number has exponent 0xFE == 254 and not 0xFF.

  Then with the bias, it becomes:

  254 - 127 == 127
  

• FLT_MIN is the smallest normal number. But there are smaller subnormal ones! Those take up the -127 exponent slot.

All asserts of the following program pass on Ubuntu 18.04 amd64:

#include 
#include 
#include 
#include 
#include 
#include 

float float_from_bytes(
    uint32_t sign,
    uint32_t exponent,
    uint32_t fraction
) {
    uint32_t bytes;
    bytes = 0;
    bytes |= sign;
    bytes <<= 8;
    bytes |= exponent;
    bytes <<= 23;
    bytes |= fraction;
    return *(float*)&bytes;
}

int main(void) {
    /* All 1 exponent and non-0 fraction means NaN.
     * There are of course many possible representations,
     * and some have special semantics such as signalling vs not.
     */
    assert(isnan(float_from_bytes(0, 0xFF, 1)));
    assert(isnan(NAN));
    printf("nan                  = %e\n", NAN);

    /* All 1 exponent and 0 fraction means infinity. */
    assert(INFINITY == float_from_bytes(0, 0xFF, 0));
    assert(isinf(INFINITY));
    printf("infinity             = %e\n", INFINITY);

    /* ANSI C defines FLT_MAX as the largest non-infinite number. */
    assert(FLT_MAX == 0x1.FFFFFEp127f);
    /* Not 0xFF because that is infinite. */
    assert(FLT_MAX == float_from_bytes(0, 0xFE, 0x7FFFFF));
    assert(!isinf(FLT_MAX));
    assert(FLT_MAX < INFINITY);
    printf("largest non infinite = %e\n", FLT_MAX);

    /* ANSI C defines FLT_MIN as the smallest non-subnormal number. */
    assert(FLT_MIN == 0x1.0p-126f);
    assert(FLT_MIN == float_from_bytes(0, 1, 0));
    assert(isnormal(FLT_MIN));
    printf("smallest normal      = %e\n", FLT_MIN);

    /* The smallest non-zero subnormal number. */
    float smallest_subnormal = float_from_bytes(0, 0, 1);
    assert(smallest_subnormal == 0x0.000002p-126f);
    assert(0.0f < smallest_subnormal);
    assert(!isnormal(smallest_subnormal));
    printf("smallest subnormal   = %e\n", smallest_subnormal);

    return EXIT_SUCCESS;
}

GitHub upstream.

Compile and run with:

gcc -ggdb3 -O0 -std=c11 -Wall -Wextra -Wpedantic -Werror -o subnormal.out subnormal.c
./subnormal.out

Output:

nan                  = nan
infinity             = inf
largest non infinite = 3.402823e+38
smallest normal      = 1.175494e-38
smallest subnormal   = 1.401298e-45