Bits in a Float, and Infinity, NaN, and denormal

Bits in a Floating-Point Number

void print_bits(float f) {
	int i=*reinterpret_cast<int *>(&f); /* read bits with "pointer shuffle" */
	std::cout<<" float "<<std::setw(10)<<f<<" = ";
	for (int bit=31;bit>=0;bit--) {
		if (i&(1<<bit)) std::cout<<"1"; else std::cout<<"0";
		if (bit==31) std::cout<<" ";
		if (bit==23) std::cout<<" (implicit 1).";
	}
	std::cout<<std::endl;
}

int foo(void) {
	print_bits(0.0);
	print_bits(-1.0);
	print_bits(1.0);
	print_bits(2.0);
	print_bits(4.0);
	print_bits(8.0);
	print_bits(1.125);
	print_bits(1.25);
	print_bits(1.5);
	print_bits(1+1.0/10);
	return sizeof(float);
}

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/* IEEE floating-point number's bits:  sign  exponent   mantissa */
struct float_bits {
	unsigned int fraction:23; /**< Value is binary 1.fraction ("mantissa") */
	unsigned int exp:8; /**< Value is 2^(exp-127) */
	unsigned int sign:1; /**< 0 for positive, 1 for negative */
};

/* A union is a struct where all the fields *overlap* each other */
union float_dissector {
	float f;
	float_bits b;
};

float_dissector s;
s.f=8.0;
std::cout<<s.f<<"= sign "<<s.b.sign<<" exp "<<s.b.exp<<"  fract "<<s.b.fraction<<"\n";
return 0;

• Floats can store fractional numbers.
  
• Floats never overflow; they hit "infinity" as explored below.
  
• "double" has more bits than "int" (but less than "long").

Normal (non-Weird) Floats

Weird: Zeros and Denormals

Weird: Infinity

float z=0.0;
float f=1.0/z;
std::cout<<f<<"\n";
return (int)f;

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float x=1.0;
while (true) {
	float old_x=x;
	x=x+x;
	std::cout<<x<<"\n";
	if (x==old_x) {
		std::cout<<"Finally hit "<<x<<" and stopped.\n";
		return 0;
	}
}

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Weird: NaN

• 0.0/0.0 (neither zero nor infinity, because we'd want (x/x)==1.0; but not 1.0 either, because we'd want (2*x)/x==2.0...)
• infinity-infinity (might cancel out to anything)
• infinity*0

float f=sqrt(-1.0);
std::cout<<f<<"\n";
return (int)f;

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Performance impact of special values

enum {n_vals=1000};
double vals[n_vals];

int average_vals(void) {
	for (int i=0;i<n_vals-1;i++) 
		vals[i]=0.5*(vals[i]+vals[i+1]);
	return 0;
}

int foo(void) {
	int i;
	for (i=0;i<n_vals;i++) vals[i]=0.0; 
	printf(" Zeros: %.3f ns/float\n",time_function(average_vals)/n_vals*1.0e9);
	for (i=0;i<n_vals;i++) vals[i]=1.0; 
	printf(" Ones: %.3f ns/float\n",time_function(average_vals)/n_vals*1.0e9);
	for (i=0;i<n_vals;i++) vals[i]=1.0e-310; 
	printf(" Denorm: %.3f ns/float\n",time_function(average_vals)/n_vals*1.0e9);
	float x=0.0;
	for (i=0;i<n_vals;i++) vals[i]=1.0/x; 
	printf(" Inf: %.3f ns/float\n",time_function(average_vals)/n_vals*1.0e9);
	for (i=0;i<n_vals;i++) vals[i]=x/x; 
	printf(" NaN: %.3f ns/float\n",time_function(average_vals)/n_vals*1.0e9);
	return 0;
}

Bonus: Arithmetic Tables for Special Floating-Point Numbers

Addition

Subtraction

Multiplication

Division

Equality

Less-Than