**Note:** The examples below are abridged; the book contains more details.

- Rounding
- Overflow
- Underflow
- Comparing Real Numbers
- The Log-Scale Trick
- Use of Assembly Language in High-Level Languages

Rounding occurs when a real number `x`

cannot be precisely matched
to a fixed- or a floating- point representation. The result of rounding
is typically either the nearest representation of `x`

or a truncated version
of `x`

obtained by dividing and dropping the remainder. Here’s an example in R:

```
# Round 0.333.. to a nearby floating-point
print(1 / 3, digits = 22) # print 22 digits of fp(1/3)
## 0.3333333333333333148296
```

Overflow occurs when a number has a big absolute value that’s outside the range of possible binary representations. In the case of integers, this occurs when the number of bits in the representation is too small to represent the corresponding number. When overflow occurs, the number is replaced by either the closest binary representation or is marked as an overflow and considered unavailable. Here’s an example in R:

```
print(10 ^ 100) # no overflow
print(10 ^ 500) # overflow, marked by Inf value
## 1e+100
## Inf
```

Underflow occurs when the number `x`

is closer to `0`

than any of the possible
binary representations. In most cases, the number `x`

is then replaced by zero.
An underflow example using R code appears below:

```
print(10 ^ -200, digits = 22) # roundoff, but no underflow
print(10 ^ -400, digits = 22) # underflow
## 9.999999999999999821003e-201
## 0
```

Comparing whether or not two real numbers are identical is problematic due to rounding errors. See the below C++ example for an illustration:

```
// { autofold
#include <iostream>
#include <math.h>
using namespace std;
// }
int main() {
cout << (sqrt(3) * sqrt(3) - 3) << endl
<< (sqrt(3) * sqrt(3) == 3) << endl
<< (fabs(sqrt(3) * sqrt(3) - 3) < 0.0000001) << endl;
}
```

A common trick for avoiding overflow and underflow is to work in a logarithmic scale, rather than the original scale:

```
# The R code below graphs log(x) as a function of x
curve(log, from = 0, to = 1000, xlab = 'x',
ylab = 'log(x)', main = 'The logarithm function')
print(3 ^ -800, digits = 22) # underflow
print(log(3 ^ -800), digits = 22) # log of underflow
print(-800 * log(3), digits = 22) # avoiding underflow using the log-scale
# Addition, when using the log-scale, replaces multiplication
print(3 ^ -600 * 3 ^ -100 * 3 ^ 150, digits = 22) # underflow
# Avoid underflow and return results in the log-scale
print(log(3) * (-600 - 100 + 150), digits = 22)
# Avoid underflow and return results in original scale
print(exp(log(3) * (-600 - 100 + 150)), digits = 22)
```

This example simply executes the RDTSC assembly instruction, which reads
the CPU’s timestamp counter, then stores the higher 32 bits into the `EDX`

register and the lower ones into the `EAX`

register. The value of `EAX`

is then
assigned to the `timeStampCounter`

variable in C++ and printed out:

```
// { autofold
#include <iostream>
using namespace std;
// }
int main() {
uint timeStampCounter = 0;
asm("rdtsc": "=a" (timeStampCounter));
std::cout << timeStampCounter << std::endl;
return 0;
}
```