To start, why we need this optimization in the first place. You see, when you perform decimal calculations using floating-point numbers (which is what most programming languages use), there are inherent limitations to their accuracy due to the way they work. This can lead to some pretty hilarious results if you’re not careful.
For example, let’s say we want to calculate 0.1 + 0.2 on a Linux machine using bc (which is a popular calculator program). Here’s what happens:
# Set the scale to 5 decimal places for more accurate calculation
$ echo "scale=5; 0.1+0.2" | bc -l
# The -l flag is not necessary as it is used for loading the math library, which is not needed for this calculation
# The result is not accurate due to the limitations of bc, which uses floating point arithmetic
# To get a more accurate result, we can use the printf command to format the output
$ printf "%.3f\n" $(echo "0.1+0.2" | bc)
# The printf command formats the output to 3 decimal places, giving us a more accurate result of 0.300
# This is because printf uses the decimal arithmetic instead of floating point arithmetic used by bc
Hmm, that doesn’t look right… why did we get a decimal with four digits instead of two? Well, it turns out that floating-point numbers have limited precision due to the way they are represented in memory. In this case, bc is using 53 bits (or about 16 decimal places) for its mantissa and 1 bit for its exponent.
So what can we do to optimize our decimal calculations? Here are a few tips:
1. Use a higher precision calculator program like gawk or GNU Parallel’s awk. These programs use arbitrary-precision arithmetic, which means they don’t have the same limitations as floating-point numbers. For example, let’s try our 0.1 + 0.2 calculation again using gawk:
# This script uses gawk to perform a calculation with higher precision
# The "echo" command prints the string "BEGIN { printf \"%.50f\n\", (0.1+0.2)" to the terminal
# The "BEGIN" keyword indicates that the following code will be executed before any input is processed
# The "printf" function formats and prints the result of the calculation with a precision of 50 decimal places
# The "gawk" command runs the gawk program, which is a higher precision calculator
# The "-v" option allows us to set a variable, in this case "OFS" which stands for "output field separator"
# The "." after "OFS=" sets the output field separator to a period, which will be used to separate the integer and decimal parts of the result
# The calculation (0.1+0.2) is enclosed in parentheses and will be evaluated by gawk
echo "BEGIN { printf \"%.50f\n\", (0.1+0.2)" | gawk -v OFS=.
# The result is printed to the terminal:
# 0.300000000000000000000000000000000000000000000000000000000000000
Wow, that’s a lot of decimal places! But do we really need all those digits? Probably not. In fact, most calculations don’t require more than 16 or so decimal places to be accurate enough for practical purposes. So let’s try our calculation again with gawk and limit the output to just 16 decimal places:
# Set the beginning of the awk script
BEGIN {
# Use printf to format the output with 16 decimal places
printf "%.16f\n", (0.1+0.2)
} |
# Use gawk to perform the calculation and limit the output to 16 decimal places
gawk -v OFS=.
# Print the result
0.30000000000000007
Hmm, that’s a little different than what we got with bc… why is there an extra 7 at the end? Well, it turns out that gawk uses rounding to handle decimal calculations. This can lead to some interesting results if you’re not careful. For example:
# This script uses gawk to perform a decimal calculation and print the result with 16 decimal places.
# The BEGIN statement indicates the start of the gawk program.
# The printf function is used to format and print the result of the calculation.
# The "%.16f" format specifier specifies that the result should be printed with 16 decimal places.
# The calculation is performed within the parentheses, subtracting 0.2 from 0.3.
# The -v option is used to set the output field separator (OFS) to a period, to format the result as a decimal number.
echo "BEGIN { printf \"%.16f\n\", (0.3-0.2)" | gawk -v OFS=.
# The echo command is used to print the gawk program to the standard output.
# The gawk command is used to execute the gawk program.
# The -v option is used to set the value of the OFS variable.
# The OFS variable is used to specify the output field separator, which is used to format the result as a decimal number.
# The result is printed to the standard output.
Just remember that floating-point numbers are inherently limited in their accuracy due to the way they work, so be careful when performing complex calculations with them. And if you’re really serious about precision, consider using a library like GMP (which is a popular open source multiple-precision arithmetic library) instead of relying on floating-point numbers altogether.