Floating Point Arithmetic: Errors

Floating Point Arithmetic

• Floating-point arithmetic involves operations on numbers represented in a format similar to scientific notation, consisting of a mantissa (or significand) and an exponent.

Errors in Floating Point Arithmetic

• Errors can arise in floating-point arithmetic due to the limitations in representing real numbers with finite precision.
• Types of errors include data errors, truncation errors, round-off errors, and computational errors.

Error Definition

• Error represents the difference between the actual and approximate values.
• Formula: Error (E) = Actual Value - Approximate Value or E = x - xa where x is the actual value and xa is the approximate value.

Truncation Error

• Truncation error occurs when digits are discarded from a number.
• This commonly happens when representing numbers in normalized floating-point form due to limited mantissa digits or during the conversion from one number system to another.
• For instance, truncating 0.90012346 to four digits yields 0.9001.

Round-Off Error

• Round-off error arises from rounding digits.
• Rounding is akin to truncation but involves adjusting the last digit depending on the truncated digit.
• If rounding to the nth decimal place, 1 is added to the nth digit if the (n+1)th digit is 5-9; otherwise, the nth digit remains unchanged.
• Rounding 0.90012346 to four digits results in 0.9001.

Normalized Floating Point Form

• Normalized floating-point form requires the digit to the left of the decimal point to be zero and the digit to the right should be non-zero.
• The notation En means multiplication by 10 to the power n and abc E5 = abc × 10^5
• In the number abc E5, 'abc' denotes the mantissa, while '5' represents the exponent.
• 0.0312 E3 in normalized form is 0.312 E2.

Addition of Floating Point Numbers

• For adding floating-point numbers, exponents must match, select the largest.
• Express the addition in standard floating-point form.
• For 0.3254 E2 + 0.5462 E5:
  • Convert 0.3254 E2 to 0.0003 E5, then 0.0003 E5 + 0.5462 E5 = 0.5465 E5

Subtraction of Floating Point Numbers

• To subtract floating-point numbers, exponents should be equal (adjust to the larger of the two).
• The subtraction should be converted into standard floating point form.
• Subtract 0.7254 E5 from 0.7288 E5: 0.7288 E5 - 0.7254 E5 = 0.0034 E5 = 0.3400 E3.

Multiplication of Floating Point Numbers

• Multiply the mantissa values and add the exponents.
• The result should be expressed in standard floating-point format.
• 0.6543 E5 multiplied by 0.2253 E3 gives (0.6543 × 0.2253) E(5 + 3) = 0.1474 E8.

Division of Floating Point Numbers

• Divide the mantissa of the first number by the mantissa of the second.
• The exponent of the second number is subtracted from the exponent of the first number.
• The quotient is expressed in standard form.
• 0.8888 E5 divided by 0.2000 E3 becomes (0.8888 ÷ 0.2000) E(5 - 3) = 4.4444 E2 = 0.4444 E3.

Absolute Error (EA)

• Absolute error represents the positive difference between the actual and approximate values of an observation.
• Formula: EA = |Actual Value - Approximated Value| or EA = |x - ​​xa| when x stands for true and xa stands for approximate.

Relative Error (ER)

• Relative error is the ratio of the error to the actual value of a variable.
• Formula: ER = (x - xa) / x, where x is the actual value and xa the approximated value.