This document discusses numerical inaccuracies that can occur when representing real numbers in computers using floating point types. It then describes some ways to avoid errors, such as using double precision instead of float. The document also covers arithmetic overflow and underflow issues. Furthermore, it introduces common mathematical functions in C like trigonometric, logarithmic and exponential functions. It provides examples of using the printf() and scanf() functions to output and input data in C programs.

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Lecture 5
Mathematical Errors and Functions
input / output functions (scanf, printf)
CS106
Introduction to Programming
Methodology & Abstractions
Spring 20023
Balochistan University
of I.T & M.S
Faculty of System Sciences
Sadique Ahmed Bugti

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Numerical Inaccuracies
• One of the problems with floating point types is
that sometimes errors occur in representing real
numbers.
• In the decimal number system, the only numbers
which can be represented exactly are those
numbers whose denominators contain only
factors of two (2) and five (5);
e.g. 1/2 = 0.5, 1/5 = 0.2, 7/10 = 0.7, 9/20 = 0.45,
3/25 = 0.12, etc.

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Avoiding Round-Off Errors
• Use double instead of float.
– This solution provides approximately twice the
accuracy, as well as a greater range

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Arithmetic Underflow and Overflow
• Arithmetic underflow occurs when a very small
computational result is represented as zero
– e.g. 0.00007 X 0.000005 = 0
• Arithmetic overflow is an error which results
from an attempt to represent a computational
result that is too large
– e.g. 4000 X 4000 (int) = 16,000,000
– 32,767 is the max for type int!
• Results are compiler / machine / OS dependant.

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Math Functions
The C math library has the header file math.h, and
contains the following useful functions:
acos(d) arc cosine of d (in range 0 to pi)
asin(d) arc sine of d (in range -pi/2 to pi/2)
atan(d) arc tangent of d (in range -pi/2 to pi/2)
atan2(d1,d2) arc tangent of d1/d2 (in range -pi to pi)
cbrt(d) cube root of d
cos(d) cosine of d
cosh(d) hyperbolic cosine of d
exp(d) exponential of d
fabs(d) absolute value of d
hypot(d1,d2) sqrt(d1 * d1 + d2 * d2)
log(d) natural logarithm of d

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Math Functions
log10(d) logarithm (base 10) of d
pow(d1,d2) 1 raised to the power d2
sin(d) sine of d
sinh(d) hyperbolic sine of d
sqrt(d) square root of d
tan(d) tangent of d
tanh(d) hyperbolic tangent of d

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Math Functions
• A program that makes use of the C math library
would contain the statement
#include <math.h>
close to its start. In the body of the program, a
statement like
x = cos(y);
would cause the variable x to be assigned a
value which is the cosine of the value of the
variable y (both x and y should be of type
double).

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Math Functions
• There is no built-in exponentiation operator in C:
– Normally represented by x^y, which implies x
raised to the power of y.
– Instead, there is a library function pow which
carries out this operation:
z = pow(x,y);
would cause the variable z to be assigned a value
which is x raised to the power of y.
Note: x2 is best represented as x * x

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Math Constants
• The C math library comes with a useful set of
predefined mathematical constants:
M_PI Pi
M_PI_2 Pi divided by two
M_PI_4 Pi divided by four
M_1_PI The reciprocal of pi (1/pi)
M_SQRT2 The square root of two
M_SQRT1_2 The reciprocal of the square root of two
M_E The base of natural logarithms

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The Output Function: printf()
• The function printf (print-eff) is used for
writing data to standard output (usually, the
terminal).
• A call to this function takes the general form:
printf(format_string, arg1, arg2, …)
– where format_string refers to a character
string containing certain required formatting
information, and arg1, arg2, etc., are
arguments that represent the individual input
data items.

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The Output Function: printf()
• The format string consists of individual
groups of characters, with one character
group for each data input item.

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Conversion Specifications
• The percent sign (%) symbol introduces a
conversion specification, or format. A
single conversion specification is a string
that begins with % and ends with a
conversion character.
• Explicit formatting may be included in a
conversion specification.
– Otherwise defaults are used.

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Conversion Characters
Conversion
character
How the corresponding argument is printed
c as a character
d, i as an integer
u as an unsigned integer
e as a floating-point number, e.g. 3.141590e+00
E as a floating-point number, e.g. 3.141590E+00
f as a floating-point number, e.g. 3.141590
s as a string
% with the format %% a single % is written to
the output stream.

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printf() Examples: Characters
& Strings
char c=‘M’, s[]=“blue moon”;

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printf() Examples: Numbers
int i = 123;
double x = 32.17865753173828125

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The Input Function: scanf()
• The function scanf (scan-eff) is used for reading
data from standard input (usually the terminal).
• A call to this function takes the general form:
scanf(format_string, &arg1, &arg2, …)
– where format_string refers to a character string
containing certain required formatting
information, and arg1, arg2, etc., are arguments
that represent the individual input data items.

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The Input Function: scanf()
• The format string consists of individual groups of
characters, with one character group for each
data input item.
• In its simplest form, each character group
consists of a percent sign (%), followed by a set
of conversion characters which indicate the type
of the corresponding data item.
• The arguments are a comma-separated list of
addresses.

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The End