This article is just a simplification of the IEEE 754 standard. Here, we will see how floating-point no stored in memory, floating-point exceptions/rounding, etc. But if you will want to find more authoritative sources then go for

- What Every Computer Scientist Should Know About Floating-Point Arithmetic
- https://en.wikipedia.org/wiki/IEEE_754-1985
- https://en.wikipedia.org/wiki/Floating_point.

**Floating-point numbers stored by encoding significand & the exponent (along with a sign bit)**

- Above line contains 2-3 abstract terms & I think you will unable to understand the above line until you read further.

Contents

## Floating Point Number Memory Layout

+-+--------+-----------------------+ | | | | +-+--------+-----------------------+ ^ ^ ^ | | | | | +-- significand(width- 23 bit) | | | +------------------- exponent(width- 8 bit) | +------------------------ sign bit(width- 1 bit)

A typical single-precision 32-bit floating-point memory layout has the following fields :

- sign
- exponent
- significand(AKA mantissa)

### Sign

- The high-order bit indicates a sign.
`0`

indicates a positive value,`1`

indicates negative.

### Exponent

- The next 8 bits are used for the exponent which can be positive or negative, but instead of reserving another sign bit, they’re encoded such that
`1000 0000`

represents`0`

, so`0000 0000`

represents`-128`

and`1111 1111`

represents`127`

. - How does this encoding work? go to exponent bias or see it in next point practically.

### Significand

- The remaining 23-bits used for the significand(AKA mantissa). Each bit represents a negative power of 2 countings from the left, so:

01101 = 0 * 2^-1 + 1 * 2^-2 + 1 * 2^-3 + 0 * 2^-4 + 1 * 2^-5 = 0.25 + 0.125 + 0.03125 = 0.40625

OK! We are done with basics.

## Let’s Understand Practically

- So, we consider very famous float value
`3.14`

(PI) example. **Sign**: Zero here, as PI is positive!

### Exponent calculation

`3`

is easy:`0011`

in binary- The rest,
`0.14`

0.14 x 2 = 0.28, 0 0.28 x 2 = 0.56, 00 0.56 x 2 = 1.12, 001 0.12 x 2 = 0.24, 0010 0.24 x 2 = 0.48, 00100 0.48 x 2 = 0.96, 001000 0.96 x 2 = 1.92, 0010001 0.92 x 2 = 1.84, 00100011 0.84 x 2 = 1.68, 001000111 And so on . . .

- So,
`0.14 = 001000111...`

If you don’t know how to convert decimal no in binary then refer this float to binary. - Add
`3`

,`11.001000111... with exp 0 (3.14 * 2^0)`

- Now shift it (normalize it) and adjust the exponent accordingly
`1.1001000111... with exp +1 (1.57 * 2^1)`

- Now you only have to add the bias of
`127`

to the exponent`1`

and store it(i.e.`128`

=`1000 0000`

)`0 1000 0000 1100 1000 111...`

- Forget the top
`1`

of the mantissa (which is always supposed to be`1`

, except for some special values, so it is not stored), and you get:`0 1000 0000 1001 0001 111...`

- So our value of
`3.14`

would be represented as something like:

0 10000000 10010001111010111000011 ^ ^ ^ | | | | | +--- significand = 0.7853975 | | | +------------------- exponent = 1 | +------------------------- sign = 0 (positive)

- The number of bits in the exponent determines the range (the minimum and maximum values you can represent).

### Summing up Significand

- If you add up all the bits in the significand, they don’t total
`0.7853975`

(which should be, according to 7 digit precision). They come out to`0.78539747`

. - There aren’t quite enough bits to store the value exactly. we can only store an approximation.
- The number of bits in the significand determines the precision.
- 23-bits gives us roughly 6 decimal digits of precision. 64-bit floating-point types give roughly 12 to 15 digits of precision.

**Strange! But Fact**

- Some values cannot represent exactly no matter how many bits you use. Just as values like 1/3 cannot represent in a finite number of decimal digits, values like 1/10 cannot represent in a finite number of bits.
- Since values are approximate, calculations with them are also approximate, and rounding errors accumulate.

## Let’s See Things Working

#include <stdio.h> #include <string.h> /* Print binary stored in plain 32 bit block */ void intToBinary(unsigned int n) { int c, k; for (c = 31; c >= 0; c--) { k = n >> c; if (k & 1) printf("1"); else printf("0"); } printf("\n"); } int main(void) { unsigned int m; float f = 3.14; /* See hex representation */ printf("f = %a\n", f); /* Copy memory representation of float to plain 32 bit block */ memcpy(&m, &f, sizeof (m)); intToBinary(m); return 0; }

- This C code will print binary representation of float on the console.

f = 0x3.23d70cp+0 01000000010010001111010111000011

## Where the Decimal Point Is Stored?

- The decimal point not explicitly stored anywhere.
- As I wrote a line
`Floating-point numbers stored by encoding significand & the exponent (along with a sign bit)`

, but you don’t get it the first time. Don’t worry 99% people don’t get it first, including me.

## A Bit More About Representing Numbers

- According to
`IEEE 754-1985`

worldwide standard, you can also store zero, negative/positive infinity and even `NaN`(Not a Number). Don’t worry if you don’t know what is`NaN`

, I will explain shortly(But be worried, if you don’t know infinity).

### Zero Representation

- sign = 0 for positive zero, 1 for negative zero.
- exponent = 0.
- fraction = 0.

### Positive & Negative Infinity Representation

- sign = 0, for positive infinity, 1 for negative infinity.
- exponent = all 1 bits.
- fraction = all 0 bits.

### NaN Representation

- sign = either 0 or 1.
- exponent = all 1 bits.
- fraction = anything except all 0 bits (since all 0 bits represents infinity)

## Why Do We Need `NaN`

?

- Some operations of floating-point arithmetic are invalid, such as dividing by zero or taking the square root of a negative number.
- The act of reaching an invalid result called a floating-point exception(next point). An exceptional result is represented by a special code called a
`NaN`

, for “Not a Number”.

## Floating-Point Exceptions

- The
`IEEE 754-1985`

standard defines five exceptions that can occur during a floating-point calculation named as

**Invalid Operation**: occurs due to many causes like multiplication of infinite with zero or infinite, division of infinite by zero or infinite & vice-versa, square root of operand less than zero, etc.**Division by Zero**: occurs when “as its name sounds”**Overflow**: This exception raised whenever the result cannot represent a finite value in the precision format of the destination.**Underflow**: The underflow exception raised when an intermediate result is too small to calculate accurately, or if the operation’s result rounded to the destination precision too small to normalized**Inexact**: raised when a rounded result not exact.

## Rounding in Floating-Point

- As we saw floating-point numbers have a limited number of digits, they cannot represent all real numbers accurately: when there are more digits than the format allows, the leftover ones are omitted – the number is rounded.
- There are 4 rounding modes :

1. Round to Nearest: rounded to the nearest value with an even (zero) least significant bit, which occurs 50% of the time.

2. Round toward 0 – simply truncate the extra digits.

3. Round toward +∞ – rounding towards positive infinity.

4. Round toward −∞ – rounding towards negative infinity.

## Misc points

- In older time, embedded system processors do not use floating-point numbers as they don’t have such hardware capabilities.
- So there is some alternative to a floating-point number, called Fixed Point Numbers.
- A fixed-point number is usually used in special-purpose applications on embedded processors that can only do integer arithmetic, but decimal fixed point(‘.’) is manipulated by software library.
- But nowadays, the microcontroller has separate FPU’s too, like STM32F series.

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