Floating-Point Arithmetic

Performing an arithmetic operation $\Rightarrow$ no exact answers unless only integers or exact powers of 2 are involved,
Floating-point (real numbers) $\rightarrow$ not integers,
Resembles scientific notation,
IEEE standard $\rightarrow$ storing floating-point numbers (see the Table 2.1).

Table 2.1: Floating $\rightarrow$ Normalised.
floating normalised (shifting the decimal point)

13.524

-0.0442
the sign $\pm$
the fraction part (called the mantissa)
the exponent part
What about the sign of the exponent? Rather than use one of the bits for the sign of the exponent, exponents are biased.
For single precision (we have 8 bits reserved for the exponent):
- =256
- 0 $\longrightarrow$ 00000000 = 0
- 255 $\longrightarrow$ 11111111=255
- 0 (255) $\Longrightarrow$ -127 (128). An exponent of -127 (128) stored as 0 (255).
- So biased $\longrightarrow 2^{128}=3.40282E+38$ , mantissa gets 1 as maximum
- Largest: 3.40282E+38; Smallest: 5.87747E-39 (!)
- For double and extended precision the bias values are 1023 and 16383, respectively.
- $\frac{0}{0},0*\infty,\sqrt{-1}\Longrightarrow NaN$ : Undefined.

**Table 2.1:** Floating $\rightarrow$ Normalised.
floating	normalised (shifting the decimal point)
13.524
-0.0442

There are three levels of precision (see the Fig. 2.4)

**Figure 2.4:** Level of precision.

week3_HandsOn.py

    1 import sys
    2 print(sys.float_info)
    3 # sys.float_info(max=1.7976931348623157e+308, max_exp=1024, max_10_exp=308, min=2.2250738585072014e-308, min_exp=-1021,
    4 # min_10_exp=-307, dig=15, mant_dig=53, epsilon=2.220446049250313e-16, radix=2, rounds=1)
    5 print(sys.float_info.max)
    6 # 1.7976931348623157e+308
    7 print("%10.6e " % 2**128)
    8 # 3.402824e+38
    9 print("%10.6e " % 2**1023)
   10 # 8.988466e+307f

EPS: short for epsilon $\longrightarrow$ used for represent the smallest machine value that can be added to 1.0 that gives a result distinguishable from 1.0! $eps\longrightarrow\varepsilon\Longrightarrow(1+\varepsilon)+\varepsilon=1~but~1+(\varepsilon+\varepsilon)>1$ Two numbers that are very close together on the real number line can not be distinguished on the floating-point number line if their difference is less than the least significant bit of their mantissas.

    1 import sys
    2 print(sys.float_info.epsilon)
    3 # 2.220446049250313e-16
    4 eps=sys.float_info.epsilon
    5 print(1+eps*0.5)
    6 # 1.0
    7 print(1+eps*0.5)
    8 # 1.0
    9 print((1+eps*0.5)+eps*0.5)
   10 # 1.0
   11 print(1+eps*0.6)
   12 # 1.0000000000000002