-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathFloating Point Arithmetic.py
307 lines (233 loc) · 9.65 KB
/
Floating Point Arithmetic.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
#!/usr/bin/env python
# coding: utf-8
# Before you turn this problem in, make sure everything runs as expected. First, **restart the kernel** (in the menubar, select Kernel$\rightarrow$Restart) and then **run all cells** (in the menubar, select Cell$\rightarrow$Run All).
#
# Make sure you fill in any place that says `YOUR CODE HERE` or "YOUR ANSWER HERE", as well as your name and collaborators below:
# In[21]:
NAME = "Voyager0001"
COLLABORATORS = ""
# ---
# # Floating Point Representation
#
# ## 8 bit Floating Point Representation
#
# We will implement an 8 bit Floating point number system in this part, where First bit is a sign bit, next three bits are exponents,
# and the rest 4 bits are mantissas (significands).
#
# Numbers are divided into two categories: normals and subnormals.
#
# For normals the encoding is:
# \begin{equation}
# (-1)^{sign} \times 2^{(e-3)} \times 1.fraction
# \end{equation}
# For subnormals the encoding is:
# \begin{equation}
# (-1)^{sign} \times 2^{(1-3)} \times 0.fraction
# \end{equation}
#
# Example:
# \begin{aligned}
# 0 001 0000 &= 2^{(1-3)} \times 1.0 = 0.25\\
# 0 010 0000 &= 2^{(2-3)} \times 1.0 = 0.5\\
# 0 011 0000 &= 2^{(3-3)} \times 1.0 = 1.0\\
# 0 011 0100 &= 2^{(3-3)} \times 1.25 = 1.25
# \end{aligned}
#
# If the exponent is 0 then the number is considered subnormal.
#
# Example:
# \begin{aligned}
# 0 000 0000 &= 2^{-2} \times 0 = 0.0\\
# 0 000 1000 &= 2^{-2} \times 0.5 = 0.125\\
# 0 000 1100 &= 2^{-2} \times 0.75 = 0.1875
# \end{aligned}
#
# If the exponent is 7 then and mantissa is zero then the number is
# considered infinity (inf), otherwise it is not-a-number (nan).
#
# Example:
# \begin{aligned}
# 0 111 0000 &= +\infty\\
# 1 111 0000 &= -\infty\\
# X 111 XXXX &= NaN
# \end{aligned}
# In[22]:
import math
import itertools
import matplotlib.pyplot as plt
import numpy as np
# In[23]:
class Float8():
def __init__(self, bitstring):
'''Constructor
takes a 8-bit string of 0's and 1's as input and stores the sub-strings
accordingly.
Usage: Float8('00011110')
'''
# Make sure the input consists of exactly 8-bits.
assert(len(bitstring)==8)
# Assign the sign bit
# self.sign = bitstring[?]
# Assign the exponent part
# self.exponent = bitstring[?]
# Assign the mantissa
# self.mantissa = bitstring[?]
# YOUR CODE HERE
self.sign=bitstring[0]
self.exponent=bitstring[1:4]
self.mantissa=bitstring[4:8]
self.val = self.calculate_value()
def __str__(self):
return f'Sign bit value: {self.sign}\n' + f'Exponent value: {self.exponent}\n' + f'Mantissa value: {self.mantissa}\n' + f'Floating value: {self.val}\n'
def tobitstring(self):
return self.sign + self.exponent + self.mantissa
def toformattedstring(self):
return ' '.join([self.sign, self.exponent, self.mantissa])
def calculate_value(self):
'''Calculate the value of the number from bits'''
# Initialize with zero
val = 0.0
# YOUR CODE HERE
if int(self.exponent,2)==7 and int(self.mantissa,2)==0:
if self.sign=="0":
val="inf"
elif self.sign=="1":
val="-inf"
elif int(self.exponent,2)==7 and int(self.mantissa,2)!=0:
val="nan"
elif int(self.exponent,2)!=0:
val=((-1)**(int(self.sign)))*(2**(int(self.exponent,2)-3))*float((1+(int(mantissa, 2) / (2.**len(mantissa)))))
else:
val=((-1)**(int(self.sign)))*(2**-2)*float((0+(int(mantissa, 2) / (2.**len(mantissa)))))
return val
# In[24]:
# 28 test cases for 8-bit floating points
data = [ '00000000', '00000001', '00001001', '00010000',
'00010001', '00011000', '00011011', '00100000',
'00101101', '00110000', '00110101', '01000011',
'01000000', '01010000', '01011100', '01100000',
'01110111', '01110000', '10000000', '10000001',
'11110001', '11110000', '10110001', '10111101',
'11100000', '11101011', '11010000', '11000000']
result = ['(0, 1)', '(1, 64)', '(9, 64)', '(1, 4)', '(17, 64)', '(3, 8)', '(27, 64)',
'(1, 2)', '(29, 32)', '(1, 1)', '(21, 16)', '(19, 8)', '(2, 1)', '(4, 1)',
'(7, 1)', '(8, 1)', 'nan', 'inf', '(0, 1)', '(-1, 64)', 'nan', '-inf',
'(-17, 16)', '(-29, 16)', '(-8, 1)', '(-27, 2)', '(-4, 1)', '(-2, 1)']
test = [Float8(x).val for x in data]
for index in range(len(test)):
d = test[index]
try:
test[index] = str(d.as_integer_ratio())
except Exception:
test[index] = str(d)
np.testing.assert_array_equal(test, result)
# ## 16 Bit Floating Point Representation
#
# Over here, we will implement a 16 bit Floating point number system,
# where the first bit is a sign bit, next four bits are exponents,
# and the rest 11 bits are mantissas (significands).
#
# Numbers are divided into two categories: normals and subnormals.
#
# For normals the encoding is:
# \begin{equation}
# (-1)^{sign} \times 2^{(e-7)} \times 1.fraction
# \end{equation}
# For subnormals the encoding is:
# \begin{equation}
# (-1)^{sign} \times 2^{(1-7)} \times 0.fraction
# \end{equation}
#
# Example:
# \begin{aligned}
# 0 0101 00000000000 &= 2^{(5-7)} \times 1.0 = 0.25\\
# 0 0110 00000000000 &= 2^{(6-7)} \times 1.0 = 0.5\\
# 0 0111 00000000000 &= 2^{(3-7)} \times 1.0 = 1.0\\
# 0 0111 01000000000 &= 2^{(3-7)} \times 1.25 = 1.25
# \end{aligned}
#
# If the exponent is 0 then the number is considered subnormal.
#
# Example:
# \begin{aligned}
# 0 0000 00000000000 &= 2^{-6} \times 0 = +0.0\\
# 0 0000 10000000000 &= 2^{-6} \times 0.5 = +0.0078125\\
# 0 0000 11000000000 &= 2^{-6} \times 0.75 = +0.01171875
# \end{aligned}
#
# If the exponent is 15 (all 1's) then and mantissa is zero then the number is considered infinity (inf), otherwise it is not-a-number (nan).
#
# Example:
# \begin{aligned}
# 0 1111 00000000000 &= +\infty\\
# 1 1111 00000000000 &= -\infty\\
# X 1111 XXXXXXXXXXX &= NaN
# \end{aligned}
#
# ### Instructions
# In this task, you need to implement the `Float16()` class. Some parts of the class are already written for you. You only need to modify `__init__(self, bitstring)` and `calculate_value(self)` fuctions, where it says `# YOUR CODE HERE`. You may ignore `raise NotImplementedError()`.
# In[25]:
class Float16():
def __init__(self, bitstring):
'''Constructor
takes a 16-bit string of 0's and 1's as input and stores the sub-strings
accordingly.
Usage: Float16('0001111000011110')
'''
# Make sure the input consists of exactly 16-bits.
assert(len(bitstring)==16)
# Assign the sign bit
# self.sign = bitstring[?]
# Assign the exponent part
# self.exponent = bitstring[?]
# Assign the mantissa
# self.mantissa = bitstring[?]
# YOUR CODE HERE
self.sign=bitstring[0]
self.exponent=bitstring[1:5]
self.mantissa=bitstring[5:16]
self.val = self.calculate_value()
def __str__(self):
return f'Sign bit value: {self.sign}\n' + f'Exponent value: {self.exponent}\n' + f'Mantissa value: {self.mantissa}\n' + f'Floating value: {self.val}\n'
def tobitstring(self):
return self.sign + self.exponent + self.mantissa
def toformattedstring(self):
return ' '.join([self.sign, self.exponent, self.mantissa])
def calculate_value(self):
'''Calculate the value of the number from bits'''
# Initialize with zero
val = 0.0
# YOUR CODE HERE
if int(self.exponent,2)==15 and int(self.mantissa,2)==0:
if self.sign=="0":
val="inf"
elif self.sign=="1":
val="-inf"
elif int(self.exponent,2)==15 and int(self.mantissa,2)!=0:
val="nan"
elif int(self.exponent,2)!=0:
val=((-1)**(int(self.sign)))*(2**(int(self.exponent,2)-7))*float((1+(int(mantissa, 2) / (2.**(len(mantissa)))))
else:
val=((-1)**(int(self.sign)))*(2**-6)*float((0+(int(mantissa, 2) / (2.**len(mantissa)))))
return val
# In[26]:
# 28 test cases for 16-bit floating points
data = [ '0011100000000010', '0100000000000000', '1100000000000000', '0100010000000000',
'1100010000000000', '0100100000000000', '1100100000000000', '0100101000000000',
'1100101000000000', '0100110000000000', '1100110000000000', '0101101110000000',
'0010010000000000', '0000000000000001', '0000011111111111', '0000100000000000',
'0111011111111111', '0000000000000000', '1000000000000000', '0111100000000000',
'1111100000000000', '0111100000000001', '0111110000000001', '0111111111111111',
'0010101010101011', '0100010010010001', '0011100000000000', '0011100000000001']
result = ['(1025, 1024)', '(2, 1)', '(-2, 1)', '(3, 1)', '(-3, 1)', '(4, 1)', '(-4, 1)',
'(5, 1)', '(-5, 1)', '(6, 1)', '(-6, 1)', '(23, 1)', '(3, 16)', '(1, 131072)',
'(2047, 131072)', '(1, 64)', '(4095, 16)', '(0, 1)', '(0, 1)', 'inf', '-inf',
'nan', 'nan', 'nan', '(2731, 8192)', '(3217, 1024)', '(1, 1)', '(2049, 2048)']
test = [Float16(x).val for x in data]
for index in range(len(test)):
d = test[index]
try:
test[index] = str(d.as_integer_ratio())
except Exception:
test[index] = str(d)
np.testing.assert_array_equal(test, result)