1 ====================================== 2 Python IEEE 754 floating point support 3 ====================================== 4 5 >>> from sys import float_info as FI 6 >>> from math import * 7 >>> PI = pi 8 >>> E = e 9 10 You must never compare two floats with == because you are not going to get 11 what you expect. We treat two floats as equal if the difference between them 12 is small than epsilon. 13 >>> EPS = 1E-15 14 >>> def equal(x, y): 15 ... """Almost equal helper for floats""" 16 ... return abs(x - y) < EPS 17 18 19 NaNs and INFs 20 ============= 21 22 In Python 2.6 and newer NaNs (not a number) and infinity can be constructed 23 from the strings 'inf' and 'nan'. 24 25 >>> INF = float('inf') 26 >>> NINF = float('-inf') 27 >>> NAN = float('nan') 28 29 >>> INF 30 inf 31 >>> NINF 32 -inf 33 >>> NAN 34 nan 35 36 The math module's ``isnan`` and ``isinf`` functions can be used to detect INF 37 and NAN: 38 >>> isinf(INF), isinf(NINF), isnan(NAN) 39 (True, True, True) 40 >>> INF == -NINF 41 True 42 43 Infinity 44 -------- 45 46 Ambiguous operations like ``0 * inf`` or ``inf - inf`` result in NaN. 47 >>> INF * 0 48 nan 49 >>> INF - INF 50 nan 51 >>> INF / INF 52 nan 53 54 However unambigous operations with inf return inf: 55 >>> INF * INF 56 inf 57 >>> 1.5 * INF 58 inf 59 >>> 0.5 * INF 60 inf 61 >>> INF / 1000 62 inf 63 64 Not a Number 65 ------------ 66 67 NaNs are never equal to another number, even itself 68 >>> NAN == NAN 69 False 70 >>> NAN < 0 71 False 72 >>> NAN >= 0 73 False 74 75 All operations involving a NaN return a NaN except for nan**0 and 1**nan. 76 >>> 1 + NAN 77 nan 78 >>> 1 * NAN 79 nan 80 >>> 0 * NAN 81 nan 82 >>> 1 ** NAN 83 1.0 84 >>> NAN ** 0 85 1.0 86 >>> 0 ** NAN 87 nan 88 >>> (1.0 + FI.epsilon) * NAN 89 nan 90 91 Misc Functions 92 ============== 93 94 The power of 1 raised to x is always 1.0, even for special values like 0, 95 infinity and NaN. 96 97 >>> pow(1, 0) 98 1.0 99 >>> pow(1, INF) 100 1.0 101 >>> pow(1, -INF) 102 1.0 103 >>> pow(1, NAN) 104 1.0 105 106 The power of 0 raised to x is defined as 0, if x is positive. Negative 107 values are a domain error or zero division error and NaN result in a 108 silent NaN. 109 110 >>> pow(0, 0) 111 1.0 112 >>> pow(0, INF) 113 0.0 114 >>> pow(0, -INF) 115 Traceback (most recent call last): 116 ... 117 ValueError: math domain error 118 >>> 0 ** -1 119 Traceback (most recent call last): 120 ... 121 ZeroDivisionError: 0.0 cannot be raised to a negative power 122 >>> pow(0, NAN) 123 nan 124 125 126 Trigonometric Functions 127 ======================= 128 129 >>> sin(INF) 130 Traceback (most recent call last): 131 ... 132 ValueError: math domain error 133 >>> sin(NINF) 134 Traceback (most recent call last): 135 ... 136 ValueError: math domain error 137 >>> sin(NAN) 138 nan 139 >>> cos(INF) 140 Traceback (most recent call last): 141 ... 142 ValueError: math domain error 143 >>> cos(NINF) 144 Traceback (most recent call last): 145 ... 146 ValueError: math domain error 147 >>> cos(NAN) 148 nan 149 >>> tan(INF) 150 Traceback (most recent call last): 151 ... 152 ValueError: math domain error 153 >>> tan(NINF) 154 Traceback (most recent call last): 155 ... 156 ValueError: math domain error 157 >>> tan(NAN) 158 nan 159 160 Neither pi nor tan are exact, but you can assume that tan(pi/2) is a large value 161 and tan(pi) is a very small value: 162 >>> tan(PI/2) > 1E10 163 True 164 >>> -tan(-PI/2) > 1E10 165 True 166 >>> tan(PI) < 1E-15 167 True 168 169 >>> asin(NAN), acos(NAN), atan(NAN) 170 (nan, nan, nan) 171 >>> asin(INF), asin(NINF) 172 Traceback (most recent call last): 173 ... 174 ValueError: math domain error 175 >>> acos(INF), acos(NINF) 176 Traceback (most recent call last): 177 ... 178 ValueError: math domain error 179 >>> equal(atan(INF), PI/2), equal(atan(NINF), -PI/2) 180 (True, True) 181 182 183 Hyberbolic Functions 184 ==================== 185 186
