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The inexactness is not a problem speci c to Python all programming languages have this problem with oating-point numbers Python 31 produces much more sensible-looking output:
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>>> 00, 54, -25, 89e-4 (00, 54, -25, 000089)
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When Python 31 outputs a oating-point number, in most cases it uses David Gay s algorithm This outputs the fewest possible digits without losing any accuracy Although this produces nicer output, it doesn t change the fact that computers (no matter what computer language is used) effectively store oating-point numbers as approximations If we need really high precision there are two approaches we can take One approach is to use ints for example, working in terms of pennies or tenths of a penny or similar and scale the numbers when necessary This requires us to be quite careful, especially when dividing or taking percentages The other approach is to use Python s decimalDecimal numbers from the decimal module These perform calculations that are accurate to the level of precision we specify (by default, to 28 decimal places) and can represent periodic numbers like 01 exactly; but processing is a lot slower than with floats Because of their accuracy, decimalDecimal numbers are suitable for nancial calculations Mixed mode arithmetic is supported such that using an int and a float produces a float, and using a float and a complex produces a complex Because decimalDecimals are of xed precision they can be used only with other decimal Decimals and with ints, in the latter case producing a decimalDecimal result If an operation is attempted using incompatible types, a TypeError exception is raised
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All the numeric operators and functions in Table 22 (55 ) can be used with floats, including the augmented assignment versions The float data type can be called as a function with no arguments it returns 00, with a float argument it returns a copy of the argument, and with any other argument it attempts to convert the given object to a float When used for conversions a string argument can be given, either using simple decimal notation or using exponential notation It is possible that NaN ( not a number ) or in nity may be produced by a calculation involving floats unfortunately the behavior is not consistent across implementations and may differ depending on the system s underlying math library Here is a simple function for comparing floats for equality to the limit of the machine s accuracy:
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Table 25 The Math Module s Functions and Constants #1
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Syntax mathacos(x) mathacosh(x) mathasin(x) mathasinh(x) mathatan(x) mathatan2(y, x) mathatanh(x) mathceil(x) mathcopysign(x,y) mathcos(x) mathcosh(x) mathdegrees(r) mathe mathexp(x) mathfabs(x) mathfactorial(x) mathfloor(x) mathfmod(x, y)
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Returns the arc cosine of x in radians Returns the arc hyperbolic cosine of x in radians Returns the arc sine of x in radians Returns the arc hyperbolic sine of x in radians Returns the arc tangent of x in radians Returns the arc tangent of y / x in radians Returns the arc hyperbolic tangent of x in radians Returns x , ie, the smallest integer greater than or equal to x as an int; eg, mathceil(54) == 6 Returns x with y s sign Returns the cosine of x in radians Returns the hyperbolic cosine of x in radians Converts float r from radians to degrees The constant e; approximately 2718 281 828 459 045 1 Returns ex, ie, mathe ** x Returns | x |, ie, the absolute value of x as a float Returns x! Returns x , ie, the largest integer less than or equal to x as an int; eg, mathfloor(54) == 5 Produces the modulus (remainder) of dividing x by y; this produces better results than % for floats Returns a 2-tuple with the mantissa (as a float) and e the exponent (as an int) so, x = m 2 ; see mathldexp() Returns the sum of the values in iterable i as a float Returns x2 + y2 Returns True if float x is inf ( ) Returns True if float x is nan ( not a number ) Returns m 2 ; effectively the inverse of mathfrexp() Returns logb x; b is optional and defaults to mathe Returns log10x Returns loge(1 + x); accurate even when x is close to 0 Returns x s fractional and whole parts as two floats
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mathfrexp(x) mathfsum(i) mathhypot(x, y) mathisinf(x) mathisnan(x) mathldexp(m, e) mathlog(x, b) mathlog10(x) mathlog1p(x) mathmodf(x)
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