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FAQ: Difference between revisions
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→Why is this floating point computation wrong?
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Floating point arithmetic is an approximation '''in binary''' to arithmetic on real or complex numbers. Just like you cannot represent 1/3 exactly in decimal arithmetic (0.333333 is only a rough approximation to 1/3), you cannot represent some fractions like <math>1/10</math> exactly in base 2. In binary, the representation to one tenth is <math>0.0\overline{0011}_b</math> where the bar indicates that it repeats infinitely (like how <math>1/6 = 0.1\overline{6}_d</math> in decimal). Because this infinite repetition cannot be represented exactly with a finite number of digits, rounding errors occur for values that appear to be exact in decimal but are in fact approximations in binary, such as for example how 0.3 - 0.2 - 0.1 is not equal to zero. | Floating point arithmetic is an approximation '''in binary''' to arithmetic on real or complex numbers. Just like you cannot represent 1/3 exactly in decimal arithmetic (0.333333 is only a rough approximation to 1/3), you cannot represent some fractions like <math>1/10</math> exactly in base 2. In binary, the representation to one tenth is <math>0.0\overline{0011}_b</math> where the bar indicates that it repeats infinitely (like how <math>1/6 = 0.1\overline{6}_d</math> in decimal). Because this infinite repetition cannot be represented exactly with a finite number of digits, rounding errors occur for values that appear to be exact in decimal but are in fact approximations in binary, such as for example how 0.3 - 0.2 - 0.1 is not equal to zero. | ||
This isn't an Octave bug. It happens with any program that uses [http://en.wikipedia.org/wiki/IEEE_754 IEEE 754 floating point arithmetic]. The reason why Octave and other programs use IEEE 754 floats is that they are ''fast'', because they are implemented in hardware. Unless you are using very exotic hardware, Octave will use your computer's processor for floating point arithmetic. | In addition, some advanced operations are computed by approximation and are not guaranteed to be accurate, see [https://en.wikipedia.org/wiki/Rounding#Table-maker.27s_dilemma Table-maker's dilemma]. Their results are system-dependent. | ||
This isn't an Octave bug. It happens with any program that uses [http://en.wikipedia.org/wiki/IEEE_754 IEEE 754 floating point arithmetic]. To be fair, IEEE 754 also specifies decimal floating point arithmetic, which has never seen wide adoption. The reason why Octave and other programs use IEEE 754 binary floats is that they are ''fast'', because they are implemented in hardware or system libraries. Unless you are using very exotic hardware, Octave will use your computer's processor for basic floating point arithmetic. | |||
Like death and taxes, rounding errors are a fact of life. You cannot avoid them. You can only move a rounding error from one part of a computation to another, or you can use more precision and delay the rounding error. One way to delay the rounding error is to use arbitrary precision arithmetic, which is inevitably slower as it has to be implemented in software instead of hardware. You may use the vpa function from the symbolic package for this purpose. | Like death and taxes, rounding errors are a fact of life. You cannot avoid them. You can only move a rounding error from one part of a computation to another, or you can use more precision and delay the rounding error. One way to delay the rounding error is to use arbitrary precision arithmetic, which is inevitably slower as it has to be implemented in software instead of hardware. You may use the vpa function from the symbolic package for this purpose. | ||