Stability | experimental |
---|---|

Maintainer | erkokl@gmail.com |

Safe Haskell | None |

Several examples involving IEEE-754 floating point numbers, i.e., single
precision `Float`

(`SFloat`

) and double precision `Double`

(`SDouble`

) types.

Note that arithmetic with floating point is full of surprises; due to precision
issues associativity of arithmetic operations typically do not hold. Also,
the presence of `NaN`

is always something to look out for.

- assocPlus :: SFloat -> SFloat -> SFloat -> SBool
- assocPlusRegular :: IO ThmResult
- nonZeroAddition :: IO ThmResult
- multInverse :: IO ThmResult

# FP addition is not associative

assocPlus :: SFloat -> SFloat -> SFloat -> SBoolSource

Prove that floating point addition is not associative. We have:

`>>>`

Falsifiable. Counter-example: s0 = -Infinity :: SFloat s1 = Infinity :: SFloat s2 = -9.403955e-38 :: SFloat`prove assocPlus`

Indeed:

`>>>`

`let i = 1/0 :: Float`

`>>>`

NaN`((-i) + i) + (-9.403955e-38) :: Float`

`>>>`

NaN`(-i) + (i + (-9.403955e-38)) :: Float`

But keep in mind that `NaN`

does not equal itself in the floating point world! We have:

`>>>`

False`let nan = 0/0 :: Float in nan == nan`

assocPlusRegular :: IO ThmResultSource

Prove that addition is not associative, even if we ignore `NaN`

/`Infinity`

values.
To do this, we use the predicate `isFPPoint`

, which is true of a floating point
number (`SFloat`

or `SDouble`

) if it is neither `NaN`

nor `Infinity`

. (That is, it's a
representable point in the real-number line.)

We have:

`>>>`

Falsifiable. Counter-example: x = 1.5775295e-30 :: SFloat y = 1.92593e-34 :: SFloat z = -2.1521e-41 :: SFloat`assocPlusRegular`

Indeed, we have:

`>>>`

1.5777222e-30`(1.5775295e-30 + 1.92593e-34) + (-2.1521e-41) :: Float`

`>>>`

1.577722e-30`1.5775295e-30 + (1.92593e-34 + (-2.1521e-41)) :: Float`

Note the loss of precision in the second expression.

# FP addition by non-zero can result in no change

nonZeroAddition :: IO ThmResultSource

Demonstrate that `a+b = a`

does not necessarily mean `b`

is `0`

in the floating point world,
even when we disallow the obvious solution when `a`

and `b`

are `Infinity.`

We have:

`>>>`

Falsifiable. Counter-example: a = -4.0 :: SFloat b = 4.5918e-41 :: SFloat`nonZeroAddition`

Indeed, we have:

`>>>`

True`-4.0 + 4.5918e-41 == (-4.0 :: Float)`

But:

`>>>`

False`4.5918e-41 == (0 :: Float)`

# FP multiplicative inverses may not exist

multInverse :: IO ThmResultSource

The last example illustrates that `a * (1/a)`

does not necessarily equal `1`

. Again,
we protect against division by `0`

and `NaN`

/`Infinity`

.

We have:

`>>>`

Falsifiable. Counter-example: a = 1.3625818045773776e-308 :: SDouble`multInverse`

Indeed, we have:

`>>>`

`let a = 1.3625818045773776e-308 :: Double`

`>>>`

0.9999999999999999`a * (1/a)`