Copyright | (c) Levent Erkok |
---|---|

License | BSD3 |

Maintainer | erkokl@gmail.com |

Stability | experimental |

Safe Haskell | None |

Language | Haskell2010 |

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.

## Synopsis

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

# FP addition is not associative

assocPlus :: SFloat -> SFloat -> SFloat -> SBool Source #

Prove that floating point addition is not associative. For illustration purposes,
we will require one of the inputs to be a `NaN`

. We have:

`>>>`

Falsifiable. Counter-example: s0 = 0.0 :: Float s1 = 0.0 :: Float`prove $ assocPlus (0/0)`

Indeed:

`>>>`

`let i = 0/0 :: Float`

`>>>`

NaN`i + (0.0 + 0.0)`

`>>>`

NaN`((i + 0.0) + 0.0)`

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 ThmResult Source #

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

/`Infinity`

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

, 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 = 3.1705284e18 :: Float y = 5.634997e12 :: Float z = -1.503316e20 :: Float`assocPlusRegular`

Indeed, we have:

`>>>`

`let x = 3.1705284e18 :: Float`

`>>>`

`let y = 5.634997e12 :: Float`

`>>>`

`let z = -1.503316e20 :: Float`

`>>>`

-1.4716107e20`x + (y + z)`

`>>>`

-1.4716106e20`(x + y) + z`

Note the difference in the last digit before the exponent!

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

nonZeroAddition :: IO ThmResult Source #

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 = -2.032879e-20 :: Float b = 4.887041e-39 :: Float`nonZeroAddition`

Indeed, we have:

`>>>`

`let a = -2.032879e-20 :: Float`

`>>>`

`let b = 4.887041e-39 :: Float`

`>>>`

True`a + b == a`

`>>>`

False`b == 0`

# FP multiplicative inverses may not exist

multInverse :: IO ThmResult Source #

This 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 = -6.72794746807321e-309 :: Double`multInverse`

Indeed, we have:

`>>>`

`let a = -6.72794746807321e-309 :: Double`

`>>>`

0.9999999999999999`a * (1/a)`

# Effect of rounding modes

roundingAdd :: IO SatResult Source #

One interesting aspect of floating-point is that the chosen rounding-mode
can effect the results of a computation if the exact result cannot be precisely
represented. SBV exports the functions `fpAdd`

, `fpSub`

, `fpMul`

, `fpDiv`

, `fpFMA`

and `fpSqrt`

which allows users to specify the IEEE supported `RoundingMode`

for
the operation. This example illustrates how SBV can be used to find rounding-modes
where, for instance, addition can produce different results. We have:

`>>>`

Satisfiable. Model: rm = RoundNearestTiesToAway :: RoundingMode x = 1.0 :: Float y = -0.43749997 :: Float`roundingAdd`

(Note that depending on your version of Z3, you might get a different result.)
Unfortunately we can't directly validate this result at the Haskell level, as Haskell only supports
`RoundNearestTiesToEven`

. We have:

`>>>`

0.5625`1.0 + (-0.43749997) :: Float`

While we cannot directly see the result when the mode is `RoundNearestTiesToAway`

in Haskell, we can use
SBV to provide us with that result thusly:

`>>>`

Satisfiable. Model: s0 = 0.56250006 :: Float`sat $ \z -> z .== fpAdd sRoundNearestTiesToAway 1.0 (-0.43749997 :: SFloat)`

We can see why these two resuls are indeed different: The `RoundNearestTiesToAway`

(which rounds away from zero) produces a larger result. Indeed, if we treat these numbers
as `Double`

values, we get:

> 1.0 + (-0.43749997 :: Double)

- 56250003

we see that the "more precise" result is larger than what the `Float`

value is, justifying the
larger value with `RoundNearestTiesToAway`

. A more detailed study is beyond our current scope, so we'll
merely note that floating point representation and semantics is indeed a thorny
subject, and point to http://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/02Numerics/Double/paper.pdf as
an excellent guide.