sbv-7.0: SMT Based Verification: Symbolic Haskell theorem prover using SMT solving.

Data.SBV.Examples.Misc.Floating

Description

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

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

>>> prove $assocPlus (0/0) Falsifiable. Counter-example: s0 = 0.0 :: Float s1 = 0.0 :: Float  Indeed: >>> let i = 0/0 :: Float >>> i + (0.0 + 0.0) NaN >>> ((i + 0.0) + 0.0) NaN  But keep in mind that NaN does not equal itself in the floating point world! We have: >>> let nan = 0/0 :: Float in nan == nan False  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: >>> assocPlusRegular Falsifiable. Counter-example: x = 1.9259302e-34 :: Float y = -1.9259117e-34 :: Float z = -1.814176e-39 :: Float  Indeed, we have: >>> ((1.9259302e-34) + ((-1.9259117e-34) + (-1.814176e-39))) :: Float 3.4438e-41 >>> (((1.9259302e-34) + ((-1.9259117e-34))) + (-1.814176e-39)) :: Float 3.4014e-41  Note the difference between two additions! FP addition by non-zero can result in no change 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: >>> nonZeroAddition Falsifiable. Counter-example: a = 2.424457e-38 :: Float b = -1.0e-45 :: Float  Indeed, we have: >>> (2.424457e-38 + (-1.0e-45)) == (2.424457e-38 :: Float) True  But: >>> -1.0e-45 == (0 :: Float) False  FP multiplicative inverses may not exist This example illustrates that a * (1/a) does not necessarily equal 1. Again, we protect against division by 0 and NaN/Infinity. We have: >>> multInverse Falsifiable. Counter-example: a = 1.119056263978578e-308 :: Double  Indeed, we have: >>> let a = 1.119056263978578e-308 :: Double >>> a * (1/a) 0.9999999999999999  Effect of rounding modes 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. (Also see the class RoundingFloat.) This example illustrates how SBV can be used to find rounding-modes where, for instance, addition can produce different results. We have: >>> roundingAdd Satisfiable. Model: rm = RoundTowardPositive :: RoundingMode x = -256.0 :: Float y = 4.6475088e-10 :: Float  (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: >>> (-256.0 + 4.6475088e-10) :: Float -256.0  While we cannot directly see the result when the mode is RoundTowardPositive in Haskell, we can use SBV to provide us with that result thusly: >>> sat$ \z -> z .== fpAdd sRoundTowardPositive (-256.0) (4.6475088e-10 :: SFloat)
Satisfiable. Model:
s0 = -255.99998 :: Float


We can see why these two resuls are indeed different: The RoundTowardsPositive (which rounds towards positive-infinity) produces a larger result. Indeed, if we treat these numbers as Double values, we get:

>>> (-256.0 + 4.6475088e-10) :: Double
-255.99999999953525


we see that the "more precise" result is larger than what the Float value is, justifying the larger value with RoundTowardPositive. 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 https://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/02Numerics/Double/paper.pdf as an excellent guide.