02157FunctionalProgram-

ming

Michael R. Hansen02157 Functional ProgrammingProperty-based testing: Floating-point properties

Michael R. Hansen

1 DTU Compute, Technical University of Denmark Property-based testing: Floating-point properties MRH 14/11/2019

02157FunctionalProgram-

ming

Michael R. Hansen

Overview

Testing properties of ”programs” involving floating-point values (oftype float)

Many obstacles, for example:• Approximations• Functions may be undefined for some arguments• .....

An example based presentation guided by complication spotted byFsCheck.

The technicalities concerning floats are not special to F#.

Floating-point values must always be treated with care!!

2 DTU Compute, Technical University of Denmark Property-based testing: Floating-point properties MRH 14/11/2019

02157FunctionalProgram-

ming

Michael R. Hansen

Testing that addition is associative

#r ".......FsCheck.dll"open FsCheck

let assoc(x:float,y,z) = x+(y+z) = (x+y) + z;;

Check.Verbose assoc;;

(*...3:(0.0, nan, 0.0)Falsifiable, after 4 tests (0 shrinks) (StdGen (1487969817,296669693)):Original:(0.0, nan, 0.0)

*)

What is nan?

3 DTU Compute, Technical University of Denmark Property-based testing: Floating-point properties MRH 14/11/2019

02157FunctionalProgram-

ming

Michael R. Hansen

Not A Number: nan:float

The domain of log is the positive floating-point values:

log -1.0;;val it : float = nan

nan = nan;;val it : bool = false

Ups — nan is indeed a special float value

It appears in randomly generated floats:

Gen.sample 10 10 Arb.generate<float>;;val it : float list =

[-6.556494257; 4.940656458e-324; ... ; nan;5.534954001; ... ; -6.136783418; 9.039187115]

4 DTU Compute, Technical University of Denmark Property-based testing: Floating-point properties MRH 14/11/2019

02157FunctionalProgram-

ming

Michael R. Hansen

Taking care of nan:float in tests

The law of associativity is only checked for floats that are not NaNs.

let assocV1(x,y,z) =(System.Double.IsNaN x|| System.Double.IsNaN y || System.Double.IsNaN z)|| x+(y+z) = (x+y) + z;;

But

Check.Verbose assocV1;;(* Falsifiable, after 17 tests (4 shrinks) (StdGen (830896496,296669696)):Original:(-1.355241514, 4.249447687, -3.425315758)Shrunk:(1.355241514, 4.0, 3.0) *)

What is the problem now?

5 DTU Compute, Technical University of Denmark Property-based testing: Floating-point properties MRH 14/11/2019

02157FunctionalProgram-

ming

Michael R. Hansen

Representation of float and float32 values

1.0 + 0.1 + 0.1 = 1.2;;val it : bool = false

NOT NICE!!!! WHY?

A floating point value (type float or float32) is represented bybinary numbers mant and exp, that represent

mant · 2exp

Problem:110 cannot be written in the form m · 2n with integers m and n.

The float representation of 110 is just an approximation.

6 DTU Compute, Technical University of Denmark Property-based testing: Floating-point properties MRH 14/11/2019

02157FunctionalProgram-

ming

Michael R. Hansen

Understanding the counterexample

The property being checked:

x + (y + z) = (x + y) + z

Check.Verbose assocV1;;(* Falsifiable, after 17 tests (4 shrinks) (StdGen (830896496,296669696)):Original:(-1.355241514, 4.249447687, -3.425315758)

Have a look at

let res1 = -1.355241514 + (4.249447687 + -3.425315758);;let res2 = (-1.355241514 + 4.249447687) + -3.425315758;;let diff = res1 - res2;;val diff : float = 2.220446049e-16

7 DTU Compute, Technical University of Denmark Property-based testing: Floating-point properties MRH 14/11/2019

02157FunctionalProgram-

ming

Michael R. Hansen

Equality within a tolerance

let assocV3 tol (x:float,y,z) =(System.Double.IsNaN x || System.Double.IsNaN y || .... )|| abs((x+(y+z)) - ((x+y) + z)) < tol;;

Check.Verbose (assocV3 1.0e-6);;

Falsifiable, after 2 tests (1 shrink) (StdGen (376632198,296669700)):Original:(1.797693135e+308, -infinity, 0.0)Shrunk:(0.0, -infinity, 0.0)

When does it stop?

8 DTU Compute, Technical University of Denmark Property-based testing: Floating-point properties MRH 14/11/2019

02157FunctionalProgram-

ming

Michael R. Hansen

A test Considering: -infinity, infinity and MaxValue

Are all special values:

> 1.0 / 0.0;;val it : float = infinity

> -1.0 / 0.0;;val it : float = -infinity

> System.Double.MaxValue;;val it : float = 1.797693135e+308

But there is no float for the textual representation of MaxValue

1.797693135e+308;;

1.797693135e+308;;ˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆˆ

stdin(6,1): error FS1153: Invalid floating point number

a problem for the testing

9 DTU Compute, Technical University of Denmark Property-based testing: Floating-point properties MRH 14/11/2019

02157FunctionalProgram-

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Michael R. Hansen

A test considering -infinity, infinity and MaxValue

let assocV5 tol (x,y,z) =(System.Double.IsNaN x || . ...)||(System.Double.IsInfinity x || .... )|| x = System.Double.MaxValue || y = .....|| abs((x+(y+z)) - ((x+y) + z)) < tol;;

Check.Verbose (assocV5 1.0e-14);;

Ok, passed 100 tests.

Can I be more confident?

let _ = Check.One ({ Config.Quick with MaxTest = 400; },assocV5 1.0e-14);;

Ok, passed 400 tests.

10 DTU Compute, Technical University of Denmark Property-based testing: Floating-point properties MRH 14/11/2019

02157FunctionalProgram-

ming

Michael R. Hansen

Summary

Property-based testing was used to reveal aspect involve concerningfloating-point programs

Floating-point values should be used with care!

The care needed is not specific to F#

A possible exercise:(Re)consider Exercise 6.8. Restrict the type Fexpr to theconstructors: X, Add, Sub and Log and restrict you solution to 6.8accordingly.

• Declare a function Eval fe x, for fe: Fexpr and x:float. Eval fe xcomputes the floating-point value of fe when X has value x

• Use FsCheck to test the correctness of the translation toinstructions in the sense that: for all fe, for all x

Eval fe x = intpProg(trans(fe,x))

11 DTU Compute, Technical University of Denmark Property-based testing: Floating-point properties MRH 14/11/2019