Converting Timing to TimeObject

Question

What's the proper way to get Timing and post-process it for example to convert to a TimeObject as a defined function?

In line, this replacement works as intended:

Inverse[Table[RandomReal[], {3000}, {3000}]]; // Timing // 
 Replace[{timeSec_, value_} :> {TimeObject[{0, 0, timeSec}], value}]

gives (full form):

List[TimeObject[List[0,0,3.450302`],"Instant",None],Null]

but when trying to define a function to perform both the timing and conversion, the actual timing extracted is not that of the intended input expression:

unitTiming[expr_] := Timing[expr] // Replace[{timeSec_,value_}:> {TimeObject[{0,0,timeSec}],value}]

Testing on the matrix inversion above:

Inverse[Table[RandomReal[], {3000}, {3000}]]; // unitTiming

gives:

List[TimeObject[List[0,0,0.000011`],"Instant",None],Null]

I've tried several variations on the definition (eg using RightComposition) but can't get the desired results.

Answer 1

This answer addresses Szabolcs' points in comments, including avoiding TimeObject

(* returns S, MS, HMS depending on the magnitude of the input *)
minimalTimeQuantity[time_] := Replace[time, {
    t_ /; (t < 60) :> Quantity[MixedMagnitude[{t}], MixedUnit[{"Seconds"}]],
    t_ /; (t < 3600) :> 
     Quantity[MixedMagnitude[{0, t}], MixedUnit[{"Minutes", "Seconds"}]],
    t_ :> Quantity[MixedMagnitude[{0, 0, t}], 
      MixedUnit[{"Hours", "Minutes", "Seconds"}]]}
   ];

Then package output for both types of timing:

(* data structure to return both timing and absolute timing *)
timingsAssociation[t_, absT_] := <|"Timing" -> minimalTimeQuantity[t], 
  "Absolute Timing" -> minimalTimeQuantity[absT]|>

Finally, the actual timing function returning quantities:

timings[expr_] := 
 AbsoluteTiming[Timing[expr]] // 
  Replace[{{absT_, {t_, Null}} :> 
     timingsAssociation[t, 
      absT], {absT_, {t_, value_}} :> {timingsAssociation[t, absT], value}}]
SetAttributes[timings, HoldAll]

Test:

Inverse[Table[RandomReal[], {1000}, {1000}]]; // timings

gives: