Prashant Kumar answered . 2024-04-21 08:58:32

Next Rep. Value Above 0.001044397358782589435577392578125
Quantized Value       0.001044397242367267608642578125
Next Rep. Value Below 0.001044397125951945781707763671875

Let's also show the ideal mid-points between these three values. (Note, the midpoints are NOT representable in the type.)

Next Rep. Value Above 0.001044397358782589435577392578125
Mid-point Value Above 0.0010443973005749285221099853515625
Quantized Value       0.001044397242367267608642578125
Mid-point Value Below 0.0010443971841596066951751708984375
Next Rep. Value Below 0.001044397125951945781707763671875

Given these values, we can say if the decimal text's value, as interpreted in the world of ideal math, is within this range

0.0010443973005749285221099853515625 > decimalText > 0.0010443971841596066951751708984375

then MATLAB, a C compiler, etc., parsing that text would round it to this representable value of type single

0.001044397242367267608642578125

Sufficient digits and superflous digits

Let's look at our mid-point values again

Mid-point Value Above 0.0010443973005749285221099853515625
Mid-point Value Below 0.0010443971841596066951751708984375

and think about when fewer digits are sufficient to be certain we are between these two values.

Consider this example,

Mid-point Value Above 0.0010443973005749285221099853515625
                      0.0010443972xxx...
Mid-point Value Below 0.0010443971841596066951751708984375

format hex
nDigitsOfPrecision = 8;
vOrig = single(0.001044397242367267608642578125); 
vOrig = vOrig+eps(vOrig)*[1,0,-1]

vTextDig7 = mat2str(vOrig,'class',nDigitsOfPrecision-1)
vRoundTripDig7 = eval(vTextDig7)
format long
worstCaseErrorDig7 = max(abs(vRoundTripDig7 - vOrig))

format hex
vTextDig8 = mat2str(vOrig,'class',nDigitsOfPrecision)
vRoundTripDig8 = eval(vTextDig8)
format long
worstCaseErrorDig8 = max(abs(vRoundTripDig8 - vOrig))    

The output shows 7 digits was not lossless, but 8 digits was.

vOrig =
  1×3 single row vector
   3a88e429   3a88e428   3a88e427
   
vTextDig7 =
    'single([0.001044397 0.001044397 0.001044397])'
vRoundTripDig7 =
  1×3 single row vector
   3a88e426   3a88e426   3a88e426
worstCaseErrorDig7 =
  single
   3.4924597e-10

vTextDig8 =
    'single([0.0010443974 0.0010443972 0.0010443971])'
vRoundTripDig8 =
  1×3 single row vector
   3a88e429   3a88e428   3a88e427
worstCaseErrorDig8 =
  single
     0

Note 8 digits for single is sufficient to be lossless, but for individual values 8 digits may not be necessary. For example, for the individual value single(0.01), one digit is sufficient for lossless roundtrip.

nDigitsOfPrecision = 1
vOrig = single(0.01) 
format hex
vOrig
vTextDig1 = mat2str(vOrig,'class',nDigitsOfPrecision)
vRoundTripDig1 = eval(vTextDig1)
format long
vRoundTripDig1
worstCaseErrorDig1 = max(abs(vRoundTripDig1 - vOrig))

which outputs

nDigitsOfPrecision =
     1
vOrig =
  single
   0.0100000
vOrig =
  single
   3c23d70a
vTextDig1 =
    'single(0.01)'
vRoundTripDig1 =
  single
   3c23d70a
vRoundTripDig1 =
  single
   0.0100000
worstCaseErrorDig1 =
  single
     0

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