How does Mathematica determine when to use scientific notation?

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I have code that takes an initial guess for a set of parameters to use in a function that I want to minimize, calculates the function, prints the value, recalculates new parameters via the gradient descent method, recalculates the function, prints it, and so on.



This is what my output looks like:



enter image description here



I don't mind it, I'm just curious as to when Mathematica decides it should use scientific notation and when it thinks regular notation is fine. At first sight, it seems random.



MWE not necessary because this is not a question related to code.



Edit: Sorry for the huge image. I've replaced it with a smaller one.






output-formatting precision notation







share|improve this question



















  • 1




    You can use NumberForm with its ScientificNotationThreshold setting if you want more control.
    – J. M. is somewhat okay.♦
    3 hours ago










  • I know I can. As I said, I don't mind which notation Mathematica uses for this; I'm just curious as to how it determines which one to use.
    – Rain
    3 hours ago






  • 1




    I don't have your numbers, so I can only hazard a guess that it's about precision differences. See e.g. 97170., 97170.`5, 97170.`4.
    – J. M. is somewhat okay.♦
    3 hours ago











  • I suspect the message formatting is using the same thresholds as NumberForm.
    – Daniel Lichtblau
    2 hours ago














up vote
4
down vote

favorite












I have code that takes an initial guess for a set of parameters to use in a function that I want to minimize, calculates the function, prints the value, recalculates new parameters via the gradient descent method, recalculates the function, prints it, and so on.



This is what my output looks like:



enter image description here



I don't mind it, I'm just curious as to when Mathematica decides it should use scientific notation and when it thinks regular notation is fine. At first sight, it seems random.



MWE not necessary because this is not a question related to code.



Edit: Sorry for the huge image. I've replaced it with a smaller one.






output-formatting precision notation







share|improve this question



















  • 1




    You can use NumberForm with its ScientificNotationThreshold setting if you want more control.
    – J. M. is somewhat okay.♦
    3 hours ago










  • I know I can. As I said, I don't mind which notation Mathematica uses for this; I'm just curious as to how it determines which one to use.
    – Rain
    3 hours ago






  • 1




    I don't have your numbers, so I can only hazard a guess that it's about precision differences. See e.g. 97170., 97170.`5, 97170.`4.
    – J. M. is somewhat okay.♦
    3 hours ago











  • I suspect the message formatting is using the same thresholds as NumberForm.
    – Daniel Lichtblau
    2 hours ago












up vote
4
down vote

favorite









up vote
4
down vote

favorite











I have code that takes an initial guess for a set of parameters to use in a function that I want to minimize, calculates the function, prints the value, recalculates new parameters via the gradient descent method, recalculates the function, prints it, and so on.



This is what my output looks like:



enter image description here



I don't mind it, I'm just curious as to when Mathematica decides it should use scientific notation and when it thinks regular notation is fine. At first sight, it seems random.



MWE not necessary because this is not a question related to code.



Edit: Sorry for the huge image. I've replaced it with a smaller one.






output-formatting precision notation







share|improve this question















I have code that takes an initial guess for a set of parameters to use in a function that I want to minimize, calculates the function, prints the value, recalculates new parameters via the gradient descent method, recalculates the function, prints it, and so on.



This is what my output looks like:



enter image description here



I don't mind it, I'm just curious as to when Mathematica decides it should use scientific notation and when it thinks regular notation is fine. At first sight, it seems random.



MWE not necessary because this is not a question related to code.



Edit: Sorry for the huge image. I've replaced it with a smaller one.






output-formatting precision notation




output-formatting precision notation






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited 50 mins ago









Carl Woll

58.8k276150




58.8k276150










asked 3 hours ago









Rain

301110




301110







  • 1




    You can use NumberForm with its ScientificNotationThreshold setting if you want more control.
    – J. M. is somewhat okay.♦
    3 hours ago










  • I know I can. As I said, I don't mind which notation Mathematica uses for this; I'm just curious as to how it determines which one to use.
    – Rain
    3 hours ago






  • 1




    I don't have your numbers, so I can only hazard a guess that it's about precision differences. See e.g. 97170., 97170.`5, 97170.`4.
    – J. M. is somewhat okay.♦
    3 hours ago











  • I suspect the message formatting is using the same thresholds as NumberForm.
    – Daniel Lichtblau
    2 hours ago












  • 1




    You can use NumberForm with its ScientificNotationThreshold setting if you want more control.
    – J. M. is somewhat okay.♦
    3 hours ago










  • I know I can. As I said, I don't mind which notation Mathematica uses for this; I'm just curious as to how it determines which one to use.
    – Rain
    3 hours ago






  • 1




    I don't have your numbers, so I can only hazard a guess that it's about precision differences. See e.g. 97170., 97170.`5, 97170.`4.
    – J. M. is somewhat okay.♦
    3 hours ago











  • I suspect the message formatting is using the same thresholds as NumberForm.
    – Daniel Lichtblau
    2 hours ago







1




1




You can use NumberForm with its ScientificNotationThreshold setting if you want more control.
– J. M. is somewhat okay.♦
3 hours ago




You can use NumberForm with its ScientificNotationThreshold setting if you want more control.
– J. M. is somewhat okay.♦
3 hours ago












I know I can. As I said, I don't mind which notation Mathematica uses for this; I'm just curious as to how it determines which one to use.
– Rain
3 hours ago




I know I can. As I said, I don't mind which notation Mathematica uses for this; I'm just curious as to how it determines which one to use.
– Rain
3 hours ago




1




1




I don't have your numbers, so I can only hazard a guess that it's about precision differences. See e.g. 97170., 97170.`5, 97170.`4.
– J. M. is somewhat okay.♦
3 hours ago





I don't have your numbers, so I can only hazard a guess that it's about precision differences. See e.g. 97170., 97170.`5, 97170.`4.
– J. M. is somewhat okay.♦
3 hours ago













I suspect the message formatting is using the same thresholds as NumberForm.
– Daniel Lichtblau
2 hours ago




I suspect the message formatting is using the same thresholds as NumberForm.
– Daniel Lichtblau
2 hours ago










1 Answer
1






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6
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Mathematica uses 2 criteria to determine whether to show a number in scientific notation or not.




  1. For both arbitrary precision and machine precision numbers, use scientific notation if the exponent is not between 5 and -5 inclusive:



    123456.
    1234567.



    123456.



    1.23457*10^6




    and:



    1.23456*^-5
    1.23456*^-6



    0.0000123456



    1.23456*10^-6





  2. For arbitrary precision numbers, use scientific notation if the number of digits displayed is more than the precision:



    123400`4
    123400`5
    123400`6



    1.234*10^5



    1.2340*10^5



    123400.








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    1 Answer
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    up vote
    6
    down vote













    Mathematica uses 2 criteria to determine whether to show a number in scientific notation or not.




    1. For both arbitrary precision and machine precision numbers, use scientific notation if the exponent is not between 5 and -5 inclusive:



      123456.
      1234567.



      123456.



      1.23457*10^6




      and:



      1.23456*^-5
      1.23456*^-6



      0.0000123456



      1.23456*10^-6





    2. For arbitrary precision numbers, use scientific notation if the number of digits displayed is more than the precision:



      123400`4
      123400`5
      123400`6



      1.234*10^5



      1.2340*10^5



      123400.








    share|improve this answer


























      up vote
      6
      down vote













      Mathematica uses 2 criteria to determine whether to show a number in scientific notation or not.




      1. For both arbitrary precision and machine precision numbers, use scientific notation if the exponent is not between 5 and -5 inclusive:



        123456.
        1234567.



        123456.



        1.23457*10^6




        and:



        1.23456*^-5
        1.23456*^-6



        0.0000123456



        1.23456*10^-6





      2. For arbitrary precision numbers, use scientific notation if the number of digits displayed is more than the precision:



        123400`4
        123400`5
        123400`6



        1.234*10^5



        1.2340*10^5



        123400.








      share|improve this answer
























        up vote
        6
        down vote










        up vote
        6
        down vote









        Mathematica uses 2 criteria to determine whether to show a number in scientific notation or not.




        1. For both arbitrary precision and machine precision numbers, use scientific notation if the exponent is not between 5 and -5 inclusive:



          123456.
          1234567.



          123456.



          1.23457*10^6




          and:



          1.23456*^-5
          1.23456*^-6



          0.0000123456



          1.23456*10^-6





        2. For arbitrary precision numbers, use scientific notation if the number of digits displayed is more than the precision:



          123400`4
          123400`5
          123400`6



          1.234*10^5



          1.2340*10^5



          123400.








        share|improve this answer














        Mathematica uses 2 criteria to determine whether to show a number in scientific notation or not.




        1. For both arbitrary precision and machine precision numbers, use scientific notation if the exponent is not between 5 and -5 inclusive:



          123456.
          1234567.



          123456.



          1.23457*10^6




          and:



          1.23456*^-5
          1.23456*^-6



          0.0000123456



          1.23456*10^-6





        2. For arbitrary precision numbers, use scientific notation if the number of digits displayed is more than the precision:



          123400`4
          123400`5
          123400`6



          1.234*10^5



          1.2340*10^5



          123400.









        share|improve this answer














        share|improve this answer



        share|improve this answer








        edited 1 hour ago









        xzczd

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        24.3k466232










        answered 1 hour ago









        Carl Woll

        58.8k276150




        58.8k276150



























             

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