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I am optimizing an equation that involves a number of trigonometric functions and Exp[]. How do I make sure that all my calculations have an accuracy of 120-200 digits after the decimal point? This includes the accuracy of Exp[] and trig functions.

To get my point across, here is part of the equation: z[x_, y_]:= Exp[Sin[60.0*x]] + Sin[50.0*Exp[y]]

Mathematica lets you control Precision of computations (which is total number of digits in the number) with two global variables - $MinPrecision and $MaxPrecision. However, I am not looking for precision.

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Do you know that keeping expressions in terms of integers and rationals will keep exact values? Try NestList[16 # (1 - #)/3 &, 1/5, 4] ? Though for heavy computations you will lose speed. –  Vitaliy Kaurov Jul 1 '12 at 17:53
I am aware that Mathematica will give me exact values for integer & rational calculations, but my calculations are far from exact. –  newprint Jul 1 '12 at 18:00
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3 Answers

How about something like

z[x_, y_] := Exp[Sin[60*x]] + Sin[50*Exp[y]]
z[SetAccuracy[20., 200], SetAccuracy[20., 200]] // Accuracy

does this not do what you need?

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In[8]:= z[SetAccuracy[20., 200], SetAccuracy[20., 200]] // Accuracy Out[8]= 13.6508 Don't think it worked. –  newprint Jul 1 '12 at 18:25
@newprint Sorry, I forgot to add the modified definition of z I used. Or, is it that you simply don't know the coeffs? Because that's different... –  acl Jul 1 '12 at 18:47
Thanks, I understood the SetAccuracy and // Accuracy –  newprint Jul 2 '12 at 1:13
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If you use inexact constant in your equation it helps if you increase their accuracy as well. You can do that easily using the backtick notation:

z[x_, y_] := Exp[Sin[60.0`200*x]] + Sin[50.0`200*Exp[y]]
z[SetAccuracy[20., 200], SetAccuracy[20., 200]] // Accuracy


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Initially, I used the backtick for floating point values as well(I read about in Mathematical Cookbook), though it becomes somewhat cumbersome within long equation. –  newprint Jul 2 '12 at 1:19
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Short answer

If you want Mathematica to return arbitrary-precision calculations with (approximately) a given Accuracy you could ask for it in MantissaExponent form:

MantissaExponent[Exp[150] 123``50][[1]] // Accuracy

Supporting information

Accuracy of input numbers can be set with double backticks:

12345.5678``50   // Accuracy
12345``50        // Accuracy

Single backticks define Precision:

12345.5678`50   // Precision
12345`50        // Precision

A "constant accuracy" doesn't make much sense for floating point calculations. Mathematica tracks and preserves Precision. This is called "arbitrary-precision" after all.

Exp[150] 123`50  // Precision
Exp[150] 123``50 // Accuracy
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I think in the "Short answer" section you use wrong formulation because MantissaExponent does not alter the accuracy of the result (I know, you know it - just for correctness). The right formulation could be: "If you want Mathematica to give you the accuracy of the result of calculations you could ask for it using MantissaExponent form: ..." –  Alexey Popkov Jul 2 '12 at 17:43
@Alexey I wasn't sure how to say what I wanted to. I'll try to rephrase it as you suggest later today. –  Mr.Wizard Jul 2 '12 at 18:00
One problem is that Wolfram's understanding of precision and accuracy is significantly misleading and the second one is that you use in this sentence the term "accuracy" with another meaning: the number of precise digits right to the decimal point in ScientificForm. In other words, it is something always approximately equal to Precision minus 1. –  Alexey Popkov Jul 2 '12 at 18:14
Discussion on Precision and Accuracy in Mathematica: groups.google.com/d/msg/comp.soft-sys.math.mathematica/… –  Alexey Popkov Jul 2 '12 at 18:20
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